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Physics LibreTexts

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

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Exercise 7.1.1

Answer

Constant t, r, and ϕ so we have

ds=a(t)rdθ

Exercise 7.1.2

Answer

Similar to 7.1.1 above, we now have constant t, r, and θ, which gives

ds=a(t)rsinθdϕ

Exercise 7.1.3

Answer

This is simply a2(t)r2sinθdθdϕ.

Exercise 7.1.4

Answer

Using the result from 7.1.3 above, we now just integrate over θ and ϕ, so the area is

A=2π0π0a(t)rsinθdθdϕ=2π02a2(t)r2dϕ=4πa2(t)r2

Exercise 7.1.5

Answer

We know that Luminosity = (Flux)×(Surface Area). Making the appropriate substitutions we do indeed find that F=L/(4πd2a2).

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(tr)/a(te). The same argument applies to the rate of arrival of photons. We also saw that wavelength would be stretched out by a factor 1+zλrλe=a(tr)/a(te). 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=hcλ. Substituting this into the definition of redshift, we find that

Er/Ee=11+z

Exercise 7.3.1

Answer

Solving Equation 7.4 for dlum we get

dlum=L4πF

Exercise 7.3.2

Answer

Substituting in Equation 7.3 to our result above we get

d2lum=L4π4πd2(1+z)2Ldlum=d(1+z)


This page titled 2.7: S07. Distances as Determined by Standard Candles - SOLUTIONS is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Lloyd Knox.

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