Estimating the temperature of a star

A quick energy-based method to estimate the temperature of a star given only its mass and radius. Although based on a simple model, the equation we derive shows surprisingly good agreement with the temperature of the Sun!
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About me: I studied Physics at the University of Cambridge, then stayed on to get a PhD in Astronomy. During my PhD, I also spent four years teaching Physics undergraduates at the university. Now, I'm working as a private tutor, teaching Physics & Maths up to A Level standard.
My website: benyelverton.com/
#physics #mathematics #stars #temperature #gravity #energy #equipartitiontheorem #boltzmannconstant #density #integration #calculus #sphere #thermalenergy #potential #mass #gravitationalpotentialenergy #star #conservationofenergy #hydrogen #atoms #thermalphysics #ionisation #physicsproblems #maths #math #science #education

[@logician1234]

Prof takes a break and then the first video he posts is calculating temperature of the star 💀

[@DrBenYelverton]

I had to think of something impressive for my return!

[@shrivatsa8604]

Great to have you back, sir. I had previously worked with the same problem, which after solving yielded the same result. And then tried to solve it in a non-classical, relativistic way .my initial goal was to find the potential energy of the entire system using relativistic way calculating energy (Swartzchild interior solution) , which was complicated. Based on your expertise in this field, I believe working on relativistic physics in your upcoming videos will be very brilliant and intresting like this one.

[@DrBenYelverton]

Thanks for your support. Solving this one relativistically certainly sounds complicated!

[@yahavhazut]

Great video that shows the power of estimation theory!
Thanks!

[@DrBenYelverton]

Thank you - it's amazing how well simple models work sometimes!

[@mohsenrezaei5965]

brilliant! your work in this channel is priceless, thank you

[@DrBenYelverton]

Thanks for your kind words! Glad you are enjoying the videos.

[@paxshild4924]

Hey Dr. Ben, it's great to see you back! I really enjoyed your recent video on the temperature of a star, especially after such a lon break. Your content always brings out some of the most interesting problems in physics!
I was thinking, as a follow-up to this thermodynamics topic, what if you did a video exploring how temperature changes over time for an object in a completely empty vacuum? Specifically, if we consider an object with a certain initial temperature in an ideal vacuum (with no surrounding objects), how does its temperature evolve over time as it radiates heat away? I've been trying to come up with a satisfying equation for this, but it seems like Newton's law of cooling might not fully apply in this scenario. I feel like this could be a really fascinating continuation of your recent video.
Also, I wanted to say that I'm deeply sorry for your loss, and I admire your resilience in continuing to share such amazing content. Looking forward to more thought-provoking problems from you!

[@umylten4142]

Just for fun (as almost all assumptions are incorrect), assume:
- constant density r
- constant specific heat capacity c
- uniform temperature T(t)
- black body radiation
Then, thermal energy is (up to a constant) r•c•V•T(t) (where V is the volume) and radiated power is s•A•T(t)⁴ (where s is Stephan-Bolzmann constant and A is the area of the object).
Equating the loss of thermal energy with the radiated power:
r•c•V•(dT/dt) = -s•A•T⁴
This is a simple differential equation which provides T(t) :^) If I didn't mess up, the result is
T(t) = T(0)/[(1 + bt)^(1/3)]
where b = 3•s•A•T(0)³/(r•c•V).

[@paxshild4924]

@@umylten4142 I tried to solve it a long time ago with the approach similar to yours which kinda worked but I'm not sure of it's accuracy hence I wanted to ask a professional

[@Herogamergaming59]

Sir after doing some analysis using the Stefan-Boltzmann law I found out that this model suggests that the intensity released by a star is proportional to the mass of the star raised to the power of 4 ! I find that quite interesting

[@DrBenYelverton]

Apparently, that is actually a pretty good approximation for Sun-like stars! See e.g. en.wikipedia.org/wiki/Mass%E2%80%93luminosity_relation

[@mxminecraft9410]

Sir I have been doing a lot of physics problems some of them are very hard where can I share the problems with you?

[@DrBenYelverton]

Feel free to email me at [e-mail]!

[@mxminecraft9410]

@@DrBenYelverton ok

[@mxminecraft9410]

@@DrBenYelverton question sent

[@DrBenYelverton]

@@mxminecraft9410 Got it, looks interesting! I will have a go at solving it when I get the chance.

[@mxminecraft9410]

@@DrBenYelverton ok can you also upload a solution too?

[@drscott1]

👍🏼