Calculating star's radius and temperature.

[ted1986]

Hi all,

I've been trying to solve the attached question for a long time, but it didn't help. I don't know how to start solving it.

Could someone help me please :(

Thnx

Ted

 

[Ken G]

I've been trying to solve the attached question for a long time, but it didn't help. I don't know how to start solving it.

I don't know how accurate you are expected to be, so I'll guide you into a rough "order of magnitude" calculation, since in the end you are just trying to show that the Sun requires some other energy source (fusion). The virial theorem says the total internal kinetic energy, which depends on T (core temp, not surface temp) and M (stellar mass), must be of order the total internal gravitational potential energy (which depends on M and R). So use the fact that kT is approximately the internal kinetic energy of each particle to find the total kinetic energy, and then find the total gravitational energy (use M and R and the constant G). Setting them equal gives you an interesting constraint on the R as a function of T (taking M a constant, eventually to be set to be the mass of the Sun). This relation holds generally, and is even true of the Sun right now (approximately, any way).

Then you need to bring in the time dependence. Do that by equating the luminosity L (taken to be constant, also eventually set to the Sun's current L) to the time rate of change of internal kinetic energy (it's really the time rate of change of the total energy, but we are doing order of magnitude, and are not worried that the internal kinetic energy is increasing with time-- L still gives the order of magnitude of its rate of change). Now you can easily find T(t), the temperature as a function of time, and use your above expression to give R(t) as well. The interesting "payoff" of this important problem is that when you look at the R of the Sun now, and find what t it takes to reach it, it's way less than the age of the Earth. How did the Sun and the Earth end up on different clocks like that? (Hint: something happened to the Sun that put it into an equilibrium state, almost like a state of suspended animation in its evolution, if you will, and that is what allows us to have the same "clock" for both Sun and Earth).

By the way, if you care, the virial theorem actually says the internal kinetic energy, which is positive, is half the magnitude of the gravitational energy, which is negative. Then L is the rate of change of total energy. You can throw in the resulting factors of 2 if you really want to.

 

[ted1986]

Hi Ken,

Thank you for your answer!

How do I evaluate the luminosity formula so that it'll be time dependent?

I know that L=4*pi*R^2*sigma*T^4, but I can't find the relation between L and the internal energy.

thnx again,

Ted.

 

[qraal]

Hi Ken,

Thank you for your answer!

How do I evaluate the luminosity formula so that it'll be time dependent?

I know that L=4*pi*R^2*sigma*T^4, but I can't find the relation between L and the internal energy.

thnx again,

Ted.

Luminosity is the energy radiated per unit time so it's already a time dependent variable. The trick is relating change in radius with time to the change in luminosity with time.

 

[Ken G]

How do I evaluate the luminosity formula so that it'll be time dependent?

It's not the luminosity that's time dependent in your problem, it's the T and R. The problem asks you to keep L fixed. That's not very physically realistic, but that's why this is just a homework problem and not a research paper! What's more, you mentioned T in a formula for L, but that's the surface T-- all the T this problem is dealing with are internal (core) T, the surface T is irrelevant. Generally, surface T is set by L, not the other way around, so the trick is knowing L not surface T. Here you are given L, so there's no worry, and you never need to even mention surface T.

 

[qraal]

So much for my comprehension skills! Sorry! Look at what's changing - R and T. T rises as the internal temperature of the cloud rises as it contracts, and you can compute that via the Virial Theorem.

 

[ted1986]

It's not the luminosity that's time dependent in your problem, it's the T and R. The problem asks you to keep L fixed. That's not very physically realistic, but that's why this is just a homework problem and not a research paper! What's more, you mentioned T in a formula for L, but that's the surface T-- all the T this problem is dealing with are internal (core) T, the surface T is irrelevant. Generally, surface T is set by L, not the other way around, so the trick is knowing L not surface T. Here you are given L, so there's no worry, and you never need to even mention surface T.

Ok, so how do I evaluate T as a function of time?

So far I've got

T = (G*M)/(r*K) up to a constant

I don't know what to do next.

 

[Ken G]

Ok, so how do I evaluate T as a function of time?

So far I've got

T = (G*M)/(r*K) up to a constant

I don't know what to do next.

So far so good (K is usually denoted k, the Boltzmann constant, yes?). Now ask yourself how much total energy is in the Sun at that T (to within an order-unity constant again, and note here you'll need to figure out something about the number of particles in the Sun), and recognize this is changing at the rate L if the Sun is gravitationally contracting (yes, L is a loss, and the T will be increasing, but the magnitudes are right if not the sign-- that's the virial theorem). Note that T started out essentially zero, and r essentially infinite, and use the constant L to tell you what T is doing as a function of time, then look at current values for the Sun to see how much time that would take-- if gravitational contraction were all that is going on.

 

[ted1986]

Now ask yourself how much total energy is in the Sun at that T (to within an order-unity constant again, and note here you'll need to figure out something about the number of particles in the Sun), and recognize this is changing at the rate L if the Sun is gravitationally contracting .

The total energy (ideal gas approx. ) is 3/2nkT, so from here I estimate the number of particles in the sun?

Do I also have to use the connection between L and time (L=energy/time?)

I'm sorry I don't know to solve this question from this point :-/

 

[Ken G]

You're right there-- find the number of particles in the Sun (using its mass), and then find the total energy in the Sun from your above expression. Then say the rate of change of that energy is characterized by L, and figure out the timescale for the Sun to change dramatically if gravitational collapse was all that was going on. You can even find the age of the Sun if gravitational collapse was all that was going on, because you have a constant L (given), and a fixed number of particles (from M), so you immediately know T(t) (if you are bothered that L is an energy loss, not a gain, note that the virial theorem tells you that the energy associated with T(t) is always 1/2 the energy lost by L, which after time t is = Lt).

 

[ted1986]

You're right there-- find the number of particles in the Sun (using its mass), and then find the total energy in the Sun from your above expression. Then say the rate of change of that energy is characterized by L, and figure out the timescale for the Sun to change dramatically if gravitational collapse was all that was going on. You can even find the age of the Sun if gravitational collapse was all that was going on, because you have a constant L (given), and a fixed number of particles (from M), so you immediately know T(t) (if you are bothered that L is an energy loss, not a gain, note that the virial theorem tells you that the energy associated with T(t) is always 1/2 the energy lost by L, which after time t is = Lt).

The mass is M. I assume that most of the particles are hydrogen (with mass m), so the number of particles is M/m.
The luminosiry (as you say) has to be: L=(dE/dt).
If I put the last term in the total energy equation I'll get
L*dt=(3/2)*(M/n)*KT ?
Is it correct?

 

[Ken G]

Yes, although you can't take the 3/2 seriously, there are many other order-unity factors like that being dropped. (For example, hydrogen is ionized, so there are 2 particles per hydrogen mass, and the "virial theorem" says the kinetic energy will only be 1/2 of L*t. Finally, the whole idea that we have a single T is approximate, as is the idea that we have a constant L). Also, I wouldn't call "t" by the name "dt", the latter sounds like a tiny time but you just want time. But these are all minor issues-- you have the guts of it now, you just need to connect T to R to be able to solve for R(T), and look up the current R, to get t.