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_{B} 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 10^{4} L_{V,Sun},
while at K its
luminosity is much larger compared to the Sun:
L_{K} = 4.1 x 10^{5} 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*(N_{1} - 3.5) works slightly better.
In part c, the metallicity is Z/ Z_{Sun} = (N_{2} + 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 V_{0}(d/R).

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

_{+}and &theta

_{-}

## 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 r^{3} /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 M^{2}/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_{r}^{2} r_{c}/G.

We can't prove Equation 3.44, but must fudge it by writing
L_{tot} = 4 &pi r_{c}^{2} 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 10^{6}M_{o}, not 4 x 10^{6}M_{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 R_{0} 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.

^{2}.

In Problem 3.23, the term in square brackets in Equation 3.103 should be - &Phi(x) - 0.5 v

^{2}, and the distribution function f

_{P}= k (-E)

^{7/2}, not k (-E)

^{5/2}.

## 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 10^{10} L_{Sun}, not
3.3 x 10^{10} 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 V^{2}_{max}
= V^{2}(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 ^{2} V_{H} / 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_{prol} > or = A/B', not `< or = A/B'.

Problem 6.6: instead of `Q < 0.95 and -21 < M_{B} < -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_{xx} << PE_{zz}', 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
&pi^{2} or about 10.

In Problem 6.15, the galaxy mass M = 3 x 10^{12} M_{o},
not 1.5 x 10^{12} M_{o}.

In Problem 6.20, the Plummer-sphere model yields
M = 10^{15} h^{-1} M_{o}, not
10^{15} 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/&pi^{2})
(&pi c_{c}^{2} / G &rho) ], or 2r = &lambda_{J}
to within sqrt(15/&pi^{2}) 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

&Omega_{0}/[(2 (1 - &Omega_{0})) *
(cosh &eta - 1)]^{-1}.

Alternatively,
1/(1+z) = R(t)/R(t_{0}) =
&[Omega_{0}/(2 (1 - &Omega_{0}))] * (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 &Omega_{0}=1, then redshift z=5 corresponds to
R(t_{0}) &sigma_{e} = 1.18 in units of c/H_{0},
while for &Omega_{0}=0,
R(t_{0}) &sigma_{e} = 2.92.
For any given density n(z) of quasars, use Equation 8.24 to show that if
&Omega_{0}=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
&Omega_{0}=1. What is this ratio at z=3?'

In Problem 8.11, n_{0} from Figure 1.16 is proportional to
h^{3}, so &sigma scales like h^{-2} and the radius like
h^{-1}.

For &Omega_{0}=0, &sigma >= 1,700 h^{-2} kpc^{2},
not 17,000 h^{-2} kpc^{2}.
Taking dN/dz=0.15 from Problem 8.12 and &Omega_{0}=1,
&sigma >= 4,400 h^{-2} kpc^{2} and radius >50kpc for
H_{0} = 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}(t_{0}). The correct expression is

&Omega_{g} =
[(&mu m_{H} H_{0})/(&rho_{crit}(t_{0})c)]
N(HI).dN/dz. [H(z) / H_{0} (1+z)^{2}].

Show that the term in square brackets is
1.2 x 10^{-23} h^{-1} cm^{2}.
Taking dN/dz = 0.15, and an average
N(HI) = 10^{21} 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
&Omega_{0}= 1, while it is about half as large if
&Omega_{0}= 0.

In Problem 8.16, correct answers for m_{bol} are 25.9 for
&Omega_{0}=1, and 27.3 for &Omega_{0}=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.