lecture 4 stellar masses. spectroscopy obtaining a spectrum of a star allows you to measure:...
DESCRIPTION
Doppler shifts Doppler shifts of the spectral lines yield the radial (i.e. toward the observer) velocity of the starTRANSCRIPT
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Lecture 4Lecture 4
Stellar masses
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SpectroscopySpectroscopyObtaining a spectrum of a star allows you to
measure:1. Chemical composition2. Distance (via spectral parallax)3. Effective temperature4. Radial velocity5. Magnetic field strength
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Doppler shiftsDoppler shiftsDoppler shifts of the spectral lines yield the radial (i.e. toward the observer) velocity of the star
1 if
zzcv
z
r
restrest
restobs
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1. Typical stars in the solar neighbourhood have velocities ~30 km/s. What is the size of their Doppler shift at 500 nm?
Doppler shifts: examplesDoppler shifts: examples
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The Zeeman effectThe Zeeman effectIn the presence of an external magnetic field (which defines a preferred spatial direction) the orbital energy depends on the field strength and on the quantum number ml
ml +10 -1
0
cmeB e4
cmeB e4
emeB
4
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Example: the Zeeman effectExample: the Zeeman effectPulsars are rapidly spinning neutron stars which beam light in opposite directions. They have huge magnetic fields of 104 – 108 Tesla. How large is the Zeeman splitting?
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Kepler’s LawsKepler’s LawsJohannes Kepler derived the following 3
empirical laws, based on Tycho Brahe’s careful observations of planetary positions (astrometry).
1. A planet orbits the Sun in an ellipse, with the Sun at one focus
2. A line connecting a planet to the Sun sweeps out equal areas in equal time intervals
3. P2=a3, where P is the period and a is the average distance from the Sun.
32 aP
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EllipsesEllipses
Ellipticity: Relates the semi-major (a) and semi-minor (b) axes: 21 eab
Equation of an ellipse:
cos1
1 2
eear
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Centre of massCentre of mass
rm
r
rm
r
rrr
22
11
12
21
21
mmmm
Where we have defined the reduced mass:
More generally, it is the centre of mass that is at one focus of the ellipse
For the Earth-Sun system, how far is the Sun from the centre of mass?
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Energy and Angular momentumEnergy and Angular momentum
2
21.. vEK
vrL
The two-body problem may be treated as a one-body problem with reduced mass orbiting a fixed mass M=m1+m2
rMGEP
..
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Kepler’s Second LawKepler’s Second Law2. A line connecting a planet to the Sun sweeps out equal areas in
equal time intervals
This is just a consequence of angular momentum conservation.zrvprLˆ
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Since L is constant,
Example: how much faster does Earth move at perihelion compared with aphelion? Recall e=0.0167
Angular momentum conservationAngular momentum conservation
ee
rr
vv
vrvr
LL
a
p
p
a
ppaa
pa
11
034.19833.00167.1
11
ee
vv
a
p
i.e. 3.4% faster
(aphelion=perihilion)
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BreakBreak
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Kepler’s First LawKepler’s First LawThe radius r connecting two bodies
describes an ellipse, with eccentricity and semimajor axis related to the energy and angular momentum
Now, since: the mass m1 also moves in an ellipse with semi-major axis a1 and the same eccentricity, e, and period P.
1
2
1
2
2
1
aa
rr
mm
rm
rrm
r
22
11 ,
cos1
1 2
eear
am
a1
1
21
2
11
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11
1
aa
mma
mmmmama
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ExamplesExamplesTwo stars are separated by 3 A.U. One star is three times more massive than the other. Plot their orbits for e=0.
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Orbital angular momentumOrbital angular momentumWe know the angular momentum is constant; but what is its value?
zrvprLˆ
dtdAL 2
Since L is constant, we can take A and t at any time, or over any time interval.
Pea
PA
L ellipse
22 12
2
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Example: the Sun-Jupiter systemExample: the Sun-Jupiter system
PeaL
22 12
What is the angular momentum of the Sun-Jupiter system, where a=5.2, e=0.048, P=11.86 yr ?
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Derivation of Generalized KIIIDerivation of Generalized KIII
GMaP
322 4
FromPeaL
22 12
322
22 21
MGELe
EGMa
2
and conservation of energy, we can derive