Linearizing Stefan-Boltzmann equation

[J_I_F]

Homework Statement

I want to linearize both the Stefan-Boltzmann equation and the adiabatic relation to draw a relationship between δL/L(0) and δR/R(0).

Homework Equations

Stefan-Boltzmann equation in the form of L=4πσ(R^2)(T^4)
Adiabatic relation TV^(γ-1) = constant

The Attempt at a Solution

I have L(0) + δL = 4πσ((R(0) + δR)^2)*((T(0) + δT)^4) for the S-B part, but don't know how to get it into this desired form:

δL/L(0) = (2δR/R(0)) + (4δT/T(0))

 

[dwintz02]

I have L(0) + δL = 4πσ((R(0) + δR)^2)*((T(0) + δT)^4)

δL/L(0) = (2δR/R(0)) + (4δT/T(0))

So you have:

$$L_0+\delta L=4\pi\sigma(R_0+\delta R)^2(T_0+\delta T)^4$$

You will have to use a Taylor Expansion to expand the terms that can be considered very small. For example:

$$(a+x)^4=a^4(1+\frac{x}{a})^4 \approx a^4(1+\frac{4x}{a})$$

This only works when $$\frac{x}{a}$$ is very small compared to 1, as is the case in your example. If you're still having trouble, show your work and we can go from there.

 

[J_I_F]

That helped a bunch, thanks.