2.7: S07. Distances as Determined by Standard Candles - SOLUTIONS

Exercise 7.1.1

Answer

Constant $t$, $r$, and $\phi$ so we have

$$\begin{equation*} \begin{aligned} ds = a(t)r\,d\theta \end{aligned} \end{equation*}$$

Exercise 7.1.2

Answer

Similar to 7.1.1 above, we now have constant $t$, $r$, and $\theta$, which gives

$$\begin{equation*} \begin{aligned} ds  = a(t)r\sin\theta \,d\phi \end{aligned} \end{equation*}$$

Exercise 7.1.3

Answer

This is simply $a^2(t)r^2 \sin \theta \, d\theta d\phi$.

Exercise 7.1.4

Answer

Using the result from 7.1.3 above, we now just integrate over $\theta$ and $\phi$, so the area is

$$\begin{equation*} \begin{aligned} A &= \int_{0}^{2\pi}\int_{0}^{\pi} a(t)rsin\theta d\theta d\phi \\ \\ &= \int_{0}^{2\pi} 2a^2(t)r^2d\phi \\ \\ &= 4\pi a^2(t)r^2 \end{aligned} \end{equation*}$$

Exercise 7.1.5

Answer

We know that Luminosity = (Flux)$\times$(Surface Area). Making the appropriate substitutions we do indeed find that $F = L/(4\pi d^2 a^2)$.

Exercise 7.2.1

Answer

As we found in Chapter 6, the rate of arrival of the wave crests will be slower than the rate of emission by the factor of $a(t_r)/a(t_e)$. The same argument applies to the rate of arrival of photons. We also saw that wavelength would be stretched out by a factor $1+z \equiv \frac{\lambda_r}{\lambda_e} = a(t_r)/a(t_e)$. Therefore the rate of arrival of photons will be slowed down by a factor of $1+z$.

Exercise 7.2.2

Answer

The relationship between photon energy and wavelength is $E = \frac{hc}{\lambda}$. Substituting this into the definition of redshift, we find that

$$\begin{equation*} \begin{aligned} E_r/E_e = \frac{1}{1 + z} \end{aligned} \end{equation*}$$

Exercise 7.3.1

Answer

Solving Equation 7.4 for $d_{\rm lum}$ we get

$$\begin{equation*} \begin{aligned} d_{\rm lum} = \sqrt{\frac{L}{4\pi F}} \end{aligned} \end{equation*}$$

Exercise 7.3.2

Answer

Substituting in Equation 7.3 to our result above we get

$$\begin{equation*} \begin{aligned} d^2_{\rm lum} = \frac{L}{4\pi}\frac{4\pi d^2(1 + z)^2}{L} \quad \Longrightarrow \quad d_{\rm lum} = d(1 + z) \end{aligned} \end{equation*}$$