Temperature needed for nuclear reactions in sun and orbit period

[creepypasta13]

Homework Statement

1. Give a calculation for the approximate temperature needed for nuclear reactions in the sun?

2. Give a calculation for the period of two stars in the shortest possible orbit?

The Attempt at a Solution

1. Our class just covered the first chapter of our book 'Stellar Interiors' by Hansen, but it doesn't cover this. According to our HW sheet, this was done in-class by the prof, but I missed the lecture due to being out of town, so I guess the only way to solve this is by having the lecture notes?

2. I was thinking of setting up the conservation of energy equation with the two stars, then set E to a min or max and solve for the period?

 

[anathema]

1. The approximate temperature needed for nuclear reactions in the sun can be calculated using the formula T = (GMm)/(3kR), where G is the gravitational constant, M and m are the masses of the sun and a nucleon, k is the Boltzmann constant, and R is the radius of the sun. Plugging in the values for these constants and the known values for the sun's mass and radius, we get a temperature of approximately 15 million Kelvin.

2. To calculate the period of two stars in the shortest possible orbit, we can use Kepler's third law, which states that the square of the orbital period is proportional to the cube of the semi-major axis. The shortest possible orbit would have the smallest possible semi-major axis, which is equal to the sum of the radii of the two stars. So, the period would be proportional to (R1+R2)^3. We can then set this equal to the period of the shortest possible orbit, which is usually taken to be the orbital period of a binary star system with a white dwarf and a neutron star, which is approximately 10 minutes. Solving for the proportionality constant, we get a period of approximately 10 minutes for two stars in the shortest possible orbit.