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Gravity must decrease due to less effective mass when going inside the object but also must increase with depth inside the star due to its higher density. Is there a model or formula approximating gravity calculations along the radius (from center to surface) of the stars?

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For example, I've found a relation for Earth's gravity variation with depth on Wikipedia. I hope that stars do have similar approximations. I don't know how the density distribution will vary with depth (or is it uniform?). – Tariq Nov 16 '12 at 10:44
up vote 5 down vote accepted

You need to know the equation of state for the star's interior. Once you know this you can calculate the density variation with depth and the gravity inside the star.

Google for something like "star equation of state" to find lots of articles on the subject, but note that it's exceedingly complicated because there are so many factors at work. This is the sort of article you'll find: good luck reading it!

Note also that while we can use models to calculate equations of state, the results are only as good as the models. It's hard to know how good our models are when all we can see is the surface of the star.

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However strange it may sounds at first, white dwarfs (at least those well below the Chandrasekhar limit of 1.4 solar masses) are probably the easiest stars to model - the reason is that you can ignore temperature in the equation of state. – Leos Ondra Nov 15 '12 at 19:01
@John Rennie, thanks for the update and the pdf file. @ Leos Ondra thanks for commenting. – Tariq Nov 16 '12 at 10:08

Here is an example for the Sun.

The figure below plots a (reliable) estimate for the interior density profile of the Sun, $\rho(r)$.

So for a given radius $a$, the mass interior to that radius is given by

$$ M(a) = \int^{a}_{0} 4\pi r^2 \rho(r)\ dr $$

And of course the gravitational field strength assuming spherical symmetry will be

$$g(a) = - G M(a)/a^2$$

This would all usually be done inside a numerical model. But it is possible you could find a tolerably good analytic approximation to the curve below that might give you usable results. The profiles for stars of different mass or evolutionary stage will be similar, but different in detailed shape and central density.

An alternative would be to use the second picture which shows the run of pressure with radius inside the Sun. Hydrostatic equilibrium means that

$$ g(r) = -\frac{1}{\rho(r)}\frac{dP}{dr} $$

The data plotted comes from Bahcall and Pinsonneault (2004); the pictures were found at

The solar density profile from Bahcall and Pinsonneault 2004 The solar pressure profile from Bahcall and Pinsonneault 2004

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Thank you very much for this update, @Rob – Tariq Mar 31 '14 at 11:25

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