1. Suppose you discover a binary star. The period of the binary is 20 days. The semi major axis of the binary is 0.2 AU.

a.) If one of the stars has the same mass as the sun, what is the mass of the companion in the binary?
Note: formula in text in Appendix p. A-3 is incorrect. The formula in text on p. 141 is correct. And the formula on this homework sheet is correct.

Use the formula: P² =[ 4 pi² a³ ] / [ G(M₁+M₂) ]
variables need to have same units as constant: G=6.67 x 10^-11 m³ / kg / s²
Therefore, convert the period from days to seconds and the semi major axis from AU to meters.
P=20 days = 1.728 x 10⁶ s
a=(0.2 AU)(1.5 x 10¹¹m / 1 AU ) = 2.992 x 10¹⁰m
Solve for the variable mass. Let M_total = M₁+M₂.
M_total = [ 4 pi² a³ ] / [ G P² ]
Note: all the units except kg cancel M_total =5.3 x 10³⁰ kg
M_companion= M_total - M_sun = 5.3 x 10³⁰ kg - 2 x 10³⁰ kg = 3.3 x 10³⁰ kg

b.) If the companion is a black hole, what is the radius of the black hole?
Radius_{black hole} = 2 G M / c² = 4891 m

2. Suppose an alien civilization decides to turn the sun instantly into a black hole.

a.) What will the radius of the sun be when it becomes a black hole?
Radius_{black hole} = 2 G M / c² = 2964 m

b.) What will happen to the orbit of the Earth?
The orbit of the Earth will stay the same because the Force of Gravity remains the same. (See p. 531 in text if you're curious.) In this case, the orbit depends on the mass of the central object. The Earth will preserve its angular momentum (i.e. its orbit will not change) unless there is some source of friction to get rid of angular momentum. FYI: The only way the Earth would spiral in towards any central object (whether it's the Sun or a black hole) is if that object extended out to the radius of the Earth (either by the Sun enlarging into a Red Giant or by a black hole creating an accretion disk around itself over time) so that there would be material that would create friction as the Earth orbits, therefore reducing the Earth's angular momentum, enabling the Earth to spiral in.

c.) What will happen to the climate of the Earth?
The black hole would not be emitting light as the Sun does, so the climate on Earth would become drastically cold and it would freeze over.

3. What are the three possible end points of stellar evolution (i.e. what do stars become when they die)?

Compare the radii, masses, and densities of different stellar remnants. (i.e. which has the smallest radius, what has the largest density, etc.)
White dwarfs : smallest mass (Stars that are born with an initial mass M_{main sequence} < 8 M_sun will end up as a white dwarf with a final mass of M_{white dwarf} <= of 1.4 M_sun) , largest radii (comparable to the radius of the Earth), lowest density. (A pair of dice made of material of a white dwarf would weight 5 tons.)
Neutron stars: medium mass ( M_{neutron star} > 1.4 M_sun), medium radii (about 10 - 15 km), medium density (A paper clip worth of material from a neutron star would weigh as much as Mount Everest. See p.525 in text)
Black holes : largest masses (usually), smallest radii (a few km), greatest density

4. The space between stars in not completely empty. Gas and dust exist in interstellar space. Name two specific types of interstellar material. Give some characteristics of each. (See lecture notes for this info.)
Molecular hydrogen clouds: stars form here, gas is dense & cold, composed of mostly hydrogen.
Diffuse Hydrogen Gas : Warmer than molecular clouds. Gas may be ionized. Found throughout interstellar and intergalactic space. Very low density.
Dust : Composed of heavy elements like carbon compounds and silicates. Found mostly in the plane of the galaxy.
Dark Matter : Unkown material that is thought to exist between stars. It interacts with other material through gravity. It my be black holes, white dwarfs, brown dwarfs or some unkown massive particle.

5. A supermassive black hole at the center of a quasar is only about 10% efficient at converting mass into energy. In other words, for every 10 ergs of mass that falls into the black hole, only 1 erg of radiation is emitted by the black hole.

a.) How much energy does a black hole emit that consumes one Earth mass every day?
In part c), you will compare the luminosity (Energy / time) of this black hole to the luminosity of the Sun. For an answer of Energy in units of ergs (gm / cm² / s²), use the appropriate units.
E_{black hole} = (0.10) M_Earth c²
E_{black hole} = (0.10) (6 x 10²⁷ g ) (3 x 10¹⁰ cm/s)² = 5.4 x 10⁴⁷ erg

b.) What is the average luminosity of radiation energy emitted by that supermassive black hole in ergs/second?
Convert one day into number of seconds and divide energy calculated in part a) by number of seconds per day
Luminosity = Energy / time = 5.4 x 10⁴⁷ erg / 86400 s = 6.25 x 10⁴² ergs / s

c.) Convert this luminosity from units of ergs/second into units of the sun's luminosity.
Divide answer to part b) by L_sun=3.9 x 10³³ ergs/sec
6.25 x 10⁴² ergs / s / 3.9 x 10 ³³ ergs / s = 1.6 x 10⁹
i.e. If the black hole were at the distance of our Sun, it would be 1,600,000,000 times brighter than the Sun!