Galaxies in the Universe: errata for the first edition

`As the camera-ready book goes to press, it is completely free of any typographical errors, errors of physics, or errors of judgement. Any errors present in the final product must have crept in during the production process, and are wholly the fault of the publisher.'
E.W. Kolb and M.S. Turner, preface to the 1990 edition of `The Early Universe', Addison-Wesley

`The questions used ... have been painstakingly researched. However, the answers have not.'
Michael Feldman's 'Whad'Ya Know?' (Wisconsin Public Radio)

Please let us know by e-mail, at sparke_at_astro.wisc.edu or jsg_at_astro.wisc.edu, if you find errors in `Galaxies in the Universe' that are not already listed below. Thanks for your help!

Chapter 1: Introduction

Equation 1.13, on page 20: this equation holds when the units of flux are erg s^-1 cm^-2 Ang^-1
Page 46: the time of 887+-2 sec is the e-folding time or mean lifetime for neutron decay, and not the half-life.
In Problem 1.14, the first sentence should be replaced by `Deuterium can become abundant only when k<sub>B</sub> T < 70keV. Use Equation 1.32 to show that this temperature is reached at t=365 sec, by which time about 35% of the free neutrons have decayed.' The last sentence should read `If the mean life had been 750s, show that the predicted fraction of helium would be about 2% lower, while if it had been 1100s, we would expect to find close to 2% more helium.'

Two new homework problems:

The star Betelguese of Problem 1.3 has apparent magnitude m_V = 0 (after correcting for dust dimming: see Section 1.2), and V-K=5. Taking the distance d = 140pc, find its absolute magnitude in V and in K. Show that Betelguese has L_V = 1.7 x 10⁴ L_V,Sun, while at K its luminosity is much larger compared to the Sun: L_K = 4.1 x 10⁵ L_K,Sun.
(This star is variable: m_V changes by roughly a magnitude.)

A star cluster contains 200 main-sequence F5 stars and 20 K0III giant stars. Use Tables 1.2 and 1.3 to show that its absolute V-magnitude M_V= -3.25 and its color B-V=0.68.
(These values are similar to those of the 4-Gyr-old cluster M67: see Table 2.2)

Expanded version of problem 1.10:
... In this galaxy, show that 1" on the sky corresponds to 5pc. If the surface brightness I_B= 27, how much B-band light does one square arcsecond of the galaxy emit, compared to a star like the Sun? Show that this is equivalent to 1 L_Sun pc^-2 in the B band, but that a galaxy with I_R= 27 emits only 0.4 L_Sun pc^-2 in the R band.

Chapter 2: Mapping our Milky Way

In Figure 2.5, the area under the curve really is the number of stars in a given mass range, since the horizontal axis is logarithmic and not linear.
Problem 2.5: you can get the idea by taking the mass-luminosity relation to follow Equation 1.6 with &alpha=3.5 or so.
Problem 2.9: in part a, M_V = M_{V, sun} + 0.2*(N₁ - 3.5) works slightly better. In part c, the metallicity is Z/ Z_Sun = (N₂ + 0.5)/6

Problem 2.10: an easier and clearer version would be:
The range in apparent magnitude for Fig. 2.15 was chosen to separate stars of the thin disk cleanly from those in the halo. To see why this works, use Fig. 2.2 to represent the stars of the local disk, and assume that the color-magnitude diagram for halo stars is similar to that of the metal-poor globular cluster M30, in Fig. 2.13.
(a) What is the absolute magnitude M_V of a disk star at B-V=0.4? How far away must it be to have m_V=20? In M30, the bluest stars still on the main sequence have B-V=0.4, or B-R=0.65; use Fig 2.13 to find M_R, and hence M_V, for these stars. Show that those with apparent magnitude m_V=20 must be at distances around 20kpc.
(b) What absolute magnitudes M_V could a disk star have, if it has B-V=1.5? How far away would that star be at m_V=20. In M30, a star with B-V=1.5 corresponds to B-R= 2: what are the possible values for M_V? How distant must these stars be if m_V=20?
(c) Explain why the reddest stars in Fig 2.15 are likely to belong to the disk, while the bluest stars belong to the halo.

In Equation 2.13, both equalities are approximate only.
In Equation 2.15, the last term is V(d/R), not V₀(d/R).

On page 92, following Equation 2.25, the condition for the image &theta_- to be brighter than the source is that &beta² < (3 - 2 sqrt{2})&theta_E²/sqrt{2} or &beta < 0.348 &theta_E.

Chapter 3: The orbits of the stars

In Equation 3.5, there should be a minus in front of the integral (compare Equation 3.2).
In Problem 3.9, the expression for the radial force should be F_r = - 4 &pi G &rho r /3,
not F_r = - 4 &pi G &rho r³ /3.
In Problem 3.11, the distance increases by the factor (1-f)/(1-2f), not 1/[(1-f)(1-2f)].

On page 106, Equation 3.43 is too good to be true. The potential energy PE = - G M²/2 &eta r_c, where &eta is of order unity:
e.g. &eta = 2.6 for the Plummer sphere (Eq 3.35), and 0.96 for a homogeneous sphere (Eq 3.34).
Equation 3.43 should read M = 6 & eta &sigma_r² r_c/G.
We can't prove Equation 3.44, but must fudge it by writing L_tot = 4 &pi r_c² I(0)/3 (reasonable to 'twiddles' accuracy), or quote it from Richstone & Tremaine, AJ 92, 72; 1986; &sigma_r refers to the measured dispersion at the center of the system.

Problem 3.12 has the wrong answer: it should be 2 x 10⁶M_o, not 4 x 10⁶M_o.
On page 112, in the first paragraph we have a lower limit, not an upper limit, on the relaxation time: t_trelax is larger than 50Myr.
The lifetime of a 5 M_o star is comparable to the relaxation time, not the crossing time.
The answer to Problem 3.14 is about 0.8 Gyr, while Table 3.1 gives 5 Gyr for the central relaxation time. We are still trying to chase down the definition of the timescale used in the Table.
The middle term of Equation 3.55 is missing a factor of 1/6.
In Equation 3.74, the sign before R₀ should be +, not -.
In Problem 3.18, the first sentence should end with `B = - &Omega.', and not `B = &Omega.'
In Problem 3.19, the answers for X and R_g are 20% too small; they should be X=0.35, R_g=8.2.
In the first term of Equation 3.82, &Delta x should be &Delta v.

Equation 3.82 gives the increase in the number of stars in the center box, from stars moving with speed v.
Equation 3.83 gives the total increase in stars for that box; the square-bracket term in that equation, and the term in dv/dt of the following equation, are both +, not -. Equation 3.84 is correct.

Chapter 4: Our backyard: the Local Group

p162, Problem 4.8: about 30% of the stars have Z < Z_o/4.
Problem 4.9: clean gas flows into the system proportionally to the rate at which new stars form, not to the mass of stars already present.

Chapter 5: Spiral and S0 galaxies

In Problem 5.3, luminosity L_V = 5 x 10¹⁰ L_Sun, not 3.3 x 10¹⁰ L_Sun.
Flowing from that error: in Problem 5.7, M(HI)/L_B = 0.2, not 0.3; about the same as for M31.
Problem 5.6 needs an explanatory hint: 21cm/73m corresponds to about 10 arcmin, but structures larger than about half this size are significantly `resolved out' of interferometric maps.
Problem 5.8 should read `... show that V²_max = V²(sqrt{2} a_P) = ...' ;
the contours of V_r - V_sys should be at intervals of 0.2 V_max sin i, not 0.2 V_max.
In Problem 5.9, M/L is about 8 or 10, rather than 15.
In Problem 5.12, replace `f(R,t) = ln R + k' by `f(R,t) tan i = ln R + k', and `m spiral arms' by `m /(2 &pi tan i) spiral arms'.
On page 207, Figure 5.25: the H &alpha contours are shown in the lower left (not right) panel, and HI contours in the lower right.
In Problem 5.14, M = 6.2 x 10^5 M_sun and thus M/L = 0.25 M_sun/L_sun, not 0.4
In Problem 5.15, replace `100 years', by `100 Myr'.
On page 224, the reference in the first sentence should be to Section 3.2, not Section 3.3.
In Problem 5.17, replace `r <sup>2</sup> V<sub>H</sub> / G M' by `r ² V_H / 2 G M' in the final displayed equation. For the LMC, t_sink is about 3 Gyr.

Chapter 6: Elliptical galaxies

p241, just below Equation 6.9: the inequality should read `q<sub>prol</sub> > or = A/B', not `< or = A/B'.
Problem 6.6: instead of `Q < 0.95 and -21 < M<sub>B</sub> < -20', this should read `Q > 0.95 and...'
Equation 6.14 should read `x = a cos t, y = b sin t', and not `x = a cos 2t, y = b sin 2t'
In Equations 6.20 and 6.21, the closing bracket on the left side should come before the terms z_a and z_b, respectively.
p250: in the paragraph below Equation 6.26, for `PE<sub>xx</sub> << PE<sub>zz</sub>', read `PE_xx is much smaller in magnitude than PE_zz' -- since both these quantities are negative.
Those wanting to chase the factor of &pi/4 in Equation 6.29 should look at Binney 1978 MNRAS 183, 501.
In Figure 6.23, the cooling times are too long by a factor of &pi² or about 10.
In Problem 6.15, the galaxy mass M = 3 x 10¹² M_o, not 1.5 x 10¹² M_o.
In Problem 6.20, the Plummer-sphere model yields M = 10¹⁵ h^-1 M_o, not 10¹⁵ h M_o, and M/L = 200h, not 300h. It would have been better to use &sigma=1200km/s.

Chapter 7: Large-scale distribution of galaxies

Problem 7.1 should refer to Section 4.5, not 4.4; the Local Group's density works out to be 2.6 h^-2 of critical.
In Figure 7.6, the red galaxies are on the left and the blue ones on the right.
p294: the time lapse &Delta t_e = &lambda_e / c, not c &lambda_e; similarly, &Delta t = &lambda_obs / c.
p296: the first sentence should refer to Equation 7.15, not Equation 7.16.
p297, just below Equation 7.21: \dot{a} is proportional to a^-1/2, not to a^1/2.
At early times, both terms on the right of Equation 7.21 are large, but the first term dominates.
P302: in Equation 7.28, &delta approaches a constant at late times, but there is no equality because a decaying term is present.
Problem 7.16 doesn't work, and will be dropped or replaced.

In Equation 7.41, the kinetic energy is a factor of 3 too low. The virial theorem is satisfied when |PE| = 2xKE, which happens when
2r = sqrt[ (15/&pi²) (&pi c_c² / G &rho) ], or 2r = &lambda_J to within sqrt(15/&pi²) or about 1.23.
p309, first sentence: T_rec is proportional to R^-1, not to R itself.

Chapter 8: Active galactic nuclei and the early history of galaxies

In Equation 8.2 on page 318, &sigma_T = 6.65 x 10^-25 cm⁺², not 6.65 x 10^-25 cm^-2

In Problem 8.6, a superscript is missing. The last term in the top equation should read
&Omega₀/[(2 (1 - &Omega₀)) * (cosh &eta - 1)]^-1.
Alternatively, 1/(1+z) = R(t)/R(t₀) = &[Omega₀/(2 (1 - &Omega₀))] * (cosh &eta - 1).

In Problem 8.8, the times given are not the lookback times, but the time t_e at which the radiation was emitted.

Problem 8-9 should read:
`Show that if &Omega₀=1, then redshift z=5 corresponds to R(t₀) &sigma_e = 1.18 in units of c/H₀, while for &Omega₀=0, R(t₀) &sigma_e = 2.92. For any given density n(z) of quasars, use Equation 8.24 to show that if &Omega₀=0, we would expect to find about 15 times as many of them within a small redshift range &Delta z as we would see if &Omega₀=1. What is this ratio at z=3?'

In Problem 8.11, n₀ from Figure 1.16 is proportional to h³, so &sigma scales like h^-2 and the radius like h^-1.
For &Omega₀=0, &sigma >= 1,700 h^-2 kpc², not 17,000 h^-2 kpc². Taking dN/dz=0.15 from Problem 8.12 and &Omega₀=1, &sigma >= 4,400 h^-2 kpc² and radius >50kpc for H₀ = 75 km/s/Mpc.

In Problem 8.12, &Omega_g is incorrectly defined. It should be the ratio of the comoving gas density (what it would be if the clouds survived unchanged to the present day) to the critical density now, &rho_crit(t₀). The correct expression is
&Omega_g = [(&mu m_H H₀)/(&rho_crit(t₀)c)] N(HI).dN/dz. [H(z) / H₀ (1+z)²].
Show that the term in square brackets is 1.2 x 10^-23 h^-1 cm². Taking dN/dz = 0.15, and an average N(HI) = 10²¹ cm^-2 at z=3 (see Figure 13 of Storrie-Lombardi & Wolfe, ApJ 543, 552; 2000), show that
&Omega_g(z=3) = 10^-3 h^-1 if &Omega₀= 1, while it is about half as large if &Omega₀= 0.

In Problem 8.16, correct answers for m_bol are 25.9 for &Omega₀=1, and 27.3 for &Omega₀=0.
In Problem 8.17, the answers are 3.5h arcsec not 3.5h^-1,and 6h arcsec not 6h^-1.
In Equation 8.37, p349: at the last intermediate step, (d_L/d_A)² should be (d_A/d_L)².
In Equation 8.39, p350: 10pc/d_L should be d_L/10pc.
In Figure 8.23, the galaxy cB58 is bluer than the starburst in Figure 8.21, not redder. The figure shows F_&nu, not F_&lambda.
In Problem 8.22 on p356, the criterion h_P &nu >> k_B T holds only for wavelengths &lambda >> 500microns. The range where the flux received is almost independent of redshift is 5 < z < 20.

This page last modified: 2 March 2007 or later