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In the Kurzgesagt video Atoms As Big As Mountains — Neutron Stars Explained they state that neutron stars have an atmosphere that reaches about 10 centimeters above the surface.

How is this possible? Wouldn't the strong surface gravity pull all of the material onto the surface of the star?

Further, Kurzgesagt also states that the neutron star has a surface variation of about 5mm. Still, wouldn't gravity pull it flat? Of is the gravity not strong enough to do so?

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    $\begingroup$ 5mm of surface variations and 10 m scale height of atmosphere is pretty flat, no? Have you tried to compute an isothermal atmospheric scale-height for comparison, or are you just saying that it's not flat based on your feelings? $\endgroup$ – AtmosphericPrisonEscape Mar 15 at 2:55
  • $\begingroup$ I have tried using equations from the kurzgesagt video sources to computer atmospheric height. Using Wolfram Alpha, the atmosphere should be around 3 meters. Maybe perhaps I am not taking note of a neutron star's temperature? I'm not sure how to compute surface, but I assume it should be flatter based on my feelings. $\endgroup$ – Max0815 Mar 15 at 4:29
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    $\begingroup$ I think feelings are usually a poor basis for doing science. With me being far from a neutron star expert, I think that the quoted numbers sound pretty reasonable under enormous surface gravity $g=GM/r^2$ and r being just a few kilometers, T being possibly even $keV$, and the isothermal scale height being usually computed as $H=kT/\mu g$, this shouldn't be too far off the charts. $\endgroup$ – AtmosphericPrisonEscape Mar 15 at 4:54
  • $\begingroup$ @Max0815 You calculated incorrectly (and the calculation cannot be done without using the temperature!). cm-scales are correct. $\endgroup$ – Rob Jeffries Mar 17 at 16:38
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There is a number of issues with the question, but let me sketch out some kind of answer, so you get something out of it.

The atmosphere of a neutron star is a topic that's a bit speculative. Estimates vary a lot. Regardless, neutron stars can have an atmosphere - sure, gravity is huge, but they are also extremely hot. Some molecules are bound to jump up a bit.

X-ray measurements have shown the atmosphere of a certain neutron star to be 10 cm thick. That's not 10 meters, but it's better than nothing. Other neutron stars surely have different kinds of atmospheres, some thicker, others thinner. It looks like this is a matter for further research.

https://spacemath.gsfc.nasa.gov/weekly/6Page78.pdf

You say the relief on the surface could be up to 5 mm height differential. Earth has mountains 8.8 km tall, and ocean holes 11 km deep - that's a 20 km differential, more or less. That's 4 milion times greater than 5 mm. Keep in mind, the surface material on a neutron star also has different properties from the normal stuff on Earth, different composition, and it's under extreme compression and extreme magnetic fields, which give it different mechanical properties. So 5 mm is quite believable. However, that is an extreme, and most surface should be smooth as glass. Any surface irregularity should be unstable in the long run.

Depending on the model, in some cases, extremely hot neutron stars might have surfaces that are more like a fluid than a solid. Emphasis on "might".

None of the above is set in stone. Neutron star physics is an area of active and intense research.

I recommend the book called Dragon's Egg by physicist Robert L. Forward. It's sci-fi, but the author was a bona fide scientist with extensive work in the field of gravitational wave detection. In his own words, Dragon's Egg is "a textbook on neutron star physics disguised as a novel". It's fun to read, too, one of my favorite hard sci-fi novels.

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The scale height of a neutron star atmosphere is indeed of cm scale.

An isothermal atmosphere will decay as $$ \rho = \rho_0 \exp( -kT/mgh)\ ,$$ where $\rho$ is the density, $\rho_0$ is the density at $h=0$, where $h$ is the "height" above this datum point, $g$ is the surface gravity of the neutron star, $m$ is the mass of the particles making up the atmosphere and $T$ is the surface temperature. The exponential scale height of the atmosphere is therefore $kT/mg$.

A very young neutron star might have a surface temperature as high as $T \sim 10^6$ K. The gravity $g = GM/R^2 \sim 2 \times 10^{12}$ m/s$^{-2}$ for a $1.5 M_{\odot}$ neutron star of radius 10 km. In the best-case scenario we can assume there is a thin layer of hydrogen present (accreted from the ISM) so $m = 1.67\times 10^{-27}$ kg.

The characteristic exponential scale height is therefore $kT/mg \sim 0.4$ cm.

This scale height will be considerably less in older, colder neutron stars or where the surface composition is just heavier elements (which might be the case).

The scale height will be modified by strong magnetic fields (in an uncertain way).

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