Solution Manual Introduction to Cosmology 2nd Edition by Ryden

Solution Manual Introduction to Cosmology 2nd Edition by Ryden

Exercises

2.1 Assume you are a perfect blackbody at a temperature of T = 310K. What is

the rate, in watts, at which you radiate energy? (For the purposes of this

problem, you may assume you are spherical.)

I have a mass Mme = 70kg, and my density is comparable to that of 

water, ρme = 1000kgm

−3

. If I curl myself into a ball, my radius is 

 . 

(2.1) 

My temperature is 

Tme = 310K = 0.0534T⊙ . 

(2.2) 

(The Sun’s radius is given Section 2.1 of the text, and the Sun’s effective

temperature is given in Section 2.4.) Since the luminosity of a spherical 

 blackbody is L = 4πR

2

σsbT

4

, where σsb is the Stefan-Boltzmann constant,

my luminosity is 

Lme 

(2.3) 

= 3.838 × 10

26 

watts(3.65 × 10

−10

)

2

(0.534)

4 

= 420watts . 

(Note that the text never explicitly states that L ∝ R

2

T

4 

for a spherical 

blackbody; your students may need a prompt, depending on their

physics background.)

2.2 Since you are made mostly of water, you are very efficient at absorbing

microwave photons. If you were in intergalactic space, how many CMB

photons would you absorb per second? (The assumption that you are

spherical will be useful.) What is the rate, in watts, at which you would

absorb radiative energy from the CMB?

Since my radius is Rme = 0.256m (from the previous problem), my

geometric cross-section is 

 . 

(2.4) 

The number density of CMB photons is (from Eq. 2.35 of the text), 

nγ = 4.107 × 10

8 

m

−3 

. 

(2.5) 

At what rate will I absorb CMB photons? This is easiest to compute if I

adopt the fiction that all the CMB photons are moving in the same

direction. In that case, during a time interval dt, I would sweep up all the

photons in a cylinder of length cdt and cross-sectional area σme.

That is, the number dN of photons absorbed during time interval dt is 

dN = cdtσmenγ , 

(2.6) 

or 

 . 

(2.7) 

4 

Fundamental observations