Effects of MSW and RSFP on Neutrino Constraints and Supernova Dynamics

Tanvir Rahman

Depart ment of Physics

McGill University, Mont réal

January 2000

A Thesis submitted to the

Faculty of Graduate Studies and Research

in partial fulfillment of the requirements for the degree of

Master of Science

QTanvir Rahman, 2000

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Abstract

We have constrained the neutrino parameter space of &n2-sin2 28, by considering

the possibility of r-process nucleosynthesis occuring in the hot bubble region of a

supernova. In addition, we have calculated the effects of density fluctuations and

spin-flavor oscillations on these constraints. We also considered the delayed shock

mechanism of a supernova explosion and calculated the shock heating rate with and

without the Mikheyev-Smirnov-Wolfenstein (MSW) and Resonant spin-flavor preces-

sion (RSFP) effects. We have confirmed that both these phenomena can significantly

affect the neutrino heating rate of the supernova causing shock wave.

Résumé

L'espace des paramètres des neutrinos Am2-sin2 20, est contraint en étudiant la

nucléosynthèse de type r intervenant dans les supernovas. Nous avons aussi considéré

les effets de fluctuations de densité e t de précession spin-saveur sur ces contraintes.

D'autre part, le taux de chauffage des neutrinos lors de mécanismes de chocs retardés

est calculé lorsque les effets Mikheyev-Smirnov-Wolfenstein (MSW) et Résonant de

précession spin-saveur (RSFP) sont pris en compte. Nous confirmons que de ces

phènoménes changement de saveur affectent le réchauffement dû aux neutrinos pro-

duits par la supernova.

Acknowledgments

First, 1 would like to tbank my supervisor, Prof. Jim Cline, whose competent guid-

ance, availability and financial support made this work possible. Thank you for

introducing me to Neutrino Astrophysics and sticking with me al1 the way through. 1

have learned a great deal from you both as my supervisor and teacher over the years.

Many thanks to my parents and my brother Tazim, whose support was felt

throughout al1 of this work.

Oh so many people to thank in the Physics department. Special mentions go to

Kostas Kordas, Declan Persram, Yasher Aghababei, David Winters, Andrew Hare,

Claude Theoret, Tiago De Jesus, Guy Moore and anyone else 1 might have forgotten.

Al1 of your friendship, support and wisdom in both good and bad times have meant

a lot to me.

Outside of the Physics department, Bridgitt, Susan, Debbie, Jonathan and Cathy.

You guys are number one!! Last but definitely not least, rny friend Faisal. You have

been a beacon of inspiration to me over the last few years. Your friendship is much

appreciated.

Finally, thanks are due to the Physics Department for financing part of this work.

Contents

Abstract

Résumé

Acknowledgments

1 Introduction

2 Neutrino Oscillations 3

1 Neutrino Masses . . . . . . . . . . . . . . . . . - . . . . . . . . . . . 4

'3.1.1 Dirac Masses . . . . . . . . . . . . . . . . . . . . . . . . . . - 5

2.1.2 Majorana -Masses . . . . . . . . . . . . . . . - . . . . . . . . 5

2.2 Neutrino interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3 Vacuum Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1 Neutrino propagation t hrough matter . . . . . . . . . . . . . . . . . . 10

2.5 Spin-Flavor Precession of Weut rinos . . . . . . . . . . . . . . . . . . . 15

2.6 Afourneutrinosystem.. . . . . . . . . . . . . . . . . . . . . . . . . 19

3 Supernova mechanisms and nucleosynthesis

3.1 Supernova explosion mechanisms . . . . . . . . . . . . . . . . . . . . 23

. . . . . . . . . . . . . . . . 3.2 Supernova dynamics and the MSW effect 28

. . . . . . . . . . . . . . . . . 3.3 Neutrino constraints from a supernova 31

. . . . . . . . . . . . . . . . . . . . . . . . 3.4 RSFP and MSW combined 35

. . . . . . . . . . . . . . . . . . . . . . 3.5 Effects of density fluctuations 35

4 Results. Discussion and Conclusion 42

. . . . . . . . . . . . . . . . 4.1 Supernova dynamics and the MSW effect 42

. . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Neutrino constraints 31

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Effects of RSFP 50

4.3.1 Effects of spin-fl avor transformations on neut rino const raints . 54

. . . 4.3.2 Effeits of spin-flavor oscillations on Supernova dynamics 56

. . . . . . . . . . . . . . . . . . . . . . 4.4 Effects of density fluctuations 5S

. . . . . . . . . . . . . . . . . . . . . . . 4.5 Conclusion and Future work 59

Bi bliography 62

Chapter 1

Introduction

The role of neutrinos in supernova dynamics and nucteosyntbesis has been an area

of intense research for many years. -4 great deal of work has been done to build

accurate supernova modeIs in which neutrinos carry away much of the energy due to

gravi tational collapse of stars. In this work we focus on two particular issues regarding

neutrinos in a supernova.

First, we focus on the role of neutrinos in supernova dynamics. -4 supernova is

caused by a shock wave that is generated in the core of a collapsing star. In some

espIosion mechanism. the shock wave is not energetic enough to cause an explosion.

It has been proposed that the shock wave first stalls, and is then revived by neutrino

interactions at its wake, resulting in an explosion. We will refer to these interactions

as neutrino heating of the shock wave. Fuller et al. [3] have calculated the effects of

matter oscillations on the heating rate at the wake of the shock for a 20 Mo supernova,

0.15s after core bounce (TPB). They used a supernova model that was developed by

Mayle and Wilson r20] for their calculations. We will verify their calculations using

a similar model. We will also combine both matter and spin-flavor oscillations and

recalculate the heating rates and compare them to those obtained by Fuller e t al. [4].

In a completely different work, Fuller et al. [2] have put constraints on the neu-

trino parameter space of Am2 - sin2 26, by looking a t t h e possibility of rapid-process

CHA PTER 1. INTRODUCTION

(r-process) nucleosynthesis occurring in the 'hot bubble" region of a supernova. Hav-

ing estabiished these constraints? the- calculated the effects of spin-flavor transfor-

mations on them [dl. We will also do the same. They also considered the effects

of density fluctuations on these constraints [3]. We will verify these results too? but

using a different formalism that Ras developed by Burgess and Michaud [I4]. In order

to pro\-ide a detailed exposition to al1 of the phenomena and calculations mentioned

above. we take the following steps.

In chapter 2' we start with a brief review of neutrino physics. in particular. that

of neutrino oscillations. We start with a commentary on the incentive for massive

neutrinos which is responsible for one of the two mechanisms of oscillations that we

rvill discuss. We summarize two proposed theories of how the standard mode1 can be

extended t o give neutrinos masses. W e then discuss in detail the two flavor mixing

mechanisms of neutrinos used in Our calculations, namely, the matter and spin-flavor

oscillations.

In chapter 3; we summarize the life cycle of a star up to a supernova explosion

and point out that neutrinos of al1 species are ejected in such an explosion between

0.01 - 1.5s TPB. We will show how one can constrain the neutrino parameter space by

considering the possibility of rapid-process nucleosynthesis in a supernova. We also

derive the relevant expressions for calculating the effects of matter oscillations on the

heating rate of the shock wave by neutrinos. W e also discuss how the formalisms

that are used for computing the neutrino constraints and the shock heating rates

have to be extended when spin-flavor oscillations are included in these calculations.

We then present a summary of a formalism, developed by Burgess and Michaud [14],

that we use to study the effects of density fluctuations on Our calculations of neutrino

constraints.

In chapter 1 we present the results of a11 our calculations and discuss their lirni-

tations.

Chapter 2

Neutrino Oscillations

We start this chapter with a review of neutrino physics in the context of extensions

to the standard electroweak model that ascribe nonzero masses and magnetic mo-

ments to the neutrinos. The first topic treated is the see-saw mechanism of estending

the standard model that allows neutrinos to acquire mass. Next, we discuss vacuum

(and mat ter) oscillations and spin-flavor transformations, two neutrino rnixing phe-

nomena that are direct consequences of neutrinos being massive and having magnetic

moments, respectively.

In our discussion of neutrino vacuum and matter oscillations we present al1 the

relevant equations and expressions that we need to understand these two very similar

methods of mixing (vacuum oscillaions being a limi ting case of mat ter oscillations).

To illustrate the mixing of neutrinos in matter, we present numerical results for the

survival probability of a v, propagating through a supernova. We then discuss spin-

flavor precession of neutrinos, a phenornenon which happens only when a neutrino

has a rnagnetic moment and propagates in a rnagnetic field.

4 CHAPTER 2. NEUTRIXO OSCILL.4TION.S

2.1 Neutrino Masses

There are strong motivations for neutrino masses from cosmology and astrophysics.

According t o Langacker [ T l - the following ore worth mentioning.

1. Fermion masses are one of the great mysteries of the standard model. Obser-

vation or non observation promise new physics beyond the standard mode1 and

hence could introduce a new perspective in physics.

2 . There could be a hot dark matter component of the universe. If so, massi~e

neutrinos would be one of the important contributions to the mass of the uni-

verse.

3. The observed spectral distortion of and deficit in the solar neutrinos may be

accounted for by oscillations or conversions of massive neutrinos.

4. The ratio of rn,/m, is suggestive of neutrino oscillations. In fact. recent Su-

perriamiokande results are consistent with the existence of atmospheric neutrino

oscillations.

5 Neutrino fluxes from supernovae could be a very important probe deep into the

core of a dying star that could help us better understand supernova mechanisms.

The current limits on the masses of the three known species of neutrinos are [26]

M,, < 10-15eV a t 9.5% CL M,, < 170 keV at 90% CL

M, < 18.2 MeV a t 95% CL. (2.1)

In the next two sections we summarize two proposed forms of neutrino masses

that arise as extensions of the standard electroweak model, narnely, those of Dirac

and Majorana. W e will see that the Dirac and Majorana masses are associated with

sterile and active neutrinos respectively. Implications for supernova dynamics and

nucleosynthesis will depend on which type neutrinos actually are. Following Fuller ct

2.1. NEUTRINO M-4SSES

al. [2: 3, 11, we assume that neutrinos are of the Majorana type. However. for the

sake of completeness we present both possibilities.

2.1.1 Dirac Masses

hIhss terms allow for transitions between left (L) and right (R) Iianded neutrino

statesl. -4 Dirac m a s term couples two Weyl neutrino^^^ UL and -IrR. That is: the

right handed state NR is different from UR, the CPT partner of v ~ . Given the above,

if one irrites the Dirac field as v = (uL .VR)=, t he Lagrangian with a Dirac mass

term is given by

LDiraC = mD (pL -VR + fivL ) = ~ D Ü U . (3.2)

Thus a Dirac neutrino has four components and the mass term conserves total lepton

number, L = L , + LN.

For a conventional Dirac neutrino the UL is active (i.e. is an SU(2)L doublet)

and the NR is sterile. To generate the mass term requires SU(-) symmetry breaking

when the Higgs field acquires a vacuum espectation value (VEV) and has a Yukawa

coupling to the neutrinos of the form.

This gives us r n ~ = h , v / f i , where the VEV of the Higgs doublet is v = =

( ~ G - ) ) - ~ I ~ = 246 GeV, and h, is the Yukawa coupling constant.

2.1.2 Majorana Masses

-4 Majorana mass term arises by letting the right handed antineutrino vfi take the

place of -&. It can cause transitions from an antineutrino to a neutrino and vice

'The subscripts L and R refer to left and right handed projections. In the zero mass hmits, they

refer to left and right helicity states. 2.4 right handed particle is associated under CPT with a right handed antiparticle. The two

together constitute a Weyl spinor.

versr H e x e a Majorana rnMs terrn vioia~es le-ton numbar :vir.i LL = 2. One

cocsequence of 3lajorana neutiinos. if present. would be neuc;iooIess double bera

deca:.?. The forx of the 'vhjorsna mzFs rcrrn is g i~e r i by

The 6, unlike its Dirac neutrino counterjart, is active.

1.2 Neutrino interactions

Neiitrinos of al1 iiavors under- weak interzctions only, through the eschange of 1.V'

uid/or Zo bosons. known zs chaqed =ci neutral cur ien t irirractiocs ;es?ec:ive!y.

The Feynman d i a p m s for these ilteracrions are given beloiv (see figure 2.1 ):

Vacuum Oscillations

Figure 2.1: Neutrino interactims by the exchange of \Y' and Zo bosons.

2.3. V4CUUM OSCILLATIONS

consequeoces of attributing masses to neutrinos. A good starting point is to con-

sider the propagation of neutrinos in vacuum. In this simple picture we encounter a

phenomenon knoum as vacuum oscillations, a mechanism through which one type of

neutrino can change its flavor to another type (e -g . Ye changing to y,) as it propagates

through space. The oscillation mechanism itself is similar to the CP violation effects

observed wi t h K-mesons. Much of the interesting physics of \-acuum oscillations can

be understood by considering oscillations between two neutrino ffavors (we consider

ue and Y, only). The extension of the following analysis to the three or more neutrino

Aavor case is straightforward.

We begin by defining our notations first. We will refer to the flavor eigenstates of

ue and uP at a given time t by Ive, t) and IV,, t ) respectivel. Similarly we refer to the

mass (or equivalently the energy) eigenstates at any time t by lvl, t ) with eigenvalue

El and Ih , t ) with eigenvalue E2. -4t this point, there is no experimentai evidence

to show that the mass eigenstates and the flavor eigenstates coincide with each other

(in fact SuperKamiokande results suggest the contrary). Hence, if CP is not violated,

the flavor eigenstates Ive: t ) and lup, t) may be erpressed as a linear combination of

mass or energy eigenstates lul, t ) and 1 u2, t ) by a unitary two dimensional orthogonal

matrix U,,, L e .

where 8, is known as the vacuum mixing angle between the two flavors. Without loss

of generality we may pick its value to be O < 6, < f so that lu., t ) is closer to lul, t).

From Eq. (2.5) it is clear that if 8, = O, there is no mixing between the flavors and

hence vacuum oscillation is not possible. This fact will manifest itself throughout our

analysis as a limiting case.

Consider now the time evolution of neutrino states. If a neutrino is created at

a time to (henceforth set to O) in vacuum and propagates for an arbitrary time t ,

then the energy eigenstate Ivi) at a given time t (assuming time independence of the

8

Hamiltonian) is given by,

CHAPTER 3. N E UTRINO OSCIL LATIONS

Considering Eq. (3.6) one may then w i t e the following equation for the t ime evolution

of I &). - iEl t -iE2t Ive, t ) = cos 9,e 1vi7 to ) + sin 9,e Iv2, to ) . (2.7)

Therefore the amplitude for a ve to remain an electron neutrino after traveling a time

t in vacuum is given by

From Eq. (2.8) we may calculate the survival probability for the v, at time t to be

From Eq. (2.9) above it is clear that if the energy eigenvalues ( i .e . neutrino masses)

are equal. no flavor mixing c m take place. To see how we can rewrite t h e above

equation in the relativistic limit, we first note that the energy (in natural units h =

c = 1) is given by

Tlierefore ive may write by momentum conservation (since pi=pz).

For astrophysicd neutrinos of interest (e.9. neutrinos originating frorn the sun or a

star), the energy E is much greater than m, hence E -- p. Therefore we may replace

p by E in Eq. (2.11). If we assume that r n 2 > ml so that Am2 = lm: - m:l > O, then

we need consider only the positive sign in Eq. (2.11). Before rewriting Eq. (2.9) in

terms of Am2 and E, we first define a quantity called the vacuum oscillation length

L,, whose dimension is that of distance and whose physical significance will be clear

shortly. L, is given by

In terms of L , the v, survival probability is given by,

We may also write the transition probability of ve to v, as

where R is the distance traveled by v, between time to and t . The first thing ive note

from the aboïe equation is that the survival probability of the electron neutrino de-

pends on its energy (E), the vacuum mixing angle (8,) and the mass difference squared

(Am2). In almost al1 neu trinerelated detect ion experiments and phenomenological

research (including part of this work) great effort has been spent in trying to put

constraints on the Am2 and sin2 28, parameter space. The physical significance of L ,

is as followç. LYe note that if R << L,, then P,, x 1, i.e. almost no oscillation related

effects are observed. On the other hand if R -- Lu, then P,. = cos2(%0,). Hence, L,

is the distance over which the survival probability of the ue goes from 1 to cos2 28,.

FVe end this section by presenting the equations of motion analogous to the

Schrodinger equation for the propagation of lu,) and Iv,) through vacuum when flavor

rnixing is taken into account. If one considers the mass matrix in the electroweak basis

as the neutrino Hamiltonian, then one obtains the following coupled set of differential

equations for the propagation of neutrinos through vacuum:

- COS 28,

sin 26,

sin 24, ) ( ;;; ;; ) cos 28,

This equation can be derived easily by using Eqs. (2.5) and ( 2 . 6 ) . Once an initial

condition has been specified ( i . e . an initial composition of neutrino flavors), one can

integrate the coupled set of equations to find the survival probabiljties of neutrinos

at a given time t.

2.4 Neutrino propagation through matter

We discuss now the propagation of neutrinos through matter. We will encounter

a very interesting oscillation scenario known as the MSW effect. after Mikheyev.

Smirnov and Wolfenstein [l] that allows a complete flavor transformation of one

type of neutrino into another, unlike the vacuum oscillation case. Once again. we

consider the simplest case of just two neutrino flavors (4 and 5 ) to illustrate the

salient features of matter effects. The extension to the case of three or more neutrino

generations is straightforward.

In the presence of matter, neutrinos may undergo charged and neutral current

forward scattering (as shown in figure 2.1). Therefore, the effective Hamiltonian of

propagation must contain terms that take into account such interactions. Hoivever,

not dl types of interactions undergone by the neutrinos need to be taken into ac-

count, and simplifications to the effective Hamiltonian can be made by considering

the following facts. First, we note that neutrinos of al1 fiavors undergo neutral cur-

rent interactions to the same degree. These interactions add an overall phase to the

wave functions that does not affect the survival probabilities of neutrinos. Therefore:

such terms may be ignored in the effective Hamiltonian. Second, for neutrinos prop-

agating through the sun or a supernova. the v,,, (E,,., - IO6 eV) are not energetic

enough to undergo charged current interactions to produce p or r (m, - 105.66

MeV, m, - 1777 MeV). Hence these contributions to the effective Hamiltonian may

also be omitted. This leaves only the charged current contribution due to ve scatter-

ing. The potential term that is added to the vacuum Harniltonian is then given by

V = f i ~ ~ n , ( t ) , where ne(t) is the electron number density a t time t that a ve passes

through. By denoting a general neutrino state as I V , t ) = Ce(t)lvC7 t ) + C,(t)lup? t ) ,

the equations of motion for neutrinos in the presence of matter is given hy

i$ (, ) = & ( 2 E v - 4 m 2 c o s % Am2 sin 29,

Am2 sin 20, -2EV + Am2 cos 20,

2.4. NE UTRINO PROP-4G-4TIOX THROUGH MATTER

Eq. (2.16) can also be written in the following convenient forrn;

- COS 2 0 ~ sin 2em ) ( ) sin 26, cos 28,

u-here Dfb1 are the energy eigenvalues which can be obtained from the following rela-

t ion:

= + [ ( I V COS ie. - + (AV sin %,)*]i. (2.18)

In Eq. (2.18) 6, is known as the matter mixing angle, which satisfies

AM sin IOfv = Av sin 20,. (2.19)

Before fully exploring the physical consequences of the presence of matier on

neutrino oscillations Mie note from Eq- (2.16) that if the electron number density is

such that the diagonal terms are equal. then maximal or total conversion of one type

of neutrino to another is possible. This is known as the MSW effect. The electron

number density at which this happens is called the resonance density and is given by

- AM2 cos 28, ne,res - (2.20)

2 4 ~ ~ ~ *

-4 remarkable feature of the MS W effect is that even for a very small vacuum mixing

angle, resonant mixing between the Aavors is possible if Eq. ('2.20) is satisfied. This

makes the smallness of the vacuum mixing angle insignificant, except for the issue of

adiabaticity, which we will discuss shortly.

We will explore now in slightly greater detail the effects of matter on neutrino

oscillations. First , we define a quantity called the neut rino-electron interaction lengt h

This parameter is indicative of the distance neutrinos travel

-4nother parameter is the matter oscillation length, given by

(2.21)

before they interact .

Consider now the propagation of neutrinos in matter with a constant density profile.

The survival probability for an electron neutrino created at time to and being detect.ed

at time t is given by.

ÎïR I(veY t(v,. t,)12 = 1 - sinZ .'OLw sinZ -.

L '11

The sur\-ilal probability has the folloning three interesting limiting cases:

case 1 : « 1. Le

This corresponds to a ve traveling tlirough a region of low density. In this case we

note that Eq. (2.23) reduces to Eq. (2.113). the 1-acuum oscillation sur\-ival probabilitl-.

as is intuitively espected.

This corresponds t o a ve traveling through a region of high density. Its survi\al

probability is given by

In this case oscillatory effects are suppressed. This is because the mixing angle in

matter is small in a region of high density. as can be seen from Eqs. (2.19) and (2.17)-

case 3 : L" - - COS '30,. L e

Density equals resonance density. The survival probability is zero. Complete

con\-ersion takes place in this case (if the adiabaticity condition is satisfied. which

d l be described shortly ).

The significance of matter effects is fully realized when one considers the prop-

agation of neutrinos through a region of variable density. Since most neutrinos of

interest are created deep inside stellar cores at high density and then detected on

earth, consider the propagation of a v,, created in a high density region, which sub-

sequently goes through resonance and is detected in vacuum. The propagation of

Ive) as a function of t ime is illustrated hy figure 2.2. Initially? Ive) is close to lu2),

the heavier of the energy eigenstates. The difference between the energy eigenvalues

Ei-re 2.2: Evolution of neutrino eisenstates iri matter

is large enou- that oscillatory efrects are suppressed. This is expected h m Our

previous discussion (see - Eqns. (2.13) and (1.19) ).

-45 lu.) approaches the resonznce density, the enetey eigendulues becorne & n o s

degenerate 2r. the resonance point. leading to interference betlveea the energy e l p -

States and osciilatory eEects which changes completely to IV , ) if the density of - - - - - - - - - - - - - - - -

the region varies do+ enou& As~ th~neu t r i~ocon t inue~ ta-propagaw -as it ju,J ,=a

further oscillations are observed M the enerv eigenstates are well sepratecl again.

The discussion above is illustrated in figure 2.3 \vhere w e hzve numericz!!y intrgrwec

Eq. (2.16) for the propagation of a ve from the core of a 15 M.;, supernovr at 1.5s TPB

using the nunerical supernova mode1 of ioocle:: and IVez-;ez [SI. Fi-re 2.3 S ~ O - 3

a nearfy constant. unitary s u r v i d probability ar the beoinning - of nextzino propagz-

tien. This is expected since the presence of rriatter suppresses the e5ective mishg

angie, as w s explained previcusl. At resonance. we see a complece Zavor change.

after which point its s u r v i d probability is nearly equal to zero. This happens be-

cause the efiective mass eigenvdues are separsted sufnciezrl~ aker resonance thsr

CHAPTER 2. NE UTRINO OSCILLATIOIVS

0.6 !- sin' 2 0 - 0. 1

t I

O.' -

Figure 2.3: ue survival probability as a function of distance in a supernova.

interference effects between the eigenstates become negligi ble.

S.J. Parke [8] bas derived an analytic expression for the survival probabilities

of neutrinos in the above scenario ivith the following approximation (known as the

Landau-Zener approximation) for the electron density near the resonance region:

He found that the survival probability P(ue -t v.) is given by

I l P, (r , r o ) = - + (, - PJ) cos SeM(rO) COS 20,(r) ,

2 - where

PJ is called the jump probability, which is a measure of the probability of level crossing

as was illustrated in figure 2.2. One notes that if the density varies very rapidly near

resonance, i. e.

2.5. SPIN-FL-4VOR PRECESSION OF NEUTRINOS

then PJ = 1 and level crossing does take place. On the other hand if

then PJ = O a i t h no level crossing. Eqs. ('7.29) and (2.23) together are known as the

adiabaticity conditions that we can use to define an adiabat icity parameter.

If 7 is ver- large, the level crossing probability is almost zero. If y is very low. the

level crossing probability is almost one.

To see how accurate the Parke approximation is, we have calculated the survival

probability using Eq. (2.26) for a ve, created deep in a supernova, that travels through

a resonance region and is detected in vacuum. Once again we have used the density

profile obtained from Woosley and Néaver [9] for a 15 M.3 supernova at 1% TPB.

We have calculated the survival probability of this neutrino in vacuum for a range

of values a t a fixed mixing angle and cornpared it to those obtained from direct

integration of Eq. (1.16). These results are shown in figure 2.1. Cornparison of these

results shows good agreement and confirms the accuracy of the analytic approxima-

tion. Further confirmation of the results above are obtained when one calculates the

adiabaticity parameter y for the same range of S. This indicates for which values

of 9 the survival probability is one (when complete level crossing occurs), zero (no

level crossing ) or in between (partial level crossing). These results are shown in fig-

ure 2.5. They confirrn the fact that a t high 5 the adiabaticity parameter (y) is high ~ r n ~ and hence, the survival probability is zero. At low 2~ the adiabaticity parameter

(y) is low and the survival probability is one, as confirmed by our numerical results.

2.5 Spin-Flavor Precession of Neutrinos

We discuss now a second flavor mixing phenornenon known as spin-flavor precession.

This mechanism was suggested by Cisneros [24] as a possible solution to the Solar

CHAPTER 2- NEUTRINO OSCILL-ATIONS

1

0.8

n s 7. 0.0

i V

O

0.4

O.'

Log,, A m' / ZC (eV)

Figure 2.4: Survival probability of a ve, versus Am2, using bot h numerical integration

(left) and analytic approximations (right).

Log,, A m2 / SE, <eV)

9m2 Figure 2.5: y vs. 2~

Neutrino Problem but has subsequently been applied to neutrino dynarnics in su-

pernovae as well. If neutrinos are endowed with magnetic moments, spin precession

2.5. SPIN-FLAVOR PRECESSION OF N E UTRINOS 17

in a magnetic field can lead to flavor changes for both Majorana and Dirac neutri-

nos. -4s rnentioned earlier, in this work we assume neutrinos to be hllajorana particles.

The Lagrangian which describes the magnetic moment mediated interactions between

Majorana neutrinos and the electromagnetic field, Fap, is given by

where p.6 is the magnetic moment matrix with a; b = e. p. r or 1 .'1.3 for the flavor

or mass eigenstates res~ectively. Also, o,p = ( i / 2 ) [T,, where 7, are the Dirac

matrices. CPT invariance of Eq. (2.31) demands that the diagonal elements of the

magnetic moment matrix vanish, leaving flavor changing magnetic moments,

i.e. p,b where a # b, to be the only nonzero magnetic moments. As a result, for

Majorana neutrinos, spin precession can cause the following transitions:

We will see in the following discussions that spin-flavor transitions are analogous

to matter oscillations. To present the spin precession mechanism as we did with the

matter oscillations case, we consider the dynamics of a tivo neutrino system that

includes ü. and V, on l . Once again, the case of three or more neutrino generations

is similar and straightforward. The equations of motion for the propagation of this

neutrino system are given by [4],

and B l ( t ) is the cornponent of the magnetic field perpendicular where A =

to the direction of neutrino propagation at position t , p, is the neutrino magnetic

moment(p,=pe,), and n, is' the neutron number density a t position t. If we take a

üe to be propagating through vacuum from time to (set to zero) to time t then its

survi~al probability can be shown to be

\Ve see that P ( F e + Y,) oscil1at.e~ with frequency PB. To appreciate the effect of

matter on the survival probabilitj-. let t.he ü, travel through a region of const,ant

density and magnetic field. In this case. its survival probability is given by

We notice that the survival probability is suppressed in the presence of niatter.

sirnilar to the matter oscillation case. In fact? if ( f i ~ ~ n , ) ~ > ( . Z P B ) ~ , then precession

is almost zero. Even if this were to be the case, a complete conversion sirnilar to the

MSIV effect can still take place if the diagonal terms in Eq. (2.35) are equal L e .

where we have assumed that (n, - ne) > O . Such a complete conversion is knoivn as

resonant spin-flavor precession (RSFP ).

For neutrinos propagating through a medium of varying density, one can derive

an analytic expression for their survival probability sirnilar to the Parke formula

(Eq. (2.26)). Once again take a F, that is created in a region of high density at point

i that undergoes RSFP and is detected a t a point f . Its survival probability is given

by Lirn and Marciano (231 as

Here cos 20i and cos 20j are the matter mixing angles a t the production and detection

points respect ively. These mixing angles sat isfy the following expression [4]

sin 20, = flpu B

( ( 2 ~ " B)2 + [A - & G F ( ~ ~ - n,)l2)1 '

2.6. -4 FOUR NEUTRINO SYSTEM 19

In Eq. (2.39), the jump or level crossing probability, similar to that encountered in

matter oscillation, is given by

ir2 Sr Pj = exp(---

2 L,, ) 7

where dr and L,,, are given by

The quantities 6r and L,,, are called the resonance width and the precession length.

respectively. One can use the jump probability to define the adiabaticity conditions

for spin-flavor precession. If

then Pj = 1, and if

then PJ = O. These two conditions together may be used to define the adiabaticity

parameter W S F P as follows, br

î'RSFP = -7

Lres

whose significance is exactly the same as that of y encountered earlier. This dis-

cussion summarizes the spin-flavor precession phenornenon and draws analogies wi t h -

matter oscillations. We turn our attention now to a four neutrino system (v,, 5, v,'

üp) in which both the motter and the spin-flavor oscillation scenario are considered

simultaneously.

2.6 A four neutrino system

As will be seen in the next chapter, a supernova explosion ejects neutrinos of al1

species that may play significant roles in both the dynamics of the explosion and in

20 CHAPTER 2. N E UTRINO OSCILLATIONS

the nucleosynthesis processes that take place in such explosions. Indeed, the focus of

our research is to perform calculations related to these two processes. The question

then follows, from our discussion of MSW and RSFP effects, of hoiv the survival

probabilities of the various species are modified ivhen both these phenomena are

taken into account. To address this issue, we take a four neutrino systern consisting -

of v,, u,, v, and üp; one could as well have chosen r instead of p neutrinos in t h e

following. Our goal is to find expressions for the survival probabilities of the four

species of neutrinos.

-4 good starting point is with the equations of motion for our s>-stem. These are

given by Lim and Marciano [23] as

where a, and a, are

In this system RSFP resonance occurs when

&G& - n,) = f 4 cos 28, ('2.49)

where A=@. The plus sign ( L e . when ne - n, > O) in Eq. (2.19) corresponds to the

resonance condition for u. - v, transitions and the minus sign (textiti.e. when ne - n,

< O ) corresponds to the resonance condition for Y, - up transitions. It is obvious from

t his t hat the two resononces cannot occur simultaneously (given massive neut rinos).

Also, considering the MSW resonance condition above in Eq. (2.20), one can say that

MSW - - 2K RSFP Pr== - Ye Pres 3

2.6. -4 FOUR NEUTRINO Sk'STEhI

where Y=, is given in terms of electron and baryon number densities as

\\'e see from Eq. (2.50) above that MS\V and RSFP resonances cannot occur simulta-

neously dong a neutrino trajectory unless Ye = 3. This is important since Akhmedov

and Berezhiani [25] have shown that analytic expressions for the survival probabili-

ties of the four neutrino systern are possible only if the MSW and RSFP resonance

points are sufficiently far apart. Otherwise, one would have to numerically integrate

Eq. (5.17) to find the survival probabilities. For our calculations, this will indeed be

the case (ive present numerical results in chapter 4). Hence by using Eqs. (2.26) and

(2.39), we can estimate the survival probabilities of a ue to propagate through both

RSFP and MSW and emerge as either a ue, v, or F,. We have,

Sirnilar expressions can be derived for the other neutrinos of the system.

Chapter 3

Supernova mechanisms and

nucleosynt hesis

Our research focuses on neutrinos in supernova explosions. Specifically, we are inter-

ested in the esplosion from - 0.01 - 15s after core bounce ( W B ) of a massive star.

-4ccording to some models, this is the epoch during which the shock wave that causes

a supernova is first stalled: and is then revived by neutrinos, causing an explosion

that leaves a protoneutron star with a region known as a "hot bubble" just outside

i t . Neutrinos are cont inuously ejected during t his epoch interacting with the mat ter

through which they propagate, possibly affecting both the supernova dynamics and

nucleosynthesis. We will see later how these interactions, and the physical charac-

teristics of the regions they pass through, are important in the physical processes of

interest for our calculations. We start this chapter with a brief review of a star's life

cycle from birth to a supernova explosion. We do this because it sets up a physical

picture of a supernova explosion during the epoch of interest to us. We then discuss

implications of the MSW effect on supernova dynamics. -4fter this, we describe how

one can constrain the neutrino parameter space from a supernova. Following this we

discuss how spin-flavor precession (RSFP) can affect both supernova dynamics as well

as the neutrino parameter space constraints. Finally, we surnmarize a formalism, used

3.1. SUPERNOV4 EXPLOSION MECHI4NISMS '33

later, that takes into account the effects of density fluctuations on neutrino survival

probabili ties.

3.1 Supernova explosion mechanisms

'This section gives a brief summary of the evolution of a massive star from its for-

mation up to its explosion culminating in a supernova and the formation of a hot

bubble. Precise modeling of the dynamics of a massive staro from its formation up

to a supernova explosion, is an area of much research and controversy. -4lthough the

initial stages of star's life cycle are well understood, the exact explosion mechanism

is still not known. This is partly due to the lack of experimental data and the enor-

mous computing power that one needs to fully investigate the models. Woosley and

Weaver [9], Wilson and Mayle r90] and Nomono and Hashirnoto [IO] are some well

known modelers of precollapse stars tha.t undergo supernova esplosions. Most of our

calculations are done using the mode1 of Wilson and Mayle [XI] .

Stars are formed in interstellar gas clouds composed of hydrogen, helium and

traces of other chernical elements. It is not known exactly how stars emerge from

clouds with high kinetic energy and angular momentum to the condensed state of

stardom. However, it is clear that if the conditions for gravitational collapse are

reached, massive stellar clouds contract. This contraction leads to compression and

higher overall density. When the density is high enough, parts of the cloud are able

to contract locally, leading to the formation of a cluster of primitive stars cailed

protostars. -4 protostar reaches stardom when its gravitational collapse is halted by

the rise in interna1 pressure and temperature, which activates t hermonuclear fusion

in its core, leading to hydrostatic equilibrium. We now summarize how this takes

place.

Consider the gravitational contraction of a massive (> 9Mo) star. As the star

contracts the temperature and pressure near its core start to increase. This increase

33 CHAPTER 3. SUPERNO14 MECH.NVISMS AND NUCLEOSI'NTHESIS

continues until a temperature of -- 106 I< is reached near its core. .4t this temperature,

hydrogen undergoes exothermic thermonuclear fusion. The product of this fusion

process is 'He. In general, the fusion may take place via two possible paths. The first

path is the so-called p - p chain. t h e favorable path for stars less massive than the

Sun. For stars more massive than the Sun, as is the case in this research. a second

path, known as the Carbon, hiitrogen. Oxygen (CNO) cycle. is favorable.

The hydrogen burning near the core takes place until al1 of the hydrogen there is

used up. .4t this point, gravitational contraction restarts. -4s the core contracts, a

density and a temperature of IO5 - 10' K g mW3 and 2 x 10' K are reaclied. respectivel-

This condition is suitabie for the thermonuclear burning of Helium:

The %e then burns t o exhaustion. This pattern (fuel exhaustion, contraction and

ip i t ion of the ashes of the previous cycle) repeats several times? leading finally to the

explosive burning of 28Si to j6Fe. This gives rise to the onion skin structure of the

precollapse star in which the star's history can be told by looking at its surface inward;

there are concentric shells of H, 'He. I2C, 1 6 0 , 20Ne, 28Si with j6Fe at the center.

Consideration of the binding energy of nuclei as a function of atomic mass shows

that 56Fe is the most stable nucleus and does not undergo exothermic thermonuclear

fusion. Therefore, as the last stage of contraction begins, a catastrophic collapse is

imminent.

-4 supernova explosion occurs as a result of a shock wave that is generated in the

iron core during the 1 s t stage of contraction. Hence, we concentrate on the collapse of

the iron core only. As the iron core collapses, there is rapid heating and compression

near its center. Sufficient heating of the iron core can release u particles and free

nucleons t hrough a process called photodisintegration. When hotodisintegrat ion

occurs, tightly bound iron nuclei are broken up into more weakly bound nuclei by

thermal radiation, and energy is absorbed via

y +56 Fe + 13'He + 4n. (3.2)

3.1. SUPERNOVA EXPLOSION MECH.4NIS.S 25

It is easy to show, considering the equilibrium concentrations of the reacting particles.

that almost three quarters of the iron is dissociated when the density and temper-

ature of the core reach 1012 Kg mq3 and 10'' E;, respectively. -41~0. at even higher

temperatures the 4He nucleus is expected to dissociate, releasing free nucleons via

Considering the m a s of a stellar core (- 1.4kfa), one can show that 1.4 x IOa5 J of

energy is absorbed through photodisintegration. Another important physical process

that occurs during this stage is electron capture. This happens when the core density

reaches -- 1.1 x 1012 Kg m-3. Under normal circumstances, a neutron is an unstable

particle with a half-life of 10.25 minutes. Its decay is called the beta decay. However,

in a collapsing stellar core, the density of the electron gas rnay become so high as to

make this decay impossible because of Pauli blocking. What happens at this point is

the capture of electrons by free and bound protons via inverse beta decay,

This process is called neutronization. In reality, the protons of an evolved star are

bound in nuclei. Nevertheless, t hey can still capture energetic electrons, producing

nuclei that are increasingly rich in neutrons and which may serve as seed elernents

for r-process nucleosynthesis.

Initially, the neutrinos so produced stream out freely from the iron core carrying

energy and lepton number, but when the density reaches -- 1012 Kg cm-3, the neutri-

nos essentially get trapped within the iron core. At this point they start to scatter off

the matter through both charged and neutral current processes. The charged current

interactions produce e- and e+ which themselves produce v.,,,~, Fe,,,, through neutral

current interactions. So neutrinos of al1 species are produced in the iron core.

We mentioned earlier that the collapsing iron core has a mass of about 1.4 M-,.

Consider its innermost core with a m a s of - 0.6 - 0.9 Mo. The collapse of the

iron core continues until this innermost core reaches nuclear density (- 1017 Kg

CHAPTER 3. SUPERNOVA MECHANISMS AND NUC'LEOSYXTHESIS

m-3). Then the innermost shell of mat ter reaches supernuclear density, rebounds,

and sends a pressure wave. Subsequent shells follow and send out a series of pressure

waves, which collect a t the edge of the innermost core. 4 s the edge of the innermost

core reaches nuclear density and cornes to rest a shock wave breaks out and starts to

propagate through the iron core.

The time scale and other physical details for the propagation of the shock from

the edge of the innermost core to the outer mantle of the star is a subject of intense

study. Two basic supernova producing mechanisms have been proposed, namel-

the prompt shock rnechanisrn and the delayed shock mechanism. In the prompt

shock mechanism, the shock is able to survive the passage through the iron core with

enough energy to blow off the mantle of the star, leaving a protoneutron star and a

hot-bubble region. On the other handl the delayed mechanism begins with a failed

hydrodynamic explosion. After about 0.01 seconds of its propagation the shock wave

stalls a t a radius of about 300-500 km, due to energy lost to the disintegration of

the iron core via Eq. (3.2). Having stalled, it exists in a sort of equilibrium. gaining

energy from the matter falling in across the shock front, but losing energy to its

heating. This accretion of material leads to the formation of a high temperature. low

density region between the protoneutron star and the stalled shock. At the same time,

the protoneutron star is cooled by radiating the initially trapped energetic neut rinos

(created by electron capture, charged and neutral current reactions), which provides

an outlet for the gravitational binding energy released by the accreting material. Some

of these neutrinos begin to deposit energy in the matter just outside the protoneutron

star via charged current interactions with nucleons, u-e scattering and vü annihilation.

As energy is deposited, the temperature rises and entropy increases leading to copious

production of e--e+ pairs which can themselves scatter neutrinos, resulting in even

more energy deposition. In this way, the entropy per baryon in this region rises to

about s .- 200 and, according to the calculations of Wilson and Mayle [-O], continues

to rise as more material accretes back. After - O.%, the entropy due to neutrino

3.1. SUPERNOV4 EXPLOSION -MECHANISMS

radiation becomes so high that tremendous pressure builds up and revives the stalled

shock's outward motion and causes a supernova explosion. Immediately after the

explosion one is left with a region outside the protoneutron star with high entropy?

an ec-e- plasma, alpha particles and free nucleons through which neutrinos of all

species continue to stream out for up to 15s. This is called the "hot-bubble' region.

The following points are important to note regarding the neutrinos that are

ejected from the protoneutron star.

1. Throughout most of their migration out of the protoneutron star, the neutrinos

are in flavor equilibrium, via

.As a result , there is an approximate equipartition of energy amonp the neutrino

flavors. This translates into the lurninosities (energy carried out per second)

of al1 flavors being approximately equal, an approximation confirmed by the

calculations of Mayle and Wilson (201. This approximation will be important

in our calculations.

- 2. The ejected neutrinos have energies up to 5 50 MeV. Hence. the y,,,. v,,, are

not able to undergo charged current reactions to produce p-, p' or r-, r+.

Therefore, they decouple deeper in the iron core than u,, Fe and are hotter on

On the other hand, due to high neutron density the cross section for charged

current reactions for v, is greater than for &. Hence,

According to the calculations of Wilson and Mayle [20], the average decoupling

temperatures for the various species of neutrinos between 3-15s TPB for a 20

Me supernova are as follows;

Tvp - TuT - 8 MeV, TTC - 4.5 MeV, Th - 3.5 MeV. (3s)

Between 0.01-0.5s TPB (i. e. the shock reheating epoch), these temperatures

are given by

We will use them later in our calculations.

Before completing this section we define certain distances in the hot bubble that Ive

will refer to later. We define the v,. v,' y,-spheres to be the surfaces of their last

scattering. For ve and Te, the neutrinosphere lies at -50 km from the core between

0.01-0.5s TPB and at -10 km between 3-15s T P B [2, 221. We define the weak freeze-

out point in the hot bubble to be the distance from the core where the charged current

reaction rates (Eq. (3.10)) are lower than the material expansion rate. We take this

to be -50 km for a 20MQ supernova between 3-15s TPB. Similarly, we define the

nuclear freeze-out distance as that at which al1 nuclear reaction rates (Eq. (3.19) are

lower than the material expansion rate. This is taken to be -1000 km for a 2034: ... supernova between 3-1.5s TPB.

3.2 Supernova dynamics and the MSW effect

In the last section, we mentioned two different explosion mechanisms, narnely, the

prompt shock mechanism and the delayed shock mechanism. W e recall that in the

delayed shock mechanism, the shock aave is stalled after -0.01s l'PB [5 ] . It is then

revived by neutrino scattering at its wake. In this section we present a formalism

develo~ed by Fuller et al. [3] that we will use to see if matter oscillations would affect

the heating rate on the stalled shock.

As mentioned earlier, neutrinos of al1 species stream out of the protoneutron

star and interact with nucleons between the stalled shock and the protoneutron star.

3.2. SUPERNOV4 DYN4MICS -4ND THE MSW EFFECT 29

-4lthough neutrinos undergo both charged and neutral current interactions with nucle-

ons, it is the charged current reactions of v. and that contribute almost exclusively

to the shock's revival. This is because, firstly, the neutral current cross section is rnuch

lower than that of the charged current reactions. Secondly. the v,,, are not energetic

enough to undergo charged current reactions. Therefore, the dominant reactions are

given by

The total energy deposition rate (in eV/s) on the shock is given by

where i,, and d,, are the energy deposition rates due to the respective reactions

(Eq. (3.10))- Y; and Y i are the ratios of electron to baryon and neutron to baryon

number densi ties respectively. Assuming charge neutrali ty of the medium through

which the neutrinos propagate. we can approximate Y. iz 5- Y.Z. Qian [29] has

derived an espression for iuN which is given by

where Lu is the Iuminosity of neutrino species v and r is the distance at which

is being evduated. -41~0, ou^ a 9.6 x cm2 is the charged current cross

section [3]. f, is the normalized energy spectrum (in units of number density/eV) of

species v and is given by

In Eq. (3.13), Tu is the decoupling temperature for the neutrino species u. In the

region where neutrino heating dorninates, the reverse reaction rates of Eq. (3.10) are

negligible. This is because the average energies of the neutrinos is much higher than

30 CH.4PTER 3. SUPERNOV4 MECHANlSMS AND NUCLEOSYNTHESIS

the matter temperature. Hence, from the Boltzmann equation for the rate of change

of Y; with respect to time at the wake of the shock (Eq. (3.22)): which we assume

is constant where the shock is stalled, we c m approximate (by setting the RHS of

Eq. (3.22) to zero) [3] )X k e p - x -, (3.13) ia Le*

where A,, is the rate of the appropriate process in Eq. (3.10) mhich was derived bp

Qian 1291 to be,

The question now is, how oscillatory effects could be incorporated in t h e above equa-

tions. W e know that neutrinos of al1 species are ejected from the protoneutron star

during this epoch. If the resonance condition is satisfied for t hese out-streaming neu-

trinos between the protoneutron star and the stalled shock, then this could affect

the energy spectra f, of v. and Fe$ which appear in the equations above. This is

because the v,,, and ü,,, have higher average energies than v, and F.. Therefore.

their conversion to v, and can affect their respective energy spectra. However, one

can show from the Hamil tonian of neutrino propagation t hat resonance conditions

for both u,,, + ve and F,,, + Fe cannot take place simultaneously. In fact, only v,,;

ct Y. is possible for a 20 Mi, supernova during 0.01 - 0.5s TPB between the stalled

shock and the protoneutron star because the baryon number density satisfies the res-

onance condition for these transitions only. Hence, when MSW effects are included,

the energy spectra for ve and Fe u e given by

where f: and f:c are the v., üe spectra at the neutrinosphere. With these energy

spectra modifications, we define iIe, and x L ~ ~ ~ as the heating and reaction rates wit h

MSW effects included. They are given by

3.3. XEUTRINO CONSTR-4INTS FROM -4 SUPERNOVA

The ratio of t he total heating rate with and without the MSW effect is then given by

In chapter 1, we will present results of Our calculations of Eq. (3.18) for a range of

values of l m 2 and sin2 28.

3.3 Neutrino constraints from a supernova

The discussion in section 3.1 describes how supernovae could be the engines that

synthesize and eject most of the heavy elements that are observed in the galaxy. In

this section we present evidence that the hot bubble is the most plausible site for the

synthesis of certain relatively rare heavy elements that are observed in the gal-.

We will see that this requires a certain physicd condition in the hot bubble that can

be used to constrain the neutrino parameter space.

We have seen in section 3.1 that electron capture in the final stages of a star's

evolution can produce elements with heavy nuclei. For elements heavier than iron,

nuclear Coulomb barriers are so high that charged particle reactions (such as electron

capture) are ineffective for nucleosynthesis, leaving neutron capture as the mechanism

responsi ble for producing the heaviest nuclei,

where -4 represents a nucleus of atomic mass .4. If the neutron abundance is modest

in the region where the synthesis occurs, this capture takes place in such a way that

each newly synthesized nucleus has the opportunity to beta decay, if it is energetically

favorable to do so. Synthesis then happens dong a path of stable nuclei. This is called

slow, or s-process nucleosynthesis. However a study of the s-process nuclei in the (N,Z)

plane shows that the path misses many stable nuclei that are found in nature. This

suggests that another mechanism is also at work. This second process is called the

rapid or the r-process and is characterized by:

1- Xeutron capture rate fast compared to beta decay rates.

2. Equilibrium within a nucleus is maintained by t h e rates of (n.7 ) t, (-y.n), i.e.

neutron capture fills up acailable bound States in the nucleus until the above

equilibrium sets in.

3. The nucleosynthesis path is along exotic, neutron-rich nuclei that would be

highly unstable under normal laboratory conditions.

Using these conditions one can derive the necessary physical characteristics of a site

where r-process nucleosynthesis is feasible. The following conditions are usually stated

in the Iiterature [ i l ;

where p(n ) is the free neutron number density. Thus r-process nucleosynthesis requires

exceptional conditions and from our discussions of the physical characteristics of a

hot bubble (high temperature, low free nucelon density and duration), such sites are

perfect candidates for the r-process. Although a few other astrophysical events have

been considered as possible sites [13, 121, none of them have proved to be satisfactory.

In the next few paragraphs, we will argue for supernovae hot bubbles as the rnost

plausible sites for r-process nucleosynthesis to occur. We will also present results of

numerical calculations that the neutron to baryon nurnber density must be greater

than 0.5 in the hot bubble for r-process to occur. We will then use this condition

to constrain the neutrino parameter space. Note that the arguments and evidence

presented here are not very detailed and use some of the main findings of numerical

simulations done by supernova modelers. More detailed discussions on nucleosynthesis

and the results presented here can be found in the references to follow.

First. consider figure 3.1 which shows the soIar system abundance of r-process

elements [12!. -1 strikïng feature of this figure is the presence of s h x ? peaks in the

Figure 3.1: Solu system abundznce of r-process elements

r-process matter abundance near atomic masses -4 = 130 and -4 = 193. Such narrow

peaks suggest that rnost of the solar system r-process material th-as produced in a

single well-defined environment rather than a combination of different environments.

averaging over which would give rise to much broader abundance peaks. The hot-

bubble region outside the protoneutron star is a well defined physical region ivhose

characteristics satisfy the conditions neces sq for the r-process to occur.

Second, Hubble Space Telescope studies of very metal-poor halo stars also point

towards primary sites ( i. e. regions with fex pre-esistin; heavy element seeds for

r-process to occur, as is the case in the hot bubble) as the most plausible site for r-

process nucleosynthesis. S tudies by Corvan et al. [l;] have shown that the distribution

of r-p rocess elernents in very metal-poor stars ( where pre-exisring s-process seeds are

almost absent) is similar to that of the Sun. Since the solx r-process material did

not corne Fron ~veraging over man- environments. this fa\-ors a unique site much llke

that of a hot-bub ble for r-process nucleosynthesis.

Third. there are good theoretical argunents favorinj the r-procecs taking piace

in a prirnary site such as the hot bubble in core collapse supernovae [19]. Studies of

galactic chernical evdution indicate that the growth of r-process elemeats is consistent

wirh Ion--mass type II supernok-ae rates 2nd distributions. Also. modelers h+ve shown

that the conditions needed for the r-process mus t be realized in supernovae. more

specifically: in the hot bubble region of a supernoha.

We present now the results of numerical calculations of CVoosley et al. 1151 of the

abundance of r-process elements produced in the hot bubble region fitted to observed

r-process abundance in the solar system (figure 3.2 belolv). These resdts show a

80 100 120 O 160 180 3 0 220 Mass N-ber

Figure 3.2: Sumerical simulations of r-process material abudance from superrio\-ae

hot bubbles fitted to observed solar system abundance.

striking mat ch between the numerical results and the observed abundances [l']. This

clearly points towards the hot-buoble region N a strong candidate for a plausible site

of r-process nucleosynthesis.

In light of the evidence and a r p m e n i s presented above we rviil Esune tkzt r-

?roces nuclecsyct hecis does indeecl take ?lace in the hot bobble region oi ZL î r ? a r x n - ~

The ~ c s t point we would like to mzke Is that r-process nuclecsynrhecis is possible

ooiy i i ;Be neirtron ro baryon ratio i i ;te Sot-bubble environnent is gre+:er $22 the

electron to baryon ratio. tVe consider the results of numerical simulations, @n done

by FVoosley et al. [18] (figure 3.3) of nuckosynthesis in the hot bubble of a su-erri0k-a.

Hare ? = 2k;, and & is the electron to b-on number ratio in the hot-bubbiz resion

3 = 02co

. i f 80 LOO IZQ 140 160 L ~ O XI 210 LOO 120 :.)O 160 reo t s o 229

LIU Nuaber Yasa Number

Fi-re 3.3: Surnerical simulations done by b*oosley e t al. [18] wirh diZerear x x n n

escess in the hot bubble

deiined by Eq. (2.51). The conciition that 1; < 0.5 corresponds to thero being an

excess of neutrons in this region and inplies rhat O < 7 < 1. This condition is sacisiied

for al1 of the simulations shown in f i p e 3.3. These figures show the time evolved final

36 CHAPTER 3. SUPERNOV4 MECH.4NISMS AND NUCLEOSYXTHESIS

abundance of r-process elements with the initial values as indicated in the figures.

We note that the find figure (corresponding to an environment with initial q=0.3)

produces the peaks of r-process material observed in the solar system. Going back

to figure 3.2. w e note that these numerical results were obtained b>- averaging over a

grid of values of r ) that satisfy the above condition (O < 7 < 1). This mimicked the

production and ejection of r-process material within the hot bubble at various times

between 3-15s TPB. Given its agreement with the observed abundance. this suggests

that the r-process takes place in a hot-bubble only if

We will now show how one can constrain the neutrino parameter space from the above

condit ion.

At first we recall the physical characteristics of the hot-bubble region of a super-

nova between 3-15s TPB. Recall that the hot bubble region consists of a protoneutron

star (radius - 10 km) with a high entropy, e--e+ pair dominated plasma and free

nucleons and neutrinos of a11 species streaming through it. \Ié also recall the relative

positions of the v-sphere (r -- 10 km), weak interaction freeze out (r - 10 km) and

nuclear reaction freeze out (r - 1000 km) radii. Comparing the distances of the weak

freeze out and nuclear freeze out; it would be a good approximation to say that if

r-process nucleosynthesis takes place in t his region, i t does so predominantly between

these two freeze-out radii. This implies that ke < 0.5 in this region. Now. the elec-

tron to baryon ratio is determined by the rates of the charged current reactions of

Eq. (3.10). As was the case with supernova dynamics, the neutral current contribu-

tion and charged current reactions of v,,, and ü,,, can be neglected in the hot bubble.

Consider now an outward moving mass element in the hot bubble with velocity v ( r )

at a distance r. The rate of change of Y. in this mass element is given by

where XI = A,, + Xe+n , X2 = A I + A F ~ ~ + and each A refers to the rate of

3.3. NE UTRINO CONSTR-4INTS FROM A SUPERNOVA 37

the reaction involving its subscript. W e can sirnplifi the above equation using the

fact that the matter temperature near the weak freeze-out is small compared to the

average temperatures of ue and v, [2]. hence, Xe- , and A,+, can be ignored. -41~0'

for any mass element beyond the weak freeze out point. the matter espansion rate is

higher than the

weak freeze out

W F O and ~ N F O

(by setting the

point)

charged current reaction rates so that Y. does not change beyond the

point. Therefore Ive con approximate Y i (rivFo) = l>; ( r w ~ o ). where

are the radii of nuclear and weak freeze-out respectively. So we have

right hand side of Eq. (3.22) equd to zero at the u-eak freeze-out

Once again, the rate XUN is given by Eq. (3.15). Taking account of the fact that

the luminosities for each species of streaming neutrinos are about the same in this

epoch, and utilizing the fact that the cross sections for the reactions in Eq. (3.10) are

approximately

In the absence

equal, Eq. (3.23) can be approximated by (from Eq. (3.15)) (21.

of matter oscillations the average energies of ue and Fe are fised and

Y; = 0.41 (which satisfies Eq. (3.21)). When matter oscillation effects are taken into

account ke can be calculated from

We will see later that in our neutrino parameter space of interest (which we will

determine in chapter 4): only v, tt Y. is possible between the neutrinosphere and the

weak freeze-out point. Hence we can set P(ü, _t = O. We may now constrain

the Y parameter space by calculating K(rWFO) for a range of A n 2 and sin2 20 and

demanding that Y. < 0.5 at this freeze out point (this is because the value of Y, is

assumed constant beyond this point) and if r-process occurs in the hot bubble, Y, <

0.5 at the weak freeze-out and beyond. Our results are given in chapter 4.

35 CHAPTER 3. SUPERNOW MECHANISMS -4ND NUCLEOSYNTHESIS

3.4 RSFP and MSW combined

In chapter 2, we discussed the spin-flavor precession as the second mechanism that

can lead to neutrino mixing. Our discussions of supernova dynamics and neutrino

constraints so far have focused on matter oscillations only In this section we \vil1

consider spin-flavor transformations as well and see how our previous formalisms

have to be estended t o include these eflects. - In general. spin-%avor oscillations allom both ue - v, and ü, - v, transformations

to occur (Eq. (2.17)). However, in the region of a 20 Mo supernova that is of interest

to us: during 0.01-15s TPB, Y; is - 0.40.45 [-LI. This makes (ne-n,) < O in Eq. (2.49)

and implies that we will only need to consider Fe - v, transitions when studying the

effects of spin-flavor transitions. Transformations such as v, o Fe could change the

energy spectra fFe and f,, of both Fe and ve (see Eq. (2.47)), in a similar manner

to our discussion of matter effects on shock reheating. Therefore, to t&e spin-flavor

oscillations into account we need to write down expressions for fL and f,, in which

both PiWSw and PRsFp appear. Using Eq. (2.52) we find

Hence to take RSFP into account in our calcuIations we must use the above spectra

in Eqs. (3.23) and (3.18). Our results will be given in chapter 1.

3.5 Effects of density fluctuations

If neutrinos propagate through a region where the density fluctuates randomly, their

survival probabilities may be dec t ed by these fluctuations. Several formalisms have

been developed and many calculations have been done to study the effects of den-

sity fluctuations on the survival probabilit ies of neutrinos. Recently, Burgess and

3.3. EFFECTS OF DENSITY FLUCTUATIONS 39

Michaud [14] have considered the effects of fluctuations of heIioseismic waves on solar

neutrinos. We will use the formalism that they developed to calculate the effects of

density fluctuations on the neutrino constraints t hat we obtain from the r-process in

a supernova. We start by presenting one of their main results first and then describe

the mode1 that kvas used to obtain this result. Burgess and Micbaud [14] derived the

following espression in the presence of density fluctuations, analogous to the Pa rk

formula for

density (at

position r ) .

the survival probability of a neutrino that is created in a region of high

position r'), that goes through resonance and is detected thereafter (at

where the symbols that appear in Eq. (3.28) are defined by

Am2 sin ZO,, A d cos 20, hI1 = -Ad3 = - 9 (3.31)

4k 4k

ï is the term in the survival probability that takes into account the effects of density

fluctuations. We note that I' depends on E , l , Am2, sin2 8 and the mean electron

number density of the region through which the neutrino propagates. The parameters

E and 1 characterize the density fluctuations of the environment where the neutrino

propagates. Their meanings are given below.

The neutrino survival probability, Eq. (3.28), was derived using a so called "ce11

model". In this model, the region through which neutrinos propagate is divided

up into cells of varying lengths. In a fluctuating region, each individual neutrino

encounters a different density profile. Therefore the average survival probability is

obtained by averaging over an ensemble of density profiles. The parameter, r , in

Eq. (3.26) is the root mean square deviation from the mean density within a ce11 for

the ensemble of density profiles. The length of a particular ce11 1 is the correlation

length of the fluctuations. To derive Eq. (3.25): Burgess and Michaud [14] first

obtained a master equation (using perturbation theory) for the time evolution of the

density matris in the neutrino subsector of an abstract system of neutrinos and their

environment. This time evolution equat.ion consisted of a term t hat took into account

neutrino scattering due to average or mean properties of the en\-ironment and one

t hot included scat tering due to fluctuations in the environment. The master equat ion

n-as then applied to the ce11 mode1 of the neutrino environment from which a set

of differential equations for the time evolution of the neutrino density marris was

obtained. Using the Landau-Zener approximation, they t hen derived Eq. (3 .28) and

showed that this expression agrees well with the survival probability one obtains by

averaging over an ensemble of density profiles. There are several points t hat need to

be mentioned regarding their mode1 and Eq. (3.28).

1. Ln the absence of fluctuations ( i.e. z = 0): Eq. (3 .28) recluces to the standard

result of matter oscillations.

2 The effect of fluctuations is to damp the survival probability of neutrinos. This

is easily seen b; setting PJ = O in Eq. (:3.%).

3. Fluctuations have the largest effect a t the point of resonance (when .\if3 1 b = O

in Eq. (3.30)). -4s we will see below. this fact will be used to obtain an upper

limit on the correlation length if E is fixed.

1. Eq. (3.28) was derived using perturbation theory. Therefore, it is valid as long as

the strength of the fluctuation induced terms in the Hamiltonian is small com-

pared to the MS W contributions. Since fluctuations have maximum strength at

the resonance point, we May write down the following conditions for the validity

of this formalism:

2 2 2 GFe ne I res1 << G~nelm,, (3.33)

3.5. EFFECTS OF DENSlTY FLUCTUATIONS

or

To see horv we may apply the above conditions we note that once particular values

from the neutrino parameter space (Am2. sin2 20,) and E, have been fixed, the reso-

nance point is easily calculated. Then fising E and applying Eq. (3.31) at the point

of resonance (since this is where fluctuations could have the maximum effect) gives

us an estimate of the maximum correlation length for which the formalism is valid.

In section 3.1 we will apply this formalism and calculate the effects of density

fluctuations on the neutrino constraints we obtain from a supernova.

Chapter 4

Results, Discussion and Conclusion

So far, we have presented the t heoret ical formalisms behind our calculat ions. These

included studying the effects of matter and spin flavor oscillations on supernova dy-

namics, showing how to find neutrino constraints from r-process nucleosynt hesis, and

the effects of density fluctuations and spin-flavor oscillations on these constraints. In

this chapter we present the results of our calculations and discuss and compare them

to other known results ahen possible. We start with supernova dynamics.

4.1 Supernova dynamics and the MSW effect

In most supernova models the delayed shock mechanism predicts that the explosion

causing shock wave is stalled at a distance of about 300-500 km from the core betrveen

0.01-0.5s TPB. I t is then revived by neutrino heating at its wake. In sections 3.2

and 3.3 we have summarized this process and derived an espression for the ratio of

the beating rate of the shock wave wi th and without matter oscillations. We have

calcuiated this ratio for a range of values in the neutrino parameter space. For our

calculations we considered a 20 Mo supernova at 0.15 TPB, whose mass density profile

is shown in figure 4.1. This density profile is not an approximate representation of

the calculations of Wilson and Mayle [20] at this epoch that was also used by Fuller

4.1. SUPERNOV4 DYhXMICS AND THE M S W EFFECT 43

et al. [3]. Since we did not have access to the density profiles of [?O], we used a few

points from the density profile shown in [4] and spline fitted them to produce fig. 4.1.

' Shown in this figure are the positions of the neutrinosphere and the stalled shock.

which we take to be at 40 km and 500 km from the core. respectil-ely. To evaluate

*-Stallecl shock #-Neutrinospkiere

Figure 1.1: Density vs. distance at 0.15s TPB of a 20 MG supernova

1

h (Eq. (3.18)) we rnust determine a range of values in the lm2-sin2 20 parameter Ctot

space for which neutrino oscillations may affect the heating rate. Oscillatory effects

may be significant on the heating rate if, for specific values of Am2 and sin2 28, the

resonance condition, Eq. (2.%O), is satisfied at a position between the neutrinosphere

(r -- 50 km) and the position at which the shock is stalled. -4s mentioned in section

3.2, this is because transitions between different species of neutrinos, which are at

different average energies, would affect their energy spectra, and consequently, the

heating rate of the shock wave. We note first that the resonance position or density,

Eq. (2.20), is not sensitive to values of the vacuum mixing angle ranging between IO-'

to IO-^. This was the range that was used by Fuller et al. [3] in their calculations.

We also use the same range. From Eq. (2.20) one sees that a range for Am2 cannot be

33 CHAPTER 4. RESULTS. DISCUSSION -4ND CONCL I:SfO:V

chosen independently since the resonance condition depends on the neutrino energy

(E,,) as well as l m 2 . To malie a good choice of the Am2 range. we fix Ev to be

- '21 MeV and pick only those values for which t.he resonance condition is satisfied

between the neutrinosphere and the stalled shock. The choice of - 21 Me\- for E,,

\vas made because this is the average energy of llp.r a t this epoch. whose transitions to

v, can significantly affect the shock heating rate. Hence, the range of L m 2 thar one

finds with E+ at 21 MeV should be studied carefully. W e find this to be between 20

and 10000 e\-' which includes the range considered by Fuller et al. [3]. These values

include the case of cosmologically significant neutrinos that could make the unil-erse

closed.

The results of our calculations for the heating rates are given in figure 4.2. U e

have shown. in the Am2-sin2 20 parameter space. the contours corresponding to k cior

= 1.10. 1-20 and 1.27. Our results confirm Fuller's [3] finding that MSW oscillations

-8 -7 -6 -5 -4 -3 -2 Log,, sin228_

I

Figure 4.2: Contours of % with MSW effect included. Qot

may indeed increase the neutrino shock heating rate. .4 direct cornparison could not

be made with the results obtained by Fuller et al. [3] since they do not produce

4.1. SUPERNOVA DY-N-4 MICS -4ND THE MS W EFFECT 35

such contours. However, pointwise comparison between our results can be made. .y

The results shown in table 4.1 were obtained by pointwise calculations of 5 and

Table 4.1: .4 pointwise comparison between our data and that of ref. [3]

not by interpolating the data as was done to plot the contours shown above. Our

results show remonable agreement with those of Fuller et al. [3]. We believe that

the discrepancy between our results is due to the fact that we did not use esactly

the same density profile as they used in their calculations. This is important because

one must find the slope of the density profile to know the level crossing probabilities.

Therefore any discrepancy in the density profile could give rise to different transition

probabilities for the neutrinos and hence to different shock heating rates. However.

our results show the same trends as those of Fuller et al. [2] and for most of the

neutrino parameter space values, we are in reasonably good agreement.

Given the controversies surrounding supernova models, the following points

should be borne in mind regarding these results and their dependence on the partic-

ular supernova mode1 chosen.

1. In deriving Eq. (3.18), it was assumed that the luminosities (the energy carried

by a particular species of neutrino in eV/s at this epoch) were the same for al1

species of neutrinos streaming out of the iron core. Even though this is the case

according to the calculations of Mayle and Wilson [19]. a difference between the

luminosities of v, and ü, would modify our results.

2. The matter decoupling temperatures for the various neutrino species may \ary

between different supernova models. Even though the hierarchp of these tem-

peratures is likely to be maintained, any change in their values ivould affect the

average energies of the \arious species of neutrinos and change these results.

3 . Mie have noticed the significance of the density profile with respect to our oirn

results. It is likely that supernova modelen would disagree on the variation

of density with distance as well as the positions of the neutrinosphere and

the stalled shock. Differences in these parameters would change the neutrino

pârameter space values for which oscillations affect the heating rate: as well as

the heating rates themselves.

4. Our results have considered only transitions between ue and v,. -4 more accurate

picture ivould take into account u, and y, transitions also. However' this would

introduce more unknown parameters in the computations. This requires a more

elaborate analysis, and is a subject for further research.

5. The calculations done here did not take into account effects of possible spin-

flavor transitions between neutrinos, densi ty fluctuations, twisting effects of the

magnetic fields outside the p.rotoneutron star or the effects of flavor changing

neutral currents. These phenornena, al1 of which have been studied in terms

of their effects on solar and supernova neutrinos in many contexts, could also

play a role in the results given above. We have considered the effects of spin-

flavor oscillations on our results in section 3.3.1. We leave the others for future

considerations.

Nevert heless, cdculat ions such as t hese c m be useful for the following reasons.

They can be used to compare different supernova models and set limits on the total

energy that is deposited on the stalled shock. If restrictions from recent evidence

for neutrino oscillations are combined with neutrino parameter constraints from the

Solar Neutrino Problem or ot her ast rophysical phenomena, calculat ions such as t hese

could definitely help understand the esact supernova explosion mechanisms.

W e end this section by noting t hat . although we have shown that neutrino os-

cillations may enhance the heating rate in supernovae, Wilson and Mayle ('201 have

calculated a supernova explosion energy in agreement with the SN 1987.4 observation

by the delayed mechanism with ordinary neutrino heating alone. Thus neutrino os-

cillations rnay not be necessary for a successful supernova explosion. Hoivever. given

recent experimental evidence of neutrino oscillations, one must perforrn calculations

similar ours to see if oscillations play a role in supernova dynarnics.

4.2 Neutrino constraints

We have explained in section 3.3 how one can restrict the neutrino parameter space if

r-process nucleosynthesis takes place in the hot bubble region between 3-1 5s TPB. We

now present the results of our calculations. We consider a 20 Mg supernova mode1 of

Mayle and Wilson [?O] at 4s TPB, the m a s density profile of which is given in figure

4.3. This density profile was obtained directly from Fuller e t al. [2].

-4s was the case with supernova dynmics, we must first determine a relevant

range for the 4m2-sin228 parameter space that can be restricted by assuming that

the r-process does occur in the hot bubble. In this case, we consider those values for

which resonance occurs between the neutrinosphere and the weak freeze-out point.

This is because the values of Am2 and sin2 20 for which resonance take place beyond

the weak freeze out do not effect Y,. W e find the range of Am2 that satisfies the above

criterion to be between 3 eV2 and 1000 eV2. This was obtained assuming Eu to be 25

MeV (the average energy of v, during this epoch) in Eq. (3.8). For the mixing angle,

we take the sarne range as that used for the calculations of shock reheating. Before

CHAPTER 4. RESULTS, DISCUSSIO-N AKD CONCL LTSION

Figure 1.3: Mass density vs. distance for a 20 Mg supernova at 4s TPB.

u-e present Our results, we note that in the absence of oscillations, 1'; in Eq. (13.24) is

constant since the average energies of the various species of neutrinos remain constant.

With the matter decoupling temperatures as given by Eq. (3.8): k'; is equal to 0.41.

This satisfies the condition that would allow the r-process to occur in the hot bubble.

Our results for the neutrino constraints are presented in figure 4.4. The figure

shows contours in the Am2 - sinz 20 space of = 0.45. 0.50 and 0.55. According to

the r-process condition, neutrinos are not allowed to have values of Am2 and sin2 28

for which I; > 0.5 at the weak freeze out point. The results we obtained correspond

to those obtained by Fuller et al. [2], although they do not produce the contours

corresponding to Y , = 0.45, 0.55. W e note that in c o n t r a t to the shock reheating

case where we had some disagreements, we have used the exact density profile that

was used by Fuller et al. [2] in their calculations and obtained exactly the same

results.

The results shown above can be interpreted in two ways. First, if it is proved

that the r-process does occur in a supernova then it puts strict constraints on the

-8 -7 -6 -5 -4 -3 -2 Log., sin22ey

Figure 1.4: Neutrino constraints from a 20 M9 supernova at 1s TPB

neutrino parameter space. Conversely, if neutrino constraints from experiments were

established to be in the region for which > 0.5, then one can say that the r-process

does not take place in the hot bubble. However, the following facts should be kept

in mind regarding our interpretations of these results. First, al1 of the caveats dis-

cussed in section 1.1 regarding the dependence of these calculations on the particular

supernova mode1 used are applicable here. It should also be noted that our results

are a conservative estimate of the constra.int, since r-process can take place in the hot

bubble for values lower than 0.5. A value of Y, 1ess than 0.5 at the weak freeze-out

would then exclude a broader region in the neutrino parameter space.

It is interesting to compare our results with the experimental LSND constraints

for v,-v, oscillations (figure 4.5). We notice that they overlap for hm2 between O and

8 eV2 and sin2 28 between 4 x and 10'~.

CHAPTER 4. RESULTS, DISCUSSION AND CONCL LWON

Figure 4..5: LSND constraints on the neutrino parameter space for

4.3 Effects of RSFP

u, oscillation

In chapters 2 and 3 we introduced the spin-flavor oscillation phenomenon and dis-

cussed how the energy spectra of ut and üe may be affected when spin-flavor oscil-

lations are taken into account in addition to the MSW effect. In this section. ive

present our numerical results. However: we start with a discussion of the propagation

of neutrinos in a magnetic field, and present some numerical results that allow us

to predict for which values of Am2 and magnetic field strength, RSFP may have an

affect on the neutrino constraints. The same case, ie. the Am2 and magnetic field

strength values for the shock reheating contours we have obtained earlier would be

affected is similar and is not presented here.

We consider a Fe propagating in the hot bubble region of a 20 Mg supernova a t

4s TPB. This was the epoch frorn wkich we obtained our neutrino constraints. The

density profile a t this epoch was shown in figure 4.3. Assume that the ù, is created

at the neutrinosphere and is detected after passing through spin-flavor resonance

(RSFP). Let us also assume that the neutrino magnetic moment is pu= 10-l2 P B ,

4.3. EFFECTS OF RSFP

where p~ is the Bohr magneton. This value is close to the maximum allowed value

of 3 x 10-l' ps , a c0nstra.int that one obtains by estimating the critical mass of a

Helium flash in red giant stars [30]. Furthemore, the magnetic field strength at a

distance r from the core of the supernova is assurned to be

where ro = 10 km. This is the same magnetic field profile that was used by Fuller

et al. [?] in their calculations. -4 magnetic field of such order is plausible around a

protoneutron star given the fact that some pulsar magnetic fields are of this strengt h.

For simplici ty. we have neglected the angular (6) dependence and possible twist ing

effects of the magnetic field lines [28]. If these factors were takeen into account, the

angular dependence would affect the field strength a t a point whereas the twisting

effect would modify the resonance condition by adding a diagonal term to the effec-

tive Hamiltonian. -4 detailed analysis of these effects is left for further s t ud . In

Eq. ( M g ) , which gives the survival probabilitÿ of the &, p,B appears as a product.

Therefore. given that the transition magnetic moment of neutrinos and the magnetic

field strength outside protoneutron stars are not known, Bo above can be thought of

as parameterizing our ignorance of both the magnetic field strength and the magnetic

moment.

We have calculated (figure 4.6) t h e survival probabilities of a V, as a function

of Am2 at various values of Bo. From these figures we notice the following. If

Bo is thought of as a function of the field strength (keeping the magnetic moment

constant) only, then at low magnetic fields (< IO'* G , corresponding to Bo .- IO-'),

the survival probability stays close to one for al1 values of 4 m 2 considered, i . e . RSFP

has negligible effects. At higher field strengths ( -- 1012 G , corresponding to Bo .-

10') we start to notice effects of RSFP in the survival probabilities of Fe for some

values of AmZ. We notice that as the field strength increases, the survival probability

decreases for a wider range of Am2. One would expect from these results that if the

field strength is high enough, the survival probability would be zero for the full range

CH-4PTER 4. RESULTS, DISCUSSION .4XD CONCLIISION

Figure 4.6: ü, survival probabilities vs. Am2 at various Bo

3.3. EFFECTS OF RSFP 53

of An2 that we are interested in. Low survival probabilities for the full range of Am2

rvould imply that RSFP transitions would affect the energy spectra of the various

species of neutrinos (as was discussed in section 3.5) for this full range. Hence, u7e

espect that for magnetic field strengths of -- 1012 G , the neutrino constraints that

we had obtained earlier should be affected.

The calculations of figure 1.6 were done considering spin-flavor mixing only. W e

would now like to include both the RSFP and MSW resonances in Our calculations.

In section 1.6 we wrote down an expression: Eq. ( 2 . 5 3 , for the survi\al probability of

a neutrino which goes through both MSW and RSFP resonances. The expression was

derived assuming that the MSW and RSFP resonances do not occur simultaneousl~.

In figure 4.7, we present numerical evidence of this for Our epoch of interest. These

Figure 4.7: Resonance density vs. Am2 for both MSW and RSFP.

results clearly show that the RSFP resonance always occurs at a higher density than

the MSW resonance, ( L e . closer to the neutrinosphere) and hence, at different posi-

tions in the supernova. This was expected since we have mentioned earlier that the

mass density in our region of interest does not satisfy the condition (Y, = 5 ) neces-

sary for both MSW and RSFP to occur simultaneously. Our numerical results further

confirms this. We note that both the MSW and RSFP resonance conditions are in-

sensitive to the vacuum mixing angle for our parameter space of interest, and hence

do not affect the positions obtained here. This then validates our use of Eq. (2.32) in

su bsequent calculat ions.

4.3.1 Effects of spin-flavor transformations on neutrino con-

In figure 4.S: we show the results of taking spin-flavor transformations into account

in the neutrino constraints we obtained in section 1.2. We make the following ob-

servations from these results. First we note that a t low magnetic fields (- 10" G.

corresponding to Bo=O.l ), we essentially reproduce the constraint on the neutrino

parameter space that was obtained earlier with no RSFP. This is also expected from

our results of figure 4.6 where for Bo = 0.1, the neutrino survival probability is alrnost

equal to one for the full range of Am2 that we are interested in. We find that as the

magnet ic field increases: the excluded region (corresponding to Y, > 0.5) shrinks and

almost ail of the Am2 - sin2 20 space of interest would allow the r-process to take place

in a supernova hot bubble. This tells us that if experimental evidence were obtained

for r-process nucleosynthesis in a supernova and if neutrino magnetic moments of - 10-l2 ps were established for neutrinos and magnetic field strengths of - 10'' G

outside protoneutron stars were also found. then one could not constrain the neutrino

parameter space as we had done. Our results agree quite well with those obtained

by Fuller et al. [4]. They have similar findings at the same order of rnagni tude for

the magnetic field strength. However, we disagree on the exact value of Bo at which

almost al1 of the neutrino parameter space of interest would allow r-process to occur.

They find this t o be a t Bo=5.0, whereas we find it to be at Bo=lO.O. This discrepancy

is not surprising since we used a density profile at 4s TPB whereas they considered a

density profile a t 5.8s TPB to do the calculations. As mentioned in section 4.1, the

density profile is important since its slope at the resonance position is required for

calculating the level crossing probabilities in Eq. (2.52). In fact, a cornparison be-

4.3. EFFECTS OF RSFP

-8 -7 -6 -5 -4 -3 -2 Log,, sin22eV

0.5 t ' " ' ~ ' " ' ~ ' ' ' ' ~ ~ ' " ~ ' ' ' ' ~ ' ' ' ' ~ -8 -7 -6 -5 -4 -3 -2

Log,, sin22eV

-8 -7 -6 -5 -4 -3 -2 Log,, ~111~28~

Figure 4.8: Effects of RSFP on Y. contours of neutrino constrints

56 CHAPTER 4. RESULTSt DISCUSSION A N D CONCL USION

tween Fuller's own results of the neutrino constraints from 4s TPB [2] and 5.8s TPB

[4] show slight variations in the results due to their use of different density profiles.

This also reflects the supernova mode1 dependence of t hese calculat ions. Al t hou&

the general conclusion is likely to be the same for different superno1.a models. one

would disagree on the exact values of the field strengths at which the effects we see

are observed. If unrealistically high magnetic fields are required for similar effects

in some models, such models should be discarded provided one obtains esperimental

evidence of spin-flavor resonance as well as that of r-process nucleosynthesis occurring

in supernovae.

4.3.2 Effect s of spin-flavor oscillations on Supernova dynam-

lcs

Shown in figure 1.9 are the results of taking spin-flavor transformations into account

on the shock reheating calculations discussed in section 4.1.

W e make the following observations from figure 4.9.

1. In contrast to the case of neutrino constraints from the r-process. we notice

that lower magnetic fields are sufficient to produce appreciable changes to the

contours obtained in figure 1.2. For magnetic fields of -101° G(corresponding

to Bo up to IO), we reproduce the results of figure 4.2.

2. -4s we increase the field strength (to -10" G , corresponding to Bo > IO), we

start to notice changes in the heating rate. The maximum possible heating

rate changes from 27% to 30%. This agrees with the trend of Fuller et ai. [4]

for maximum heating rate enhancement from 35% to 40% in the presence of

spin-flavor oscillations. Because they do not produce the contours that we have-

we could not make a direct cornparison with their results. We also find that

a smaller range of the neutrino parameter space leads to a high heating rate

in the presence of RSFP. This continues until Bo is about 7.5. .kt even higher

4.3. EFFECTS OF RSFP

-8 -7 -6 -5 -4 -3 -2 Log,, sin22Bv

1

1.0 t " ' " ' ~ ' " " " ' " ' " " ' " ' p ' t J -8 -7 -6 -5 -4 -3 -2

Log, sin22Ov -8 -7 -6 -5 -4 -3 -2

Log, ~ i n ~ 2 8 ~

Figure 4.9: Effects of RSFP on Supernova shock heating ratio of * C t o z

CH-N'TER 4. RESULTS, DISCUSSION -4XD CONCL USION

d u e s . we do not expect any appreciable change since RSFP effects have been

masimized for the neutrino parameter space of interest.

Once again. we mention that the results above are dependent on the particular su-

pernova model and parameters we have chosen. Although they are reasonable choices.

they have not been experimentally verified. Nevertheless. the results above clearly

show that spin-fiavor transformations may indeed play a role in supernova dynamics

by enhancing the maximum heating rate and changing the neutrino parameter space

that corresponds to a particular heating enhancement. These results, as with the

previous ones. can be helpful in analyzing the validity of various supernova models.

4.4 Effects of density fluctuations

In chapter 3 we have summarized the formalism used by Burgess and Michaud [11]

to study the effects of fluctuations on solar neutrinos. In this section we apply this

formalism to the neutrino constraints obtained in section 4.3. In section 3.5 we

defined a term I' (Eq. (3.29)) that appears in the neutrino survival probability and

takes into account the effects of random density fluctuations. r can be computed

once particular values of A n 2 , sin2 20 and c2Z are fixed. The physical significance of E

and I was discussed in section 3.5. To reiterate, ,= is the root mean square fluctuation

about the mean density within a particular ce11 in the so called L'tell model" that ive

7 described in section 3.5, and the parameter 1 is the length of a particular ce11 and is

also equal to the correlation length of the fluctuations. We saw that once a, Am2 and

sin2 28 have been fixed, one can establish an upper limit of the correlation length for

which the formalism is valid (using Eq. (3.34)) by realizing that the largest effect of

fluctuations in the survival probability of neutrinos occur at the resonance density.

We have chosen a set of Am2 and sin2 20 values along the Y,=O.5 contour of figure

4.4 and computed r' for some values of E and their corresponding I . If i? = 1 .O, for a

particular set of values dong the Y,=0.5 contour, we can say that for those values of

4.5. CONCLUSION A N D FUTURE WORK 59

the neutrino parameter space. fluctuations with maximum correlation length lm., and

root mean square deviation e would not affect the neutrino constraints. Our results

are shown in table 3.4. It is clear from our results above that within the limits of

our formalism (specified by the maximum correlation lengths lm,,). fluctuations do

not affect the constraint space ive obtained. These results agree with those obtained

by Fuller et al. [3]. We do not speculate here on possible sources of fluctuations.

4.5 Conclusion and Future work

We conclude by summarizing our results and identifying potent i d future research

that could be followed from this work.

We have shown that matter oscillations of neutrinos could enhance the heating

rate of the shock wave in some delayed supernova explosion models. We found en-

hancement of the heating rate by up to 17% for the numerical mode1 of Mayle and

Wïlson [-O] of a 20 Mo supernova between 0.01-0.5s TPB. Our results are in reason-

ably good agreement with those of Fuller et al. [3]. We also considered the effects

of spin-flavor oscillations on the heating rates. In the presence of MSW and RSFP

oscillations, we have shown that both, the heating rate and the neutrino parameter

space that corresponds to a specific heating enhancement is different from those of

the MS W case only.

In our calculations, we considered mixing between two flavors of neutrinos only.

A more realistic scenario would include mixing between al1 three known flavors, since

neutrinos of al1 species are ejected from a supernova. Given our ignorance of supernova

explosion mechanisms, such calculations could be important in building more accurate

supernova models. Effects of flavor changing neutral currents and twisting of magnet ic

field lines were also ignored in our calculations. A more elaborate analysis should

include these effects too in the calculations. It may also be worth while to compare

the calculations we have done for different models of supernovae. We suggest al1 of

CHAPTER 4. RESULTS, DISCUSSION AND CONCL USION

5 COXCLUSION -AND FUTURE WORK 61

these as potential future research topics for studying the role of neutrinos in supernova

dynamics.

In the other half of our work: we constrained the neutrino parameter space of

lm2-sin2 '28, by considering the possibility of r-process nucleosynthesis in a super-

nova. \.Fe have verified the constraints obtained by Fuller et al. ["], and calculated the

effects of including spin-flavor transitions on t hese const raints. MIé found t hat ahen

RSFP effects are included, the neutrino parameter space that allows the r-process to

occur in the supernova hot bubble gets larger. We also found the physical conditions

for which, one could not constrain the neutrino parameter space as rve had done.

Finally. we considered the effects of density fluctuations on the constraints we had

obtained. We found that density fluctuations would not affect the constraint space

we obtained within the limits of the formalism that we used. We used a method

that was developed by Burgess and Michaud [11] to obtain our results which agreed

with those obtained by Fuller et al. [3]. As a a s the case with supernova dynamics

related calculations, we did not take into account mixing between the three known

species of neutrinos as well as possible effects due to flavor changing neutral currents

and magnetic field line twisting. We also ignored the effects of density fluctuations of

the magnetic field lines in our calculations. Incorporating al1 of these phenornena in

the calculations we did could definitely help set tle questions regarding the possi bility

of constraining the neutrino parameter space or that of the r-process occuring in a

supernova.

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