How Is Magnitude Difference Estimated from Small Variations in Energy Flux?

[Airsteve0]

Homework Statement

Suppose two objects have energy fluxes, f and f + Δ
f, where

Δf ≪ f. Derive an approximate expression for the magnitude difference

Δm between these objects. Your expression should have Δ
m
proportional to
Δf.

Homework Equations

Δm = m2 - m1 = 2.5 * log(f1/f2)

The Attempt at a Solution

So the issue is that I'm not sure which of these solutions would be the better one to work with. However, I did make an approximation with the one as shown below and I'm not sure if it is quite accurate enough. Any assistance is greatly appreciated.

Attempt 1: Δm = 2.5 * log(f / f+Δf) = 2.5 * [ log(f) - log (f + Δf)]

or,

Attempt 2: Δm = 2.5 * log(f+Δf / f) = 2.5 * log(1 + Δf/f) ≈ 2.5*Δf

The second attempt seems more correct to me but I would feel more confident with a second opinion.

 

[anathema]

it is important to consider the accuracy and validity of your approximations in any calculation. In this scenario, both of your attempts are valid approximations, but the second attempt may be more accurate. Let's break down each attempt to see why.

Attempt 1: This attempt uses the fact that the difference in magnitude (Δm) is proportional to the logarithm of the ratio of the two energy fluxes (f1/f2). This is a valid equation, but it does not take into account the small difference in flux (Δf) between the two objects. This means that this approximation may not be as accurate when Δf is significantly smaller than f.

Attempt 2: This attempt takes into account the small difference in flux (Δf) between the two objects by using the approximation log(1+x) ≈ x for small values of x. This means that the logarithm term in the equation can be approximated as Δf/f, which makes the equation simpler and more accurate. This is because the difference in magnitude (Δm) is now directly proportional to the difference in flux (Δf), which is what we want in this scenario.

In conclusion, both attempts are valid approximations, but the second attempt may be more accurate since it takes into account the small difference in flux between the two objects. However, it is always important to consider the context and limitations of your approximations in any scientific calculation.