chapter 3

3.1 • STELLAR PARALLAX

57

1 AU Id= --:::: -AU,

tanp p

Stellar ParallaxThe Magnitude ScaleThe Wave Nature ofLightBlackbody RadiationThe Quantization ofEnergyThe Color Index

The Continuous Spectrum of Light

3.13.23.33.43.53.6

Measuring the intrinsic brightness of stars is inextricably linked with determining theirdistances. This chapter on the light emitted by stars therefore begins with the problem offinding the distance to astronomical objects, one of the most important and most difficulttasks faced by astronomers. Kepler's laws in their original form describe the relative sizesof the planets' orbits in terms of astronomical units; their actual dimensions were unknownto Kepler and his contemporaries. The true scale of the Solar System was first revealedin 1761 when the distance to Venus was measured as it crossed the disk of the Sun ina rare transit during inferior conjunction. The method used was trigonometric parallax,the familiar surveyor's technique of triangulation. On Earth, the distance to the peak ofa remote mountain can be determined by measuring that peak's angular position fromtwo observation points separated by a known baseline distance. Simple trigonometry thensupplies the distance to the peak; see Fig. 3.1. Similarly, the distances to the planets can bemeasured from two widely separated observation sites on Earth.

Finding the distance even to the nearest stars requires a longer baseline than Earth'sdiameter. As Earth orbits the Sun, two observations of the same star made 6 months apartemploy a baseline equal to the diameter of Earth's orbit. These measurements reveal that anearby star exhibits an annual back-and-forth change in its position against the stationarybackground of much more distant stars. (As mentioned in Section 1.3, a star may alsochange its position as a consequence of its own motion through space. However, this propermotion, seen from Earth, is not periodic and so can be distinguished from the star's periodicdisplacement caused by Earth's orbital motion.) As shown in Fig. 3.2, a measurement ofthe parallax angle p (one-half of the maximum change in angular position) allows thecalculation of the distance d to the star.

3CHAPTER

Earth

Chapter 3 The Continuous Spectrum of Light

(3.1)

Defining a new unit of distance, the parsec (parallax-second, abbreviated pc), as I pc =2.06264806 X 105 AU = 3.0856776 X 1016 m leads to

Id~lp'lp/l

206,265de::::: AU.

p/l

FIGURE 3.2 Stellar parallax: d = 11 p" pc.

FIGURE 3.1 Trigonometric parallax: d = B / tan p.

I'd2B ------------------

L\J?

where the small-angle approximation tan p e::::: p has been employed for the parallax anglep measured in radians. Using 1 radian = 57.2957795° = 206264.806/1 to convert p to p/lin units of arcseconds produces

By definition, when the parallax angle p = I", the distance to the star is 1pc. Thus I parsecis the distance from which the radius ofEarth's orbit, I AU, subtends an angle of I/I.Anotherunit of distance often encountered is the light-year (abbreviated 1y), the distance traveledby light through a vacuum in one Julian year: 1 ly = 9.460730472 X 1015 m. One parsecis equivalent to 3.2615638 ly.

58

within 10% of the modem value 3.48 pc. 61 Cygni is one of the Sun's nearest neighbors.

Example 3.1.1. In 1838, after 4 years of observing 61 Cygni, Bessel announced his measurement of a parallax angle of 0.316" for that star. This corresponds to a distance of

1 1d = - pc = -- pc = 3.16 pc = 10.3 ly

p" 0.316 '

593.1 Stellar Parallax

Even Proxima Centauri, the nearest star other than the Sun, has a parallax angle of lessthan I". (Proxima Centauri is a member of the triple star system ct Centauri, and has aparallax angle of 0.77". If Earth's orbit around the Sun were represented by a dime, thenProxima Centauri would be located 2.4 km away!) In fact, this cyclic change in a star'sposition is so difficult to detect that it was not until 1838 that it was first measured, byFriedrich Wilhelm Bessel (1784-1846), a German mathematician and astronomer.1

From 1989 to 1993, the European Space Agency's (ESA's) Hipparcos Space AstrometryMission operated high above Earth's distorting atmosphere.2 The spacecraft was able tomeasure parallax angles with accuracies approaching 0.001" for over 118,000 stars, corresponding to a distance of 1000 pc == 1 kpc (kiloparsec). Along with the high-precisionHipparcos experiment aboard the spacecraft, the lower-precision Tycho experiment produced a catalog of more than 1 million stars with parallaxes down to 0.02" - 0.03". Thetwo catalogs were published in 1997 and are available on CD-ROMs and the World WideWeb. Despite the impressive accuracy of the Hipparcos mission, the distances that wereobtained are still quite small compared to the 8-kpc distance to the center of our Milky WayGalaxy, so stellar trigonometric parallax is currently useful only for surveying the localneighborhood of the Sun.

However, within the next decade, NASA plans to launch the Space Interferometry Mission (SIM PlanetQuest). This observatory will be able to determine positions, distances,and proper motions of stars with parallax angles as small as 4 microarcseconds (0.000004"),leading to the direct determination of distances of objects up to 250 kpc away, assuming thatthe objects are bright enough. In addition, ESA will launch the Gaia mission within the nextdecade as well, which will catalog the brightest 1 billion stars with parallax angles as smallas 10 microarcseconds. With the anticipated levels of accuracy, these missions will be ableto catalog stars and other objects across the Milky Way Galaxy and even in nearby galaxies.Clearly these ambitious projects will provide an amazing wealth of new information aboutthe three-dimensional structure of our Galaxy and the nature of its constituents.

1Tycho Brahe had searched for stellar parallax 250 years earlier, but his instruments were too imprecise to find it.Tycho concluded that Earth does not move through space, and he was thus unable to accept Copernicus's modelof a heliocentric Solar System.2Astrometry is the subdiscipline of astronomy that is concerned with the three-dimensional positions of celestialobjects.

60 Chapter 3 The Continuous Spectrum of Light

3.2 .THE MAGNITUDE SCALE

Nearly all of the information astronomers have received about the universe beyond ourSolar System has come from the careful study of the light emitted by stars, galaxies, andinterstellar clouds of gas and dust. Our modem understanding of the universe has been madepossible by the quantitative measurement of the intensity and polarization of light in everypart of the electromagnetic spectrum.

Apparent Magnitude

The Greek astronomer Hipparchus was one of the first sky watchers to catalog the starsthat he saw. In addition to compiling a list of the positions of some 850 stars, Hipparchusinvented a numerical scale to describe how bright each star appeared in the sky. He assignedan apparent magnitude m = I to the brightest stars in the sky, and he gave the dimmeststars visible to the naked eye an apparent magnitude of m = 6. Note that a smaller apparentmagnitude means a brighter-appearing star.

Since Hipparchus's time, astronomers have extended and refined his apparent magnitudescale. In the nineteenth century, it was thought that the human eye responded to the differencein the logarithms of the brightness of two luminous objects. This theory led to a scale inwhich a difference of I magnitude between two stars implies a constant ratio between theirbrightnesses. By the modern definition, a difference of 5 magnitudes corresponds exactlyto a factor of 100 in brightness, so a difference of 1 magnitude corresponds exactly to abrightness ratio of 1001/5 ::: 2.512. Thus a first-magnitude star appears 2.512 times brighterthan a second-magnitude star, 2.5122 = 6.310 times brighter than a third-magnitude star,and 100 times brighter than a sixth-magnitude star.

Using sensitive detectors, astronomers can measure the apparent magnitude of an objectwith an accuracy of ±0.01 magnitude, and differences in magnitudes with an accuracy of±0.002 magnitude. Hipparchus's scale has been extended in both directions, from ni =-26.83 for the Sun to approximately m = 30 for the faintest object detectable.3 The totalrange of nearly 57 magnitudes corresponds to over 10057/ 5 = (102)11.4 ::: 1023 for the ratioof the apparent brightness of the Sun to that of the faintest star or galaxy yet observed.

flux, Luminosity, and the Inverse Square Law

The "brightness" of a star is actually measured in terms of the radiant flux F received fromthe star. The radiant flux is the total amount of light energy of all wavelengths that crossesa unit area oriented perpendicular to the direction of the light's travel per unit time; that is,it is the number of joules of starlight energy per second (i.e., the number of watts) receivedby one square meter of a detector aimed at the star. Of course, the radiant flux receivedfrom an object depends on both its intrinsic luminosity (energy emitted per second) and itsdistance from the observer. The same star, if located farther from Earth, would appear lessbright in the sky.

3The magnitudes discussed in this section are actually bolometric magnitudes, measured over all wavelengths oflight; see Section 3.6 for a discussion of magnitudes measured by detectors over a finite wavelength region.

4If the star is moving with a speed near that of light, the inverse square law must be modified slightly.

The connection between a star's apparent and absolute magnitudes and its distance may befound by combining Eqs. (3.2) and (3.3):

61

(3.2)

(3.3)

(3.4)

3.2 The Magnitude Scale

Imagine a star of luminosity L surrounded by a huge spherical shell of radius r. Then,assuming that no light is absorbed during its journey out to the shell, the radiant flux, F,measured at distance r is related to the star's luminosity by

the denominator being simply the area of the sphere. Since L does not depend on r, theradiant flux is inversely proportional to the square of the distance from the star. This is thewell-known inverse square law for light.4

This value of the solar flux is known as the solar irradiance, sometimes also called thesolar constant. At a distance of 10 pc = 2.063 X 106 AU, an observer would measure theradiant flux to be only 1/ (2.063 X 106)2 as large. That is, the radiant flux from the Sunwould be 3.208 x 10-10 W m-2 at a distance of 10 pc.

Example 3.2.1. The luminosity of the Sun is L o = 3.839 X 1026 W. At a distance of1 AU = 1.496 X 1011 m, Earth receives a radiant flux above its absorbing atmosphere of

F = ~ = l365Wm-2 .4nr2

F (d)2100(m-M)/5 = ~ = __ ,F lOpe

Taking the logarithm of both sides leads to the alternative form:

ml - m2 = -2.5log10 (~J.

Absolute Magnitude

Using the inverse square law, astronomers can assign an absolute magnitude, M, to eachstar. This is defined to be the apparent magnitude a star would have if it were located at adistance of lOpe. Recall that a difference of 5 magnitudes between the apparent magnitudesof two stars corresponds to the smaller-magnitude star being 100 times brighter than thelarger-magnitude star. This allows us to specify their flux ratio as

The Distance Modulus


M sun = mSun - 5 log 10 (d) + 5 = +4.74,

(3.6)

(3.5)

(3.7)

(3.9)

(3.8)

I II d = 1O(m-M+5)/5 pc. I

m = M sun - 2.5log lO (~) ,FlO, 0

I M = MSun - 2.5 log10 (~) ,

I L o

i

I m - M = 5log lO (d) - 5 = 5log lO (~-).I 10 pc

where the absolute magnitude and luminosity of the Sun are MSun = +4.74 and L o =3.839 X 1026 W, respectively. It is left as an exercise for you to show that a star's apparentmagnitude m is related to the radiant flux F received from the star by

For two stars at the same distance, Eq. (3.2) shows that the ratio of their radiant fluxesis equal to the ratio of their luminosities. Thus Eq. (3.3) for absolute magnitudes becomes

Example 3.2.2. The apparent magnitude of the Sun is mSun = -26.83, and its distance is1 AU = 4.848 X 10-6 pc. Equation (3.6) shows that the absolute magnitude of the Sun is

Letting one of these stars be the Sun reveals the direct relation between a star's absolutemagnitude and its luminosity:

as already given. The Sun's distance modulus is thus mSun - Msun = -31.57.5

The quantity m - M is therefore a measure of the distance to a star and is called the star'sdistance modulus:

where FlO is the flux that would be received if the star were at a distance of 10 pc, and d isthe star's distance, measured in parsecs. Solving for d gives

where FlO,O is the radiant flux received from the Sun at a distance of 10 pc (see Example 3.2.1).

The inverse square law for light, Eq. (3.2), relates the intrinsic properties of a star(luminosity L and absolute magnitude M) to the quantities measured at a distance from

5The magnitudes m and M for the Sun have a "Sun" subscript (instead of "0") to avoid confusion with Mo , thestandard symbol for the Sun's mass.

62

3.3 .THE WAVE NATURE OF LIGHT

Much of the history of physics is concerned with the evolution of our ideas about the natureof light.

6We now know that it takes light about 16.5 minutes to travel 2 AU.7In 1905 Albert Einstein realized that the speed of light is a universal constant of nature whose value is independentof the observer (see page 88). This realization plays a central role in his Special Theory of Relativity (Chapter 4).

63

(3.10)I c = Av·1

3.3 The Wave Nature of Light

that star (radiant flux F and apparent magnitude m). At first glance, it may seem thatastronomers must start with the measurable quantities F and m and then use the distanceto the star (if known) to determine the star's intrinsic properties. However, if the starbelongs to an important class of objects known as pulsating variable stars, its intrinsicluminosity L and absolute magnitude M can be determined without any knowledge of itsdistance. Equation (3.5) then gives the distance to the variable star. As will be discussed inSection 14.1, these stars act as beacons that illuminate the fundamental distance scale ofthe universe.

The Speed of Light

The speed oflight was first measured with some accuracy in 1675, by the Danish astronomerOle Roemer (1644-1710). Roemer observed the moons of Jupiter as they passed into thegiant planet's shadow, and he was able to calculate when future eclipses of the moons shouldoccur by using Kepler's laws. However, Roemer discovered that when Earth was movingcloser to Jupiter, the eclipses occurred earlier than expected. Similarly, when Earth wasmoving away from Jupiter, the eclipses occurred behind schedule. Roemer realized thatthe discrepancy was caused by the differing amounts of time it took for light to travel thechanging distance between the two planets, and he concluded that 22 minutes was requiredfor light to cross the diameter of Earth's orbit.6 The resulting value of2.2 x 108 m S-1 wasclose to the modem value of the speed of light. In 1983 the speed of light in vacuo wasformally defined to be c = 2.99792458 X 108 m S-I, and the unit of length (the meter) isnow derived from this value.?

Young's Double-Slit Experiment

Even the fundamental nature of light has long been debated. Isaac Newton, for example,believed that light must consist of a rectilinear stream of particles, because only such astream could account for the sharpness of shadows. Christian Huygens (1629-1695), acontemporary of Newton, advanced the idea that light must consist of waves. Accordingto Huygens, light is described by the usual quantities appropriate for a wave. The distancebetween two successive wave crests is the wavelength A, and the number of waves persecond that pass a point in space is the freqnency v of the wave. Then the speed of the lightwave is given by

:=

(b)

<="~--'-':7"'""'-----L---~ tl

CIl

Slit I

,1Slit I \Path difference =d sin (j

(a)

~+~

f\f\\TV

8Actually, Young used pinholes in his original experiment.


FIGURE 3.3 Double-slit experiment.

FIGURE 3.4 Superposition principle for light waves. (a) Constructive interference. (b) Destructiveinterference.

Both the particle and wave models could explain the familiar phenomena of the reflectionand refraction of light. However, the particle model prevailed, primarily on the strengthof Newton's reputation, until the wave nature of light was conclusively demonstrated byThomas Young's (1773-1829) famous double-slit experiment.

In a double-slit experiment, monochromatic light of wavelength A from a single sourcepasses through two narrow, parallel slits that are separated by a distance d. The light thenfalls upon a screen a distance L beyond the two slits (see Fig. 3.3). The series of lightand dark inteiference fringes that Young observed on the screen could be explained onlyby a wave model of light. As the light waves pass through the narrow slits,8 they spreadout (diffract) radially in a succession of crests and troughs. Light obeys a superpositionprinciple, so when two waves meet, they add algebraically; see Fig. 3.4. At the screen, if awave crest from one slit meets a wave crest from the other slit, a bright fringe or maximum isproduced by the resulting constructive interference. But if a wave crest from one slit meetsa wave trough from the other slit, they cancel each other, and a dark fringe or minimumresults from this destructive interference.

The interference pattem observed thus depends on the difference in the lengths of thepaths traveled by the light waves from the two slits to the screen. As shown in Fig. 3.3,if L » d, then to a good approximation this path difference is d sin e. The light waveswill arrive at the screen in phase if the path difference is equal to an integral number ofwavelengths. On the other hand, the light waves will arrive 1800 out ofphase if the pathdifference is equal to an odd integral number ofhalf-wavelengths. So for L » d, the angular

64

9Another commonly used measure of the wavelength oflight is the angstrom; 1A = 0.1 nm. In these units, violetlight has a wavelength of 4000 A and red light has a wavelength of 7000 A.laThe electromagnetic wave shown in Fig. 3.5 is plane-polarized, with its electric and magnetic fields oscillatingin planes. Because E and B are always perpendicular, their respective planes of polarization are perpendicular aswell.

What is light? Since the time of Young and Fresnel we know that it is wavemotion. We know the velocity ofthe waves, we know their lengths, and we knowthat they are transverse; in short, our knowledge of the geometrical conditionsof the motion is complete. A doubt about these things is no longer possible; arefutation of these views is inconceivable to the physicist. The wave theory oflight is, from the point of view of human beings, certainty.

In either case, n is called the order of the maximum or minimum. From the measuredpositions of the light and dark fringes on the screen, Young was able to determine thewavelength of the light. At the short-wavelength end, Young found that violet light hasa wavelength of approximately 400 nm, while at the long-wavelength end, red light hasa wavelength of only 700 nm.9 The diffraction of light goes unnoticed under everydayconditions for these short wavelengths, thus explaining Newton's sharp shadows.

65

(3.11)(n = 0, 1, 2, for bright fringes)

(n = 1, 2, 3, for dark fringes).{

n'Ad sine =

(n - D'A

positions of the bright and dark fringes for double-slit interference are given by


Maxwell's Electromagnetic Wave Theory

The nature of these waves of light remained elusive until the early 1860s, when the Scottishmathematical physicistJames Clerk Maxwell (1831-1879) succeeded in condensing everything known about electric and magnetic fields into the four equations that today bear hisname. Maxwell found that his equations could be manipulated to produce wave equationsfor the electric and magnetic field vectors E and B. These wave equations predicted theexistence of electromagnetic waves that travel through a vacuum with speed v = 11JEo/.Lo,

where EO and /.La are fundamental constants associated with the electric and magnetic fields,respectively. Upon inserting the values of EO and /.La, Maxwell was amazed to discoverthat electromagnetic waves travel at the speed of light. Furthermore, these equations implied that electromagnetic waves are transverse waves, with the oscillatory electric andmagnetic fields perpendicular to each other and to the direction of the wave's propagation(see Fig. 3.5); such waves could exhibit the polarization lO known to occur for light. Maxwell wrote that "we can scarcely avoid the inference that light consists in the transversemodulations of the same medium which is the cause of electric and magnetic phenomena."

Maxwell did not live to see the experimental verification of his prediction of electromagnetic waves. Ten years after Maxwell's death, the German physicist Heinrich Hertz(1857-1894) succeeded in producing radio waves in his laboratory. Hertz determined thatthese electromagnetic waves do indeed travel at the speed of light, and he confirmed theirreflection, refraction, and polarization properties. In 1889, Hertz wrote:


E

B

v"=c

FIGURE 3.5 Electromagnetic wave.

TABLE 3.1 The Electromagnetic Spectrum.

Region Wavelength

Gamma ray A < lnmX-ray 1 nm < A < lOnmUltraviolet lOnm< A < 400nmVisible 400nm < A < 700nmInfrared 700nm < A < lmmMicrowave I mm< A < IOcmRadio lOcm< A

The Electromagnetic Spectrum

Today, astronomers utilize light from every part of the electromagnetic spectrum. The totalspectrum of light consists of electromagnetic waves of all wavelengths, ranging from veryshort-wavelength gamma rays to very long-wavelength radio waves. Table 3.1 shows howthe electromagnetic spectrum has been arbitrarily divided into various wavelength regions.

The Poynting Vector and Radiation Pressure

Like all waves, electromagnetic waves carry both energy and momentum in the directionof propagation. The rate at which energy is carried by a light wave is described by thePoynting vector, 11

1S=-ExB,

/-La

where S has units of W m-2. The Poynting vector points in the direction of the electro

magnetic wave's propagation and has a magnitude equal to the amount of energy per unittime that crosses a unit area oriented perpendicular to the direction of the propagation of

liThe Poynting vector is named after John Henry Poynting (1852-1914), the physicist who first described it.

the wave. Because the magnitudes of the fields E and B vary harmonically with time, thequantity of practical interest is the time-averaged value of the Poynting vector over one cycle of the electromagnetic wave. In a vacuum the magnitude of the time-averaged Poyntingvector, (S), is

where e is the angle of incidence of the light as measured from the direction perpendicularto the surface of area A. Alternatively, if the light is completely reflected, then the radiationpressure force must act in a direction perpendicular to the surface; the reflected light cannotexert a force parallel to the surface. Then the magnitude of the force is

67

(3.12)

(3.13)

(3.14)

F rad (reflection)

(reflection) .

(absorption),(S)A

Frad = -- cos ec

\ ,--+-------':...L.~~-.... Frad (absorption)

FIGURE 3.6 Radiation pressure force. The surface area A is seen edge on.


I(S) = -EoBo,

2lto

where Eo and Bo are the maximum magnitudes (amplitudes) of the electric and magneticfields, respectively. (For an electromagnetic wave in a vacuum, Eo and Bo are related byEo = cBo.) The time-averaged Poynting vector thus provides a description of the radiantflux in terms of the electric and magnetic fields of the light waves. However, it should beremembered that the radiant flux discussed in Section 3.2 involves the amount of energyreceived at all wavelengths from a star, whereas Eo and Bo describe an electromagneticwave of a specified wavelength.

Because an electromagnetic wave carries momentum, it can exert a force on a surfacehit by the light. The resulting radiation pressure depends on whether the light is reflectedfrom or absorbed by the surface. Referring to Fig. 3.6, if the light is completely absorbed,then the force due to radiation pressure is in the direction of the light's propagation and hasmagnitude

2(S)A 2Frad = -- cos e

c

Radiation pressure has a negligible effect on physical systems under everyday conditions.However, radiation pressure may playa dominant role in determining some aspects of thebehavior of extremely luminous objects such as early main-sequence stars, red supergiants,and accreting compact stars. It may also have a significant effect on the small particles ofdust found throughout the interstellar medium.

FIGURE 3.7 The constellation of Orion.

3.4 • BLACKBODY RADIATION

..

Rigel

Right ascension (hr)

10

8 Betelgeuse

6

4

2

o-2

-4

=:~ ./ :.-1O~

6 ----'-----'---'---'-----'-----'---'5

~

~§.~

.Su

OJ

Ci


The Connection between Color and Temperature

The connection between the color of light emitted by a hot object and its temperature wasfirst noticed in 1792 by the English maker of fine porcelain Thomas Wedgewood. All ofhis ovens became red-hot at the same temperature, independent of their size, shape, andconstruction. Subsequent investigations by many physicists revealed that any object witha temperature above absolute zero emits light of all wavelengths with varying degrees ofefficiency; an ideal emitter is an object that absorbs all of the light energy incident uponit and reradiates this energy with the characteristic spectrum shown in Fig. 3.8. Becausean ideal emitter reflects no light, it is known as a blackbody, and the radiation it emitsis called blackbody radiation. Stars and planets are blackbodies, at least to a rough firstapproximation.

Figure 3.8 shows that a blackbody of temperature T emits a continuous spectrum withsome energy at all wavelengths and that this blackbody spectrum peaks at a wavelengthAmax , which becomes shorter with increasing temperature. The relation between Amax and

Anyone who has looked at the constellation of Orion on a clear winter night has noticed thestrikingly different colors of red Betelgeuse (in Orion's northeast shoulder) and blue-whiteRigel (in the southwest leg); see Fig. 3.7. These colors betray the difference in the surfacetemperatures of the two stars. Betelgeuse has a surface temperature of roughly 3600 K,significantly cooler than the 13,OOO-K surface of Rigel. 12

12Both of these stars are pulsating variables (Chapter 14), so the values quoted are average temperatures. Estimatesof the surface temperature of Betelgeuse actually range quite widely, from about 3100 K to 3900 K. Similarly,estimates of the surface temperature of Rigel range from 8000 K to 13,000 K.

68

FIGURE 3.8 Blackbody spectrum [Planck function BA (T)].

in the ultraviolet region.

(3.15)

1000 1200 1400800

Wavelengthi\ (nm)

600400

I AmaxT = 0.002897755 m K.

200

3.4 Blackbody Radiation 69

7

6

i 51;1

iec= 4'"'e~ 3.,.S~

[:;' 2'-::(o::l

T is known as Wien's displacement law: 13

0.0029 m K -7Amax ::::: 3600 K = 8.05 x 10 m = 805 nm,

0.0029 m K -7Amax ::::: 13,000 K = 2.23 x 10 m = 223 nm,

Example 3.4.1. Betelgeuse has a surface temperature of 3600 K. If we treat Betelgeuseas a blackbody, Wien's displacement law shows that its continuous spectrum peaks at awavelength of

which is in the infrared region of the electromagnetic spectrum. Rigel, with a surfacetemperature of 13,000 K, has a continuous spectrum that peaks at a wavelength of

l3In 1911, the Gennan physicist Wilhelm Wien (1864-1928) received the Nobel Prize in 1911 for his theoreticalcontributions to understanding the blackbody spectrum.

The Stefan-Boltzmann Equation

Figure 3.8 also shows that as the temperature of a blackbody increases, it emits moreenergy per second at all wavelengths. Experiments performed by the Austrian physicist


a = 5.670400 x 10-8 W m-2 K-4 .

(3.17)

(3.16)

(3.18)

(3.19)AmaxT ~ (500 nm)(5800 K).

Example 3.4.2. The luminosity of the Sun is L(') = 3.839 X 1026 Wand its radius isR(') = 6.95508 X 108 m. The effective temperature of the Sun's surface is then

I L = 4nR2aTe

4·1

Since stars are not perfect blackbodies, we use this equation to define the effective temperature Te of a star's surface. Combining this with the inverse square law, Eq. (3.2), showsthat at the surface ofthe star (r = R), the suiface flux is

Josef Stefan (1835-1893) in 1879 showed that the luminosity, L, of a blackbody of area Aand temperature T (in kelvins) is given by

For a spherical star of radius R and surface area A = 4n R 2, the Stefan-Boltzmann equation

takes the form

Five years later another Austrian physicist, Ludwig Boltzmann (1844-1906), derived thisequation, now called the Stefan-Boltzmann equation, using the laws of thermodynamicsand Maxwell's formula for radiation pressure. The Stefan-Boltzmann constant, a, has thevalue

T(') = ( L~ ) ± = 5777 K.4nR(')a

The radiant flux at the solar surface is

Fsurf = aT6 = 6.316 x 107 W m-2.

According to Wien's displacement law, the Sun's continuous spectrum peaks at a wavelength of

0.0029 m KAmax ~ 5777 K = 5.016 X 10-7 m = 501.6 llID.

This wavelength falls in the green region (491 nm < A < 575 nm) of the spectrum ofvisible light. However, the Sun emits a continuum of wavelengths both shorter and longerthan Amax , and the human eye perceives the Sun's color as yellow. Because the Sun emitsmost of its energy at visible wavelengths (see Fig. 3.8), and because Earth's atmosphere istransparent at these wavelengths, the evolutionary process of natural selection has produceda human eye sensitive to this wavelength region of the electromagnetic spectrum.

Rounding off Amax and T(') to the values of 500 nm and 5800 K, respectively, permitsWien's displacement law to be written in the approximate form

70

-

3.5 .THE QUANTIZATION OF ENERGY

14Lord Rayleigh, as he is commonly known, was born John William Strott but succeeded to the title of third BaronRayleigh of Terling Place, Witham, in the county of Essex, when he was thirty years old.

One of the problems haunting physicists at the end of the nineteenth century was theirinability to derive from fundamental physical principles the blackbody radiation curve depicted in Fig. 3.8. Lord Rayleigh14 (1842-1919) had attempted to arrive atthe expression byapplying Maxwell's equations of classical electromagnetic theory together with the results

713.5 The Quantization of Energy

Nature and Nature's laws lay hid in night:God said, Let Newton be! and all was light.

It did not last: the Devil howling "Ho!Let Einstein be!" restored the status quo.

The Eve of a New World View

This section draws to a close at the end of the nineteenth century. The physicists andastronomers of the time believed that all of the principles that govern the physical worldhad finally been discovered. Their scientific world view, the Newtonian paradigm, was theculmination of the heroic, golden age of classical physics that had flourished for over threehundred years. The construction of this paradigm began with the brilliant observations ofGalileo and the subtle insights of Newton. Its architecture was framed by Newton's laws,supported by the twin pillars of the conservation of energy and momentum and illuminatedby Maxwell's electromagnetic waves. Its legacy was a deterministic description ofa universethat ran like clockwork, with wheels turning inside of wheels, all of its gears perfectlymeshed. Physics was in danger of becoming a victim of its own success. There were nochallenges remaining. All of the great discoveries apparently had been made, and the onlytask remaining for men and women of science at the end of the nineteenth century wasfilling in the details.

However, as the twentieth century opened, it became increasingly apparent that a crisiswas brewing. Physicists were frustrated by their inability to answer some of the simplestquestions concerning light. What is the medium through which light waves travel the vastdistances between the stars, and what is Earth's speed through this medium? What determines the continuous spectrum of blackbody radiation and the characteristic, discrete colorsof tubes filled with hot glowing gases? Astronomers were tantalized by hints of a treasureof knowledge just beyond their grasp.

It took a physicist of the stature of Albert Einstein to topple the Newtonian paradigmand bring about two revolutions in physics. One transformed our ideas about space andtime, and the other changed our basic concepts of matter and energy. The rigid clockworkuniverse of the golden era was found to be an illusion and was replaced by a randomuniverse governed by the laws of probability and statistics. The following four lines aptlysummarize the situation. The first two lines were written by the English poet Alexander Pope(1688-1744), a contemporary of Newton; the last two, by Sir J. C. Squire (1884-1958),were penned in 1926.


where a and b were constants chosen to provide the best fit to the experimental data.

(3.21)

(3.20)

(valid only if A is short)

(valid only if Ais long)

from thermal physics. His strategy was to consider a cavity of temperature T filled withblackbody radiation. This may be thought of as a hot oven filled with standing waves ofelectromagnetic radiation. If L is the distance between the oven's walls, then the permittedwavelengths of the radiation are A= 2L, L, 2L/3, 2L/4, 2L/5, ... , extending forever to increasingly shorter wavelengths. 15 According to classical physics, each ofthese wavelengthsshould receive an amount of energy equal to kT, where k = 1.3806503 X 10-23 J K- 1 isBoltzmann's constant, familiar from the ideal gas law PV = NkT. The result of Rayleigh'sderivation gave

a/ASBA(T) = eb/AT _ l'

In order to determine the constants a and b while circumventing the ultraviolet catastrophe, Planck employed a clever mathematical trick. He assumed that a standing electromagnetic wave of wavelength Aand frequency v = c/ Acould not acquire just any arbitraryamount of energy. Instead, the wave could have only specific allowed energy values that

Planck's Function for the Blackbody Radiation Curve

By late 1900 the German physicist Max Planck (1858-1947) had discovered that a modification of Wien's expression could be made to fit the blackbody spectra shown in Fig. 3.8while simultaneously replicating the long-wavelength success of the Rayleigh-Jeans lawand avoiding the ultraviolet catastrophe:

which agrees well with the long-wavelength tail of the blackbody radiation curve. However,a severe problem with Rayleigh's result was recognized immediately; his solution for BA(T)grows without limit as A~ O. The source of the problem is that according to classicalphysics, an infinite number of infinitesimally short wavelengths implied that an unlimitedamount of blackbody radiation energy was contained in the oven, a theoretical result soabsurd it was dubbed the "ultraviolet catastrophe." Equation (3.20) is known today as theRayleigh-Jeans law.16

Wien was also working on developing the correct mathematical expression for the blackbody radiation curve. Guided by the Stefan-Boltzmann law (Eq. 3.16) and classical thermalphysics, Wien was able to develop an empirical law that described the curve at short wavelengths but failed at longer wavelengths:

15This is analogous to standing waves on a string of length L that is held fixed at both ends. The permittedwavelengths are the same as those ofthe standing electromagnetic waves.16James Jeans (1877-1946), a British astronomer, found a numerical error in Rayleigh's original work; the correctedresult now bears the names of both men.

72

Planck's constant has the value h = 6.62606876 X 10-34 J s.

The Planck Function and Astrophysks.

Finally armed with the correct expression for the blackbody spectrum, we can apply Planck'sfunction to astrophysical systems. In spherical coordinates, the amount of radiant energy perunit time having wavelengths between Aand A+ dA emitted by a blackbody of temperatureT and surface area dA into a solid angle dQ == sin ede d¢ is given by

73

(3.22)

(3.23)

(3.24)

B)..(T) dAdA cosedQ = B)..(T) dAdA cose sineded¢;

3.5 The Quantization of Energy

were integral multiples of a minimum wave energy.17 This minimum energy, a quantum ofenergy, is given by h v or he/A, where h is a constant. Thus the energy of an electromagneticwave is nhv or nhe/A, where n (an integer) is the number of quanta in the wave. Given thisassumption of quantized wave energy with a minimum energy proportional to the frequencyof the wave, the entire oven could not contain enough energy to supply even one quantumof energy for the short-wavelength, high-frequency waves. Thus the ultraviolet catastrophewould be avoided. Planck hoped that at the end of his derivation, the constant h could beset to zero; certainly, an artificial constant should not remain in his final result for B).. (T).

Planck's stratagem worked! His formula, now known as the Planck function, agreedwonderfully with experiment, but only if the constant h remained in the equation: 18

17Actually, Planck restricted the possible energies of hypothetical electromagnetic oscillators in the oven wallsthat emit the electromagnetic radiation.181t is left for you to show that the Planck function reduces to the Rayleigh-Jeans law at long wavelengths(Problem 3.10) and to Wien's expression at short wavelengths (Problem 3.11).19Note that dA cos () is the area dA projected onto a plane perpendicular to the direction in which the radiation istraveling. The concept of a solid angle will be fully described in Section 6.1.20The value of the Planck function thus depends on the units of the wavelength interval. The conversion of dA.from meters to nanometers means that the values of BA obtained by evaluating Eq. (3.22) must be divided by 109.

see Fig. 3.9.19 The units of B).. are therefore W m-3 sCI. Unfortunately, these units canbe misleading. You should note that "W m-3

" indicates power (energy per unit time) perunit area per unit wavelength interval, W m-2 m-1, not energy per unit time per unitvolume. To help avoid confusion, the units of the wavelength interval d)' are sometimesexpressed in nanometers rather than meters, so the units of the Planck function becomeW m-2 nm-1 sr-1, as in Fig. 3.8.20

At times it is more convenient to deal with frequency intervals dv than with wavelengthintervals dA. In this case the Planck function has the form

(3.25)

(3.27)

(3.28)

(3.26)

--+--~y

100 crT4

BA(T)d'A =-.o n

Bl' dvdA cose drl = Bl' dvdA cose sine de d¢

z

FIGURE 3.9 Blackbody radiation from an element of surface area dA.


x

LAdA =12Jr l Jr

/

2 r BAdAdA cose sinfJded¢."'=0 8=0 JA

The angular integration yields a factor of n, and the integral over the area of the sphereproduces a factor of 4n R2

. The result is

Thus, in spherical coordinates,

LAdA = 4n 2 R2 BAdA

8n2R 2hc2IA5

= ehC/AkT _ 1 dA.

LA dA is known as the monochromatic luminosity. Comparing the Stefan-Boltzmannequation (3.17) with the result of integrating Eq. (3.26) over all wavelengths shows that

is the amount of energy per unit time of blackbody radiation having frequency between vand v + d v emitted by a blackbody of temperature T and surface area d A into a solid angledrl = sin e de d¢.

The Planck function can be used to make the connection between the observed propertiesof a star (radiant flux, apparent magnitude) and its intrinsic properties (radius, temperature).Consider a model star consisting of a spherical blackbody of radius R and temperature T.Assuming that each small patch of surface area dA emits blackbody radiation isotropically(equally in all directions) over the outward hemisphere, the energy per second havingwavelengths between Aand A+ dA emitted by the star is

74

3.6 .THE COLOR INDEX

• V, the star's visual magnitude, is measured through a filter centered at 550 nm withan effective bandwidth of 89 nm.

• V, the star's ultraviolet magnitude, is measured through a filter centered at 365 nmwith an effective bandwidth of 68 nm.

75

(3.29)L;.. 2nhc2jJ..5 (R)2

F;..dJ.. = --2 dJ.. = h JUT - dJ..,4nr e C - 1 r.

3.6 The Color Index

• B, the star's blue magnitude, is measured through a filter centered at 440 nm with aneffective bandwidth of 98 nm.

The apparent and'absolute magnitudes discussed in Section 3.2, measured over all wavelengths of light emitted by a star, are known as bolometric magnitudes and are denoted bymbol and M bo], respectively.21 In practice, however, detectors measure the radiant flux of astar only within a certain wavelength region defined by the sensitivity of the detector.

In Problem 3.14, you will use Eq. (3.27) to express the Stefan-Boltzmann constant, er, interms of the fundamental constants c, h, and k. The monochromatic luminosity is relatedto the monochromatic flux, F;.. dJ.., by the inverse square law for light, Eq. (3.2):

where r is the distance to the model star. Thus F;.. dJ.. is the number of joules of starlightenergy with wavelengths between J.. and J.. + dJ.. that arrive per second at one square meterof a detector aimed at the model star, assuming that no light has been absorbed or scatteredduring its journey from the star to the detector. Of course, Earth's atmosphere absorbs somestarlight, but measurements of fluxes and apparent magnitudes can be corrected to accountfor this absorption; see Section 9.2. The values of these quantities usually quoted for stars arein fact corrected values and would be the results of measurements above Earth's absorbingatmosphere.

UBV Wavelength Filters

The color of a star may be precisely determined by using filters that transmit the star's lightonly within certain narrow wavelength bands. In the standard VB V system, a star's apparentmagnitude is measured through three filters and is designated by three capital letters:

Color Indices and the Bolometric Correction

Using Eq. (3.6), a star's absolute color magnitudes M u , M B , and M v may be determined ifits distance d is known.22 A star's V - B color index is the difference between its ultraviolet

21A bolometer is an instrument that measures the increase in temperature caused by the radiant flux it receives atall wavelengths.22Note that although apparent magnitude is not denoted by a subscripted "m" in the UBV system, the absolutemagnitude is denoted by a subscripted "M."

and

and

U - B = M u -MB

(3.30)

(3.31)U = -2.5 log10 (100

F).,Su dJ..) + CU,

U - B = -1.47 - (-1.43) = -0.04

mba! = V + BC = -1.44 + (-0.09) = -1.53.

0.0029 mKAmax = 9970 K = 291 nm,

which is in the ultraviolet portion of the electromagnetic spectrum. The bolometric correction for Sirius is BC = -0.09, so its apparent bolometric magnitude is

Example 3.6.1. Sirius, the brightest-appearing star in the sky, has U, B, and V apparentmagnitudes of U = -1.47, B = -1.43, and V = -1.44. Thus for Sirius,

B - V = -1.43 - (-1.44) = 0.01.


B - V = MB - Mv .

and blue magnitudes, and a star's B - V color index is the difference between its blue andvisual magnitudes:

Stellar magnitudes decrease with increasing brightness; consequently, a star with a smallerB - V color index is bluer than a star with a larger value of B - V. Because a color indexis the difference between two magnitudes, Eq. (3.6) shows that it is independent of the star'sdistance. The difference between a star's bolometric magnitude and its visual magnitude isits bolometric correction BC:

I BC = mba! - V = Mba! - Mv . I

Sirius is brightest at ultraviolet wavelengths, as expected for a star with an effective temperature of Te = 9970 K. For this surface temperature,

where Cu is a constant. Similar expressions are used for a star's apparent magnitude withinother wavelength bands. The constants C in the equations for U, B, and V differ for each

The relation between apparent magnitude and radiant flux, Eq. (3.4), can be used toderive expressions for the ultraviolet, blue, and visual magnitudes measured (above Earth'satmosphere) for a star. A sensitivity function SeA) is used to describe the fraction of thestar's flux that is detected at wavelength J... S depends on the reflectivity of the telescopemirrors, the bandwidth of the U, B, and V filters, and the response of the photometer. Thus,for example, a star's ultraviolet magnitude U is given by

76

continued

BC = mbol- V

77

(3.32)

(3.33)( f FASUdA.)U - B = -2.5 log10 f + CU - B ,

FASB dJ...

3.6 The Color Index

where CU-B == Cu - CB. A similar relation holds for B - V. From Eq. (3.29), note thatalthough the apparent magnitudes depend on the radius R of the model star and its distancer, the color indices do not, because the factor of (Rjr)2 cancels in Eq. (3.33). Thus thecolor index is a measure solely of the temperature of a model blackbody star.

of these wavelength regions and are chosen so that the star Vega (a Lyrae) has a magnitudeof zero as seen through each filter.23 This is a completely arbitrary choice and does notimply that Vega would appear equally bright when viewed through the U, B, and V filters.However, the resulting values for the visual magnitudes of stars are about the same as thoserecorded by Hipparchus two thousand years ago.24

A different method is used to determine the constant Cbo1 in the expression for thebolometric magnitude, measured over all wavelengths of light emitted by a star. For apeifect bolometer, capable of detecting 100 percent of the light arriving from a star, we setS(A.) == 1:

be negative for all stars (since a star's radiant flux over all wavelengths is greater thanits flux in any specified wavelength band) while still being as close to zero as possible.After a value of Cool was agreed upon, it was discovered that some supergiant stars havepositive bolomeuic corrections. Nevertheless, astronomers have chosen to continue usingthis unphysical method of measuring magnitudes.25 It is left as an exercise for you toevaluate the constant C bo] by using the value of mbol assigned to the Sun: mSun = -26.83.

The color indices U - Band B - V are immediately seen to be

The value for Cbol originated in the wish of astronomers that the value of the bolometriccorrection

23Actually, the average magnitude of several stars is used for this calibration.24See Chapter 1 of Bohm-Vitense (1989b) for a further discussion of the vagaries of the magnitude system usedby astronomers.25Some authors, such as Bohm-Vitense (1989a, 1989b), prefer to define the bolometric correction as BC =V - mbo], so their values of BC are usually positive.

Example 3.6.2. A certain hot star has a surface temperature of 42,000 K and color indicesU - B = -1.19 and B - V = -0.33. The large negative value of U - B indicates thatthis star appears brightest at ultraviolet wavelengths, as can be confirmed with Wien'sdisplacement law, Eq. (3.19). The spectrum of a 42,000-K blackbody peaks at

0.0029 m K 69A. - - nmmax - 42,000 K - ,

and

CU - B = -0.87,

700

vB

400

u1.0

"""' 0.8CS'0..0

';:l0.6OJ..

<E

:f 0.4'"..0)

fJ)

0.2

0.0300


500 600

Wavelength (nm)

FIGURE 3.10 Sensitivity functions SeA) for U, B, and V filters. (Data from Johnson, Ap. J., 141,923,1965.)

B - V = -2.5 log10 (B440 /),AB) + CB- VB550 /),Av

-0.33 = -0.98 + CB- V

CB- V = 0.65.

It is left as an exercise for you to use these values of CU- B and CB-v to estimate thecolor indices for a model blackbody Sun with a surface temperature of 5777 K. Although

in the ultraviolet region of the electromagnetic spectrum. This wavelength is much shorterthan the wavelengths transmitted by the U, B, and V filters (see Fig. 3.10), so we will bedealing with the smoothly declining long-wavelength "tail" of the Planck function BA(T).

We can use the values of the color indices to estimate the constant CU - B in Eq. (3.33),and CB-v in a similar equation for the color index B - V. In this estimate, we will use astep function to represent the sensitivity function: SeA) = I inside the filter's bandwidth,and SeA) = 0 otherwise. The integrals in Eq. (3.33) may then be approximated by the valueof the Planck function BA at the center of the filter bandwidth, multiplied by that bandwidth.Thus, for the wavelengths and bandwidths /),A listed on page 75,

(B365 /),AU )

U - B = -2.5 log10 + CU - BB440 /),AB

-1.19 = -0.32 + CU - B

78

FIGURE 3.11 Color-color diagram for main-sequence stars. The dashed line is for a blackbody.(The data are taken from Appendix G.)

26As will be discussed in Section 10.6, main-sequence stars are powered by the nuclear fusion of hydrogen nucleiin their centers. Approximately 80% to 90% of all stars are main-sequence stars. The letter labels in Fig. 3.11 arespectral types; see Section 8.1.

79

2.01.5

MO

KO

GO

0.5 1.0

B-V0.0

"""""""""""""""""" -6>~

" <i'(><t"" (10

",,~

"""""""""""""""""

-1.0BO

-0.5

0.0I:Cl

I::::,

0.5

1.0

1.5

-0.5

3.6 The Color Index

The Color-Color Diagram

Figure 3.11 is a color-color diagram showing the relation between the V - B and B - Vcolor indices for main-sequence stars.26 Astronomers face the difficult task of connectinga star's position on a color-color diagram with the physical properties of the star itself. Ifstars actually behaved as blackbodies, the color-color diagram would be the straight dashedline shown in Fig. 3.11. However, stars are not true blackbodies. As will be discussed indetail in Chapter 9, some light is absorbed as it travels through a star's atmosphere, and theamount of light absorbed depends on both the wavelength of the light and the temperatureof the star. Other factors also playa role, causing the color indices of main-sequence andsupergiant stars of the same temperature to be slightly different. The color-color diagramin Fig. 3.11 shows that the agreement between actual stars and model blackbody stars isbest for very hot stars.

the resulting value of B - V = +0.57 is in fair agreement with the measured value ofB - V = +0.650 for the Sun, the estimate of V - B = -0.22 is quite different from themeasured value of V - B = +0.195. The reason for this large discrepancy at ultravioletwavelengths will be discussed in Example 9.2.4.


SUGGESTED READING

General

Ferris, Timothy, Coming ofAge in the Milky Way, William Morrow, New York, 1988.

Griffin, Roger, "The Radial-Velocity Revolution," Sky and Telescope, September 1989.

Heamshaw, John B., "Origins ofthe Stellar Magnitude Scale," Sky and Telescope, November1992.

Herrmann, Dieter B., The History ofAstronomy from Hershel to Hertzsprung, CambridgeUniversity Press, Cambridge, 1984.

Perryman, Michael, "Hipparcos: The Stars in Three Dimensions," Sky and Telescope, June1999.

Segre, Emilio, From Falling Bodies to Radio Waves, W. H. Freeman and Company, NewYork, 1984.

Technical

Arp, Halton, "u ~ Band B - V Colors of Black Bodies," The Astrophysical Journal, 133,874, 1961.

Bohm-Vitense, Erika, Introduction to Stellar Astrophysics, Volume I: Basic Stellar Observations and Data, Cambridge University Press, Cambridge, 1989a.

Bohm-Vitense,Erika, Introduction to Stellar Astrophysics, Volume 2: Stellar Atmospheres,Camblidge University Press, Cambridge, 1989b.

Cox, Arthur N. (ed.), Allen's Astrophysical Quantities, Fourth Edition, Springer-Verlag,New York, 2000.

Harwit, Martin, Astrophysical Concepts, Third Edition, Springer-Verlag, New York, 1998.

Hipparcos Space Astrometry Mission, European Space Agency,http://astro.estec.esa.nl/Hipparcos/.

Lang, Kenneth R., Astrophysical Formulae, Third Edition, Springer-Verlag, New York,1999.

Van HeIden, Albert, Measuring the Universe, The University of Chicago Press, Chicago,1985.

PROBLEMS

3.1 In 1672, an international effort was made to measure the parallax angle of Mars at the time ofopposition, when it was closest to Earth; see Fig. 1.6.

(a) Consider two observers who are separated by a baseline equal to Earth's diameter. If thedifference in their measurements of Mars's angular position is 33.6", what is the distancebetween Earth and Mars at the time of opposition? Express your answer both in units of mand in AU.

(b) If the distance to Mars is to be measured to within 10%, how closely must the clocks used bythe two observers be synchronized? Hint: Ignore the rotation of Earth. The average orbitalvelocities of Earth and Mars are 29.79 km s-J and 24.13 km S-I, respectively.

m = MSun - 2.5log10 (~) .FlO. 0

3.7 A 1.2 x 104 kg spacecraft is launched from Earth and is to be accelerated radially away from theSun using a circular solar sail. The initial acceleration of the spacecraft is to be 1g. Assuming aflat sail, determine the radius of the sail if it is

(a) black, so it absorbs the Sun's light.

(b) shiny, so it reflects the Sun's light.

Hint: The spacecraft, like Earth, is orbiting the Sun. Should you include the Sun's gravity inyour calculation?

3.2 At what distance from a 100-W light bulb is the radiant flux equal to the solar irradiance?

3.3 The parallax angle for Sirius is 0.379".

(a) Find the distance to Sirius in units of (i) parsecs; (ii) light-years; (iii) AU; (iv) m.

(b) Determine the distance modulus for Sirius.

3.4 Using the information in Example 3.6.1 and Problem 3.3, determine the absolute bolometricmagnitude of Sirius and compare it with that of the Sun. What is the ratio of Sirius's luminosityto that of the Sun?

81

3.5 (a) The Hipparcos Space Astrometry Mission was able to measure parallax angles down tonearly o.oor/.To get a sense of that level of resolution, how far from a dime would you needto be to observe it subtending an angle ofO.OOl"? (The diameter of a dime is approximately1.9 em.)

(b) Assume that grass grows at the rate of 5 em per week.

i. How much does grass grow in one second?

ii. How far from the grass would you need to be to see it grow at an angular rate of0.000004" (4 microarcseconds) per second? Four microarcseconds is the estimatedangular resolution of SIM, NASA's planned astrometric mission; see page 59.

3.6 Derive the relation

3.8 The average person has 1.4 m2 of skin at a skin temperature ofrougWy 306 K (92°F). Considerthe average person to be an ideal radiator standing in a room at a temperature of 293 K (68°P).

(a) Calculate the energy per second radiated by the average person in the form of blackbodyradiation. Express your answer in watts.

(b) Determine the peak wavelength Amax of the blackbody radiation emitted by the averageperson. In what region of the electromagnetic spectrum is this wavelength found?

(c) A blackbody also absorbs energy from its environment, in this case from the 293-K room.The equation describing the absorption is the same as the equation describing the emissionof blackbody radiation, Eq. (3.16). Calculate the energy per second absorbed by the averageperson, expressed in watts.

(d) Calculate the net energy per second lost by the average person via blackbody radiation.

3.9 Consider a model of the star Dschubba (<5 Sea), the center star in the head of the constellationScorpius. Assume that Dschubba is a spherical blackbody with a surface temperature of28,000 Kand a radius of 5.16 x 109 m. Let this model star be located at a distance of 123 pc from Earth.Determine the following for the star:

(a) Luminosity.

Problems


(b) Absolute bolometric magnitude.

(c) Apparent bolometric magnitude.

(d) Distance modulus.

(e) Radiant flux at the star's surface.

(0 Radiant flux at Earth's surface (compare this with the solar irradiance).

(g) Peak wavelength Amax •

3.10 (a) Show that the Rayleigh-Jeans law (Eq. 3.20) is an approximation of the Planck functionB). in the limit of A» hej kT. (The first-order expansion eX ~ 1 + x for x « 1 will beuseful.) Notice that Planck's constant is not present in your answer. The Rayleigh-Jeanslaw is a classical result, so the "ultraviolet catastrophe" at short wavelengths, produced bythe A4 in the denominator, cannot be avoided.

(b) Plotthe Planck function B). and the Rayleigh-Jeans Jaw for the Sun (To = 5777 K) on thesame graph. At roughly what wavelength is the Rayleigh-Jeans value twice as large as thePlanck function?

3.11 Show that Wien's expression for blackbody radiation (Eq. 3.21) follows directly from Planck'sfunction at short wavelengths.

3.12 Derive Wien's displacement law, Eq. (3.15), by setting d B).jdA = O. Hint: You will encounteran equation that must be solved numerically, not algebraically.

3.13 (a) Use Eq. (3.24) to find an expression for the frequency Vmax at which the Planck function Bv

attains its maximum value. (Warning: Vmax i= cjAmax .)

(b) What is the value of Vmax for the Sun?

(c) Find the wavelength of a light wave having frequency Vmax ' In what region of the electromagnetic spectrum is this wavelength found?

3.14 (a) Integrate Eq. (3.27) over all wavelengths to obtain an expression for the total luminosity ofa blackbody model star. Hint:

(b) Compare your result with the Stefan-Boltzmann equation (3.17), and show that the StefanBoltzmann constant a is given by

(c) Calculate the value of a from this expression, and compare with the value listed in AppendixA.

3.15 Use the data in Appendix G to answer the following questions.

(a) Calculate the absolute and apparent visual magnitudes, M v and V, forthe Sun.

(b) Determine the magnitudes M R , B, M u, and U for the Sun.

(c) Locate the Sun and Sirius on the color-color diagram in Fig. 3.11. Refer to Example 3.6.1for the data on Sirius.

3.16 Use the filter bandwidths for the U BV system on page 75 and the effective temperature of9600 K for Vega to determine through which filter Vega would appear brightest to a photometer

[i.e., ignore the constant C in Eq. (3.31)]. Assume that SeA) = 1 inside the filter bandwidth andthat SeA) = 0 outside the filter bandwidth.

3.17 Evaluate the constant Cbo1 in Eq. (3.32) by using mSun = -26.83.

3.18 Use the values of the constants CU - B and CB- V found in Example 3.6.2 to estimate the colorindices U - Band B - V for the Sun.

3.19 Shaula (A Scorpii) is a bright (V = 1.62) blue-white subgiant star located at the tip of thescorpion's tail. Its surface temperature is about 22,000 K.

(a) Use the values of the constants CU - B and CB- V found in Example 3.6.2 to estimate the colorindices U - Band B - V for Shaula. Compare your answers with the measured values ofU - B = -0.90 and B - V = -0.23.

(b) The Hipparcos Space Astrometry Mission measured the parallax angle for Shaula to be0.00464/1. Determine the absolute visual magnitude of the star.

(Shaula is a pulsating star, belonging to the class of Beta Cephei variables; see Section 14.2. Asits magnitude varies between V = 1.59 and V = 1.65 with a period of 5 hours 8 minutes, itscolor indices also change slightly.)

Problems 83

Astronomical Constants

Note: Uncertainties in the last digits are indicated in parentheses. For instance,

the solar radius, 1 Ro , has an uncertainty of ±O.00026 x 108 m.

5.9736 X 1024 kg6.378136 X 106 m

4.74-26.83-25.91-26.10-26.75-0.08

1.4959787066 X lOll m9.460730472 x 1015 m206264.806 AU3.0856776 x 1016 m3.26156381y (Julian)

23b56m04.0905309s

86400 s3.15581450 x 107 s365.256308 d3.155692519 x 107 s365.2421897 d3.1557600 x 107 s365.25 d3.1556952 x 107 s365.2425 d

I Mo 1.9891 X 1030 kgS 1.365(2) X 103 W m-2

I Lo 3.839(5) X 1026 WI Ro 6.95508(26) X 108 mTe,o - Lo/(4rraR~)1/4

5777(2) K

lAUI ly1 pc

Mbol

mbol

UBV

Be

Solar absolute bolometric magnitdueSolar apparent bolometric magnitudeSolar apparent ultraviolet magnitudeSolar apparent blue magnitudeSolar apparent visual magnitudeSolar bolometric correction

Earth massEarth radius (equatorial)

Solar massSolar irradianceSolar luminositySolar radiusSolar effective temperature

[C 7\7 ~'l 1 .q;;L')'0 j:rJSW

Astronomical unitLight (Julian) yearParsec

Sidereal daySolar daySidereal year

Tropical year

Gregorian year

Julian year

Astronomical and Physical ConstantsAAPPENDIX

Physical Constants

Note: Uncertainties in the last digits are indicated in parentheses. For instance, the universal

gravitational cOilstant, G, has an uncertainty of ±O.OIO x 1O-11 N m2 kg-2

k

leVh

a

hc

a

1 u

e

G 6.673(10) x 10-11 N m2 kg-2

c _ 2.99792458 x 108 m s-IfLo - 4n x 10-7 N A-2

EO - l/fLoc28.854187817 ... X 10- 12 F m- I

1.602176462(63) x 10- 19 C1.602176462(63) x 10- 19 J6.62606876(52) x 10-34 J s4.13566727(16) x 10- 15 eV s

_ h/2rr

1.054571596(82) x 10-34 J s6.58211889(26) x 10- 16 eV s1.23984186(16) x 103 eV nm1240 eV nm1.3806503(24) x 10-23 J K- 1

8.6173423(153) x 10-5 eV K- 1

2n 5k4 /(15c2h3)

5.670400(40) x 10-8 W m-2 K-4

4a/c7.565767(54) x 10-16 J m-3 K-4

1.66053873(13) x 10-27 kg931.494013(37) MeV/ c2

9.10938188(72) x 10-31 kg5.485799110(12) x 10-4 u1.6-7262r58(13) x 10-27 kg1.00727646688(13) u1.67492716(13) x 10-27 kg1.00866491578(55) u1.673532499(13) x 10-27 kg1.00782503214(35) u

6.02214199(47) x 1023 mol- 1

8.314472(15) J mol-I K- 1

ao,oo - 4rrEon,2 / mee2

5.291772083(19) x 10-11 m

aO,H - (me/fL)ao,oo5.294654075(20) x 10-11 m

_ mee4/64n3E5n3c1.0973731568549(83) x 107 m- I

- (fL/me)Roo1.09677583(13) x 107 m- I

Stefan-Boltzmann constant

Electric chargeElectron voltPlanck's constant

Radiation constant

Boltzmann's constant

Gravitational constantSpeed of light (exact)Permeability of free spacePennittivity of free space

Planck's constant x speed of light

Electron mass

Neutron mass

Hydrogen mass

Proton mass

Rydberg constant

Atomic mass unit

Avogadro's number

Gas constantBohr radius

APPENDIX

B Unit Conversions

SI to egs Unit ConversionsQuantity SI Unit egs Unit Conversion Factor"

Distance meter (m) centimeter (cm) 10-2

Mass kilogram (kg) gram (g) 10-3

Time second (s) second (s) ICurrentb ampere (A) esu S-l 3.335640952 x 10-10

Chargee coulomb (C; A s) esu 3.335640952 x 10-10

Velocity m S-l cms-1 10-2

Acceleration m s-2 cm s-2 10-2

Linear momentum kg m s-1 g cm S-1 10-5

Angular momentum kg m2 s-1 g cm2 s-1 10-7

Force newton (N; kg m s-2) dyne (g cm S-2) 10-5

Energy (work) joule (1; N m) erg (dyne cm) 10-7

Power (luminosity) watt (W; J s-I) erg s-1 10-7

Pressure pascal (Pa; N m-2) dynecm-2 10-1

Mass density kgm-3 gcm-3 103

Charge density Cm-3 esu cm-3 3.335640952 x 10-4

Current density Am-2 esu s-1 cm-2 3.335640952 x 10-6

Electric potential volt (V; J C-1) statvoh (erg esu-1 ) 2.997924580 x 102

Electric field Vm- 1 statvoh cm- I 2.997924580 x 104

Magnetic field tesla (T; N A-I m- I) gauss (G; dyne esu-1) 10-4

Magnetic flux weber (Wb; T m2) Gcm2 lO-s

"Multiply the SI unit by the conversion factor to obtain the equivalent cgs unit; e.g., 10-' m = I em.

bThe ampere is the fundamental electromagnetic unit in the SI system.

cThe esu (electrostatic unit) is the fundamental electromagnetic unit in the cgs system.

SI to Miscellaneous Unit ConversionsQuantity SI Unit Misc. Unit Conversion Factor (SI to Misc.f

Distance meter (m) angstrom (A) 10-10

Distance nanometer (nm) angstrom (A) 10-1

Spectral flux density W m-2 Hz-I jansky (ly) 10-26

dMultiply the 51 unit by the conversion factor to obtain the equivalent micellaneous unit.

SI-egs Electromagnetic Equation ConversionsSelected Equations ofElectromagnetism SI Version cgs Version

(11.10)

(3.12)

(5.9)

(11.2)

Text Equationc

(S) = 81f £oBo

F = qlq2r2

F=q(E+~XB)

B2

p=81f

I(S) = --£oBo

2/-LO

F = _1_qlq241fEo r2

F = q(E + v x B)

Poynting Vector

Magnetic pressure

Lorentz Equation

Coulomb's Law

APPENDIX

c Solar System Data

Planetary Physical DataEquatorial Average Sidereal

Massa Radiusb Density Rotation Oblateness BondPlanet (MElJ) (REJ)) (kg rn- 3) Period (d) (Re - Rp)/Re Albedo

Mercury 0.05528 0.3825 5427 58.6462 0.00000 0.Jl9Venus 0.81500 0.9488 5243 243.018 0.00000 0.750Earth 1.00000 1.0000 5515 0.997271 0.0033396 0.306Mars 0.10745 0.5326 3933 1.02596 0.006476 0.250Jupiter 317.83 11.209 1326 0.4135 0.064874 0.343Saturn 95.159 9.4492 687 0.4438 0.097962 0.342Uranus 14.536 4.0073 1270 0.7183 0.022927 0.300Neptune 17.147 3.8826 [638 0.6713 0.017081 0.290Pluto 0.0021 0.178 2110 6.3872 0.0000 0.4-0.62003 UB313 0.002? 0.188 2100? 0.6?

Planetary Orbital and Satellite DataSidereal Orbital Equatorial Number

Semimajor Orbital Orbital Inclination Inclination NaturalPlanet Axis (AU) Eccentricity Period (yr) to Ecliptic CO) to Orbit (0) Satellites

Mercury 0.3871 0.2056 0.2408 7.00 om 0Venus 0.7233 0.0067 0.6152 3.39 177.36 0Earth 1.0000 0.0167 1.0000 0.000 23.45 1Mars 1.5236 0.0935 1.8808 1.850 25.19 2Jupiter 5.2044 0.0489 11.8618 1.304 3.13 63Saturn 9.5826 0.0565 29.4567 2.485 26.73 47Uranus 19.2012 0.0457 84.0107 0.772 97.77 27Neptune 30.0476 0.0113 164.79 1.769 28.32 13Pluto 39.4817 0.2488 247.68 17.16 122.53 32003 UB313 67.89 0.4378 559 43.99 1

a Mffi = 5.9736 X 1024 kg

b REf) =6.378136 X 106 m

A-1

A-2 Appendix C Solar System Data

Data of Selected Major SatellitesOrbital Semimajor

Parent Mass Radius Density Period AxisSatellite Planet (1022 kg) (103 km) (kg m-3) (d) 003 km)

Moon Earth 7.349 1.7371 3350 27.322 384.410 Jupiter 8.932 1.8216 3530 1.769 421.6Europa Jupiter 4.800 1.5608 3010 3.551 670.9Ganymede Jupiter 14.819 2.6312 1940 7.155 1070.4Callisto Jupiter 10.759 2.4103 1830 16.689 1882.7Titan Saturn 13.455 2.575 1881 15.945 1221.8Triton Neptune 2.14 1.3534 2050 5.877 354.8

chapter 3

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