Astro-7 Fall 1999

Final Review Problems

1.

A bright region of forming stars has just been discovered in the galaxy NGC 2366, which is 3 Mpc away from us.

(a)

If the Hubble constant is $H_0=65 \, {\rm km}\, {\rm s}^{-1} /\, {\rm Mpc}$, how fast is this galaxy moving away from us?

(b)

Sketch a graph which shows the radial velocity versus distance for galaxies in general. How would this look different if the Universe were contracting instead of expanding?

(c)

The bright stars in this star-forming region have an absorption line whose rest (laboratory) wavelength is 503.5 nm. At what wavelength do we observe this absorption line?

2.

Why is a black hole black? If black holes are black, how have we detected them in the universe?

3.

Briefly describe two observations which are predicted by the theory of General Relativity and not by Newton's theory of gravity.

4.

Believe it or not, this can easily be done without a calculator! Muons are elementary particles that live only $2\times 10^{-6}$ seconds when at rest. (After that they decay into electrons and neutrinos.) For a muon travelling at a speed of $\sqrt{0.9999}\,c$, where c is the speed of light, what distance would it travel during its lifetime according to Newton? What distance would it travel according to Einstein, taking into account the predicted time dilation? Muons produced in the upper atmosphere at 10 km above ground are found to arrive at the Earth's surface. Who must be right? Why?

5.

The density of a white dwarf is $1 \times 10^{9}$kg/m³. The density of a neutron star is $4 \times 10^{17}$kg/m³.

(a)

What would be the radius of a sphere which had the density of a white dwarf and the same mass as the Earth?

(b)

What would be the radius of a sphere which had the density of a neutron star and the same mass as the Earth?

(c)

What would the Schwarzschild radius of the Earth be if it were compressed to form a black hole?

6.

Photons in the cosmic microwave background have a range of wavelengths, according to the blackbody spectrum. Although the maximum intensity of radiation is in photons with wavelength $\lambda_{max}$, given by Wien's law, some photons will have wavelengths somewhat shorter or somewhat longer than $\lambda_{max}$.

(a)

Given the present temperature of the microwave background, T₀=2.726 K, what is the wavelength $\lambda_{max}$?

(b)

One of these photons is detected today and has a wavelength of 2mm. The photon was last scattered by a particle when the universe had a scale factor 1100 times smaller than the present value of the scale factor. What was the wavelength of this photon when it was last scattered?

(c)

What was the value of the temperature of the microwave background at that time?

(d)

What was the value of $\lambda_{max}$ at that time?

7.

Thoughtful questions! Make your mind expand just like the Universe! When your less well educated friends find out you've studied the Big Bang in your astronomy course, they might ask some of these common questions. You might answer by explaining what is wrong with the question, or by trying to correct an error in your friend's notion of the Big Bang model. Practice now for the inevitable!

(a)

If the Universe is expanding, what is it expanding into?

(b)

What's at the center of the expanding Universe?

(c)

Why are all the galaxies flying away from us? Do we smell bad?

8.

Imagine that the Sun suddenly collapsed to a black hole. After the collapse, a black hole with the same mass of the Sun would remain at the location where the Sun used to be.

What would happen to the Earth? Would its orbit become larger or smaller? How long would it take to orbit around the black hole?

9.

Kepler's third law can be used to derive the equation $M=R\,v^2/G$, where M is the mass of the Milky Way enclosed within a particular radius R from the galactic center, and v is the speed of an orbiting object at radius R (this is only approximately valid because the Milky Way is not a spherical object). As you learned in class, the Sun, 8 kpc from the center of the Galaxy, orbits at a speed of 210 km/s, which implies a mass of $8\times 10^{10}$ solar masses inside the orbit of the Sun. For the following objects orbiting the galactic center, calculate the mass of the Galaxy which is inside each orbit. Your answer should be in units of solar masses. [Hint: if you are trying to convert kpc to AU or meters, you are doing too much work.]

(a)

A star at 25 pc from the center of the Galaxy orbiting at a speed of 100 km/s.

(b)

A star at 1 kpc from the center of the Galaxy orbiting at a speed of 200 km/s.

(c)

A star at 16 kpc from the center of the Galaxy, orbiting at a speed of 210 km/s.

(d)

A cloud of gas 30 kpc from the center of the Galaxy, orbiting at a speed of 240 km/s.

10.

Make two plots for:

(a)

the orbital speed (in units of km/s) vs. distance from the galactic center (in units of kpc) for the Sun and the other objects in the previous problem.

(b)

the mass you calculated in problem 9 (in units of solar masses) vs. distance from the galactic center (in units of kpc) for the same objects.

(c)

We do not observe very many stars between 16 and 30 kpc from the center of the galaxy. What does this tell us about the properties of much of the mass in our Galaxy?

11.

M100 is the brightest spiral galaxy in the Virgo Cluster. Recent Hubble Space Telescope observations of the Cepheid variable stars in this galaxy yield a distance of 17 Mpc from the Earth.

(a)

An absorption line whose wavelength is measured to be 640.00nm in the laboratory is found in several Virgo cluster galaxies with an average wavelength of 643.20nm. Is the Virgo cluster moving toward or away from us, and at what speed?

(b)

What value does this imply for the Hubble constant H₀ in km/s/Mpc?

12.

Our Galaxy's neighbor, the Andromeda galaxy (M31), is about 800 kpc away.

(a)

If there were only Hubble expansion and no peculiar motion, what would be the speed of Andromeda measured from the Earth? (You may use a Hubble constant $H_0=65\, {\rm km}\, {\rm s}^{-1} \, {\rm Mpc}^{-1}$).

(b)

Is this large compared to the orbital speed of the Sun around the Galactic Center?

(c)

In reality, Andromeda is moving toward us at a speed of about 120 km/s due to the gravitational pull of our Galaxy. If a spectral line is measured to be at a wavelength of 500.000 nm in the laboratory, what wavelength would astronomers measure for this line when they point their telescope at Andromeda?

(d)

Assuming Andromeda's speed remains 120 km/s, how many years would it take for Andromeda to collide with our Galaxy?

(e)

When this occurs, what evolutionary stage do you think our Sun will be in?