Given an expression for E, how do I incorporate dt to arrive at dE/dt?

[cloudy387]

"By shrinking, the amount of energy E_rad that the Sun can radiate into space is [MATH]E_{rad}=\frac{1}{2}\Delta U=(\frac{3}{10})\frac{GM^{2}}{R}\approx 10^{41}[/MATH] Joules. The other half of the energy goes into internal heating.

Exercise : By how much would the Sun have to shrink per year in order to maintain its present luminosity? Hint: Differentiate the expression for E_rad (assuming that the mass of the Sun remains constant), set dE_rad/dt = L_Sun and show that dR/dt = -2.4E(-6) m/s."

The E_rad expression was preceded by an expression for [MATH]\Delta U=-\frac{3GM^{2}}{5}[\frac{1}{R_{0}}-\frac{1}{R}][/MATH], which is used to determine the change in the Sun's gravitational binding energy as it contracts from some initially large radius R₀ to its currently observed radius R < R₀.

I am stuck at the very first part: differentiating E_rad to arrive at an expression for dE_rad/dt. My question is how do I incorporate time (dt) into this? I know I must arrive at a timescale of seconds. I tried using the definition of the derivative because I knew that would allow me to factor the 3GM²/10 out of the limit expression, but I'm not getting far at all.

 

[Dr.Peterson]

I'm not sure of some details here, but it looks like you may just be missing the chain rule.

If we write the equation as E = A/R, where A = 3GM^2/10, then dE/dt = d/dt(AR^-1) = (-AR^-2)dR/dt.

Can you take it from there?

 

[cloudy387]

Wow! I'm getting there but may need just a bit more help yet. For kicks I went ahead and solved for dR/dt in your expression and am getting the correct answer (thank you!!) but I'm having trouble seeing what happened to my inner derivative here.

I will work this just as I would on a sheet of scratch paper on an exam:

E = A/R = AR^-1
let A = 3GM²/10
A' = ?
dE/dA = -AR^-2 * A'

And here I think I'm not seeing something: the value for A in this problem never changes, so my gut says A'=0, but that leaves me stuck.

 

[cloudy387]

I think I might be getting somewhere. Looking the above over, I see I'm missing the product rule, or at least I wasn't properly acknowledging it before now!

E = AR^-1
dE/dt = (A')(R^-1) + (A)(-R^-2)(dR/dt)
The first term on the right goes away because A' = 0 and then I'm left with:
dE/dt = (-AR^-2)(dR/dt)
dE/dt ÷ -AR^-2 = dR/dt

The hint in the exercise gave us a value for dE/dt and so after plugging in everything back in I'm left with the correct value for dR/dt! Thank you, Dr.Peterson!

 

[Dr.Peterson]

Right, A is a constant, so it can just be pulled outside the derivative. If it were not, you would have to use the product rule -- which you didn't do! That's why what you got made no sense.

Also, you are not differentiating with respect to A! Why did you write dE/dA? You want dE/dt.

Rather, R is a function of t, so dE/dt = dE/dR * dR/dt. Try that.