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Trying to compare density of Black Holes and Neutron Stars I came up with the following:

A typical neutron star has a mass between about 1.4 and 3.2 solar masses1[3] (see Chandrasekhar Limit), with a corresponding radius of about 12 km. (...) Neutron stars have overall densities of 3.7×10^17 to 5.9×10^17 kg/m^3 [1]

and

You can use the Schwarzschild radius to calculate the "density" of the black hole - i.e., the mass divided by the volume enclosed within the Schwarzschild radius. This is roughly equal to (1.8x10^16 g/cm^3) x (Msun / M)^2 (...)

The value of the Schwarzschild radius works out to be about (3x10^5 cm) x (M / Msun) [2]

Let's take a neutron star from the top of the spectrum (3.2 Msun) and same mass black hole.

Converting units:

  • Neutron star: 5.9×10^17 kg/m^3 = 5.9 × 10^14 g/cm^3
  • Black hole: 1.8x10^16 g/cm^3 x (1/5.9)^2 = 5.2 x10^14 g/cm^3

The radius of the black hole would be (3x10^5 cm) x ( 5.2 ) = 15.6km

The 3.2Msun Neutron Star of this density would have volume of 1.08 x 10^13 m^3 which gives radius of 13.7 kilometers

According to Shell Theorem, spherical objects' gravity field strength at given distance is the same for spheres as for point masses so at the same distance from center of same mass (point - black hole, sphere - neutron star) the gravity will be the same.

That would put the surface of the neutron star below the surface of event horizon of equivalent black hole. Yet I never heard about even horizon of neutron stars.

Either I made a mistake in my calculations (and if I did, could you point it out?) or... well, why?

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There is an error: where did you get the 5.9 in the equation for the black hole and the 5.2 in the radius of the black hole? You must use 3.2. In this way you get 1.7x10^15 g/cm^3 as density and 9.6km as radius –  Francesco Montesano Oct 18 '13 at 19:14

2 Answers 2

up vote 7 down vote accepted

As Francesco Montesano points out, using the wrong mass leads to the wrong answer. Also, using the density here seems a complicated way to get to the answer; you could compute the Schwarzschild radius for the NS, and see whether it's smaller than its actual radius.

Since the density scales as ρ ~ M/R^3 and the Schwarzschild radius as Rs~M, the density of BHs scales as ρ~1/R^2; more massive BHs are less dense and simply testing whether a NS is denser than a BH alone is not sufficient - they must be of the same mass, which means that you are in fact comparing radii.

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+1, though there's another reason why this density is bad: volume is completely frame-dependent. Wiki's density figures use Euclidean volume where the geometry is strongly non-Euclidean. With the metric in the Tolman-Oppenheimer-Volkoff ansantz, a spherically symmetric simple neutron star would have volume $$V_\text{TOV} = \int_0^R \frac{4\pi r^2\,\mathrm{d}r}{\sqrt{1-\frac{2GM(r)}{rc^2}}}\text{,}$$ which is never Euclidean. In another frame, it'd be something else still. We could still use the Euclidean "overall density" to compare neutron stars, but the figure itself doesn't mean much. –  Stan Liou Dec 22 '13 at 0:33

Using density is invalid. As the radius of the event horizon for a given mass increases linearly, the volume of that radius increases as the cube and the density therefore decreases. Looking at it the other way, the density increases as the event horizon decreases.

You can calculate the size of the event horizon for any given mass. You just need to find the point at which the escape velocity exceeds the speed of light. We can use the speed of light in the formula for escape velocity and solve for the radius

Escape velocity formula enter image description here solving for r gives enter image description here

I put together a spreadsheet with the numbers. I calculate that a 3.2 solar mass black hole would have a radius of 4.752km, meaning that a neutron star of 3.2 solar masses were to become a black hole it would have to shrink to 9.504km and have a density of 7.13E18 kg/m^3. Conversely the super-massive black hole at the center of our galaxy has an event horizon radius of about 6 billion km and a density of only 4.34E6 kg/m^3. A black hole the size of a proton would need 350 million metric tons and have a density of 1.5E56 kg/m^3.

I think you are probably off on some of your numbers. Specifically you are using ranges of numbers at the top end of a spectrum and an 'about' figure for the radius of a neutron star as though 12km is a single constant radius for all neutron stars. In fact a 1.4 solar mass neutron star would nave a radius somewhere between 10.4 and 12.9 km (source)

https://heasarc.gsfc.nasa.gov/docs/nicer/nicer_about.html enter image description here

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