Astronomy-particle density/interior temp of sun check

[accountkiller]

astronomy-particle density/interior temp of sun

Homework Statement

The goal is to estimate the interior temperature of the Sun from its mass and particle density.
a)What is the average number density of particles within the Sun, given that the average mass per particle is about 10^-24 gram?

b) What is the approximate temperature necessary for gas pressure to balance gravity within the Sun, given the average particle density from a)?

c) How does your estimate compare with the internal core temperature of the Sun?

Homework Equations

It has a reference to a "Mathematical Insight" problem that uses the equation: M_balance = 18M_sun$$\sqrt{\frac{T^{3}}{n}}$$ ... M_balance being the minimum mass required to balance the force between pressure and gravity in star formation, and n being particles per unit area.

The Attempt at a Solution

a)
V = $$\frac{4}{3}$$ * $$\pi$$ * R³ = $$\frac{4}{3}$$ * $$\pi$$ * (6.95*10⁸)³ = 1.406 * 10²⁷ m³
density = mass / volume = (10^-24 g) / (1.406 * 10²⁷ m³) = 7.1 * 10^-52 g/m³

Did I do that the right way? I have a weird feeling about the mass.. was I supposed to just use the mass/particle they gave me? So then my volume is the density of one particle per m³? I'm really confused on that, I'd appreciate any clearance.

b)
Well, I'm assuming this is the part they want us to use that mathemtical equation on.. but if I solve the equation for T, what is my M_balance? Is it just what I got for volume times what I got for density? ... That would then be 9.98E-25 g - way too tiny!

c)
Not here yet.

 

[Kurdt]

For a) you need to use the sun's mass and the average mass per particle to work out the number density of particles.

 

[accountkiller]

Like this?

$$\frac{10^{-24} g}{particle}$$ * $$\frac{1}{2*10^{33} g}$$

[The sun's mass flipped so that the units cancel out]

That leaves me with $$2*10^{57}$$ particles.

Now I need particles per volume for density... so $$2*10^{57}$$ particles per volume of the sun... which is $$1.406*10^{27} m^{3}$$

So my answer is 2E57 particles / 1.406E27 $$m^{3}$$ = $$\frac{1.4E30 particles}{m^{3}}$$

Correct?

Then for part b), how do I get M_balance?

 

[Kurdt]

Part a) looks ok. You should divide mass of the sun by mass per particle. You have a good answer anyway. For part b) you will need to go back to the equation your book referenced and that you wrote down in the first post.

 

[accountkiller]

Well, I have the equation but it has two unknowns. I have the n, particles per unit area. Then it asks me to find T, the temperature, but I also don't know M_balance. Usually, we are asked to find M_balance given the temperature, but in this reverse case, am I supposed to find M_balance in some table in the index or is there actually a way I can get it?