Download Problem Set #3

Astronomy 2292
Spring Semester 2016
Problem Set #3
Due Monday, April 4 in class
Instructions
Please work out your answers on separate sheets of paper and staple them together to hand in. Write
you name on the top of each sheet of paper, in case they get separated. Please show all steps in your
calculations, including any sketches you might need. Be careful to properly evaluate and give units
where needed, and also be careful with significant figures. Calculations given without “showing the
work” will receive zero credit even if the final answer is correct.
While you may discuss these problems with other students, you must work out the details and write up
the answers yourself.
Each question is worth 25 points.
1. Consider a neutron star with a mass of 1.5 MSun.
a. What is the radius of this neutron star, expressed as a fraction of its Schwarzschild
radius? You can use Equation 18.41 from the book.
b. What is the mean density of this neutron star?
c. A carbon nucleus has a radius of 3 x 10-15 m. What is its density?
d. What is the ratio of the density of a 1.5 MSun neutron star to that of carbon nucleus?
e. Photons leaving the surface of this neutron star will be gravitationally redshifted by an
amount equal to (Δυ/υ)=-rSch/(2r), where υ is the frequency, rSch is the neutron star’s
Schwarzschild radius and r is its actual radius. What is the gravitational redshift, in
angstroms for the Hβ line at a wavelength of λ=4861Å.
2. Suppose the Milky Way consisted of 2.7 x 1011 stars, each with absolute magnitude MV=4.8.
a. What would be the absolute magnitude of the entire Milky Way?
b. What would be the total luminosity of the Milky Way in solar luminosities?
c. What would be the apparent magnitude of Milky Way if it were located at 5.2 Mpc?
d. What is the actual absolute magnitude of the Milky Way (cite your source)?
e. Explain why your answer in (a) disagrees with the actual value.
3. You are transported to a planet in the outskirts of another spiral galaxy. Being bored, you
decide to reproduce Oort’s analysis, finding a distance to the center of your newfound home
galaxy to be R0=10 kpc. You also measure the local Oort constants, finding A= 20 +/- 3
km/s/kpc and B = -20 +/- 2 km/s/kpc.
a. Is your galaxy in solid body rotation? How do you know?
b. What is the local rotation speed at your position?
c. Is the rotation curve locally flat? How do you know?
d. What is the mass of the Galaxy contained within R0?
e. Do you think this Galaxy is likely to be more or less massive than the Milky Way?
4. You’re still bored in your new galaxy, so you decide to build a large telescope with adaptive
optics capable of observing the center of the galaxy. There you notice, over a period of a few
decades, several stars orbiting what appears to be a single point at the center of the galaxy, all
in apparently Keplerian motion. You infer that there must be a supermassive black hole
(SMBH) in the center of your galaxy.
a. One star has an angular semimajor axis of 0.1 arcseconds. What is its physical
semimajor axis?
b. The same star has an orbital period around the SMBH of 10 years. What is the mass of
the SMBH?
c. If your galaxy is 10 Gyr old, at what rate must this SMBH accrete mass over the
lifetime of the Galaxy in order to obtain its current mass?
d. What is the Schwarzschild radius of this black hole?
e. Would this black hole tidally disrupt a sunlike star before it entered the event horizon?
What about a helium burning giant with a mass equal to the mass of the sun and a radius
equal to 10 RSun?