(Astrophysics) Could someone explain what theses equations mean in this paper?


[Paper link](https://arxiv.org/pdf/1903.01426.pdf)


On page 4, I am trying to understand what equations 1, 3, and 4 mean. From my understanding, equation 3 comes from the first law of thermodynamics, but could someone what they all describe?

In: Physics

You won’t get much of a good answer from here. I would recommend asking your professor, or going to /r/Physics. From a cursory look, it looks like they are relating the pressures of the fluid dynamics that occur in pulsating Cepheid type stars to the momentum and specific energies, but I may be completely off about that assessment.

I’m not sure it would be possible to ELI5 this without an awful lot of other stuff.

Equation (4) is simple enough; just Equations (2) + (3).

I think equation (1) is a Newton’s 2nd Law/conservation of momentum thing. Very, very roughly, on the left hand side you have an acceleration-type thing and on the right hand side the first term is pressure * area / mass (so force / mass), then a term for the acceleration due to the viscosity of the material, then the acceleration due to gravity (I think; they don’t seem to have defined G in their table).

And then yes, I think equation (3) is conservation of energy.

But this is a serious astrophysics paper; I think at best you’re looking for ELI5-years-into-(post-)graduate-studies-in-astrophysics.

(1) is the evolution of momentum in time, wich is basically the sum of all (newtonian) forces.

(2) is the evolution of internal energy (thermal energy for laymen) so basically the balance of heating and cooling.

These are typically the most fundamental equations to describe ANY system, so the equations depend entirely on what kind of effects you’re studying in specific

Without knowing your background it will be hard to ELI5 the details though.

Equation 1 is the conservation of momentum and equation 2 is the conservation of energy without taking into consideration turbulence. Equation 3 is basically a turbulence model, which means its a simple way to provide a description of turbulent flow, which is normally very hard to predict. Equation 4 then combines the internal energy with and without turbulence to provide an equation that includes both. I could have made some mistakes there, however all of those equations listed also have citations when they are presented in the paper, I would suggest looking at those cited papers as well for some additional background. In addition, I would suggest some research in fluid dynamics, as well as fluid dynamics applied to astrophysics for to get a better understanding of where those equations come from, and why they are in that specific form.

The equations are all material derivatives, which are a special kind of derivative used for time rate of change problems in continuum mechanics.

In this specific instance, equation 1 is the rate of change of momentum (in effect, a more complicated and generalized form of Newton’s Laws), equation 2 is the rate of change of specific internal energy (effectively, heat), equation 3 is the rate of change of specific turbulent energy (basically, the energy of eddies and flows within a given substance, from Fluid Mechanics), and equation 4 just equations 2 and 3 put together (which would be an expression of the 1st Law of Thermodynamics).