Decrease of Solar radius per year using Virial Theorem

[CharlesDamle]

Homework Statement

Calculate the decrease in Solar radius per year using the Virial Theorem

Relevant Equations

L_G = -(1/2) * (GM^2/R^2) * (dR/dt)

Hello, I am trying to solve this question:

Assume that the Sun's energy production doesn't happen by fusion processes, but is caused by a slow compression and that the radiated energy can be described by the Virial Theorem: $$L_G = - \frac{1}{2} \frac{GM^2}{R^2} \frac{dR}{dt} $$
How much must the Sun's radius decrease per year in order to uphold its energy production?

I'm not quite sure what L_G is, but I've tried inserting the gravitational term of the Virial Theorem, but that gives a decrease in solar radius of the entire Sun's radius per second... A hint would be amazing, if anyone knows what L_G represents.

Cheers

 

[gneill]

If you replace each variable on the right with the units of the variable then reduce the result, the remaining units should give you a big clue as to what $L_G$ might be.

 

[CharlesDamle]

I know it's energy, if that's what you're hinting :D

 

[TSny]

I know it's energy, if that's what you're hinting :D

$L_G$ is not energy. But, it's closely related to energy. What did you get for the overall units of the right-hand side of the expression for $L_G$?

 

[CharlesDamle]

Ahh it is J / s right?

 

[TSny]

Ahh it is J / s right?

Yes

 

[CharlesDamle]

Right, so I want to solve for (dR/dt) to get the change over time, so I can't multiply by dt on both sides to get energy on the left hand side. Not quite sure what to do with L_G

 

[TSny]

Do you know what $L_G$ stands for? The value of $L_G$ is easy to find with a web search. Or, maybe the value of $L_G$ was given in your course or textbook.

 

[CharlesDamle]

No, I have no idea. I looking for it in the Virial Theorem section, but without any luck. It hasn't been mentioned in the chapter we've been reading this week.

 

[TSny]

$L_G$ is the rate at which rotational kinetic energy plus gravitational PE is lost as the radius of the sun contracts. You are considering a hypothetical model where this loss of energy is converted into the energy radiated by the sun. So, you want $L_G$ to match the rate at which energy is radiated by the sun. The rate at which the sun radiates energy is called the sun's luminosity. So you will need to look up the luminosity of the sun.

 

[CharlesDamle]

Ahh, okay we always write it with other symbols. I have no idea why they changed it, but naturally I know what luminosity is.. Alright, so I set it equal L_G = 3.839 * 10²⁶ W and solve for(dR/dt)?
Thank you so much for explaining this!

 

[TSny]

Ahh, okay we always write it with other symbols. I have no idea why they changed it, but naturally I know what luminosity is.. Alright, so I set it equal L_G = 3.839 * 10²⁶ W and solve for(dR/dt)?
Thank you so much for explaining this!

Good. This is an interesting calculation.