Science Forums 2

[CraigD]

Originally Posted by litespeed

I invite a mathematician to calculate the impact velocity from one kilometer above the surface [of a neutron star].

You can calculate this by calculating the change in gravitational potential energy for a test body at a radius 1 meter greater than vs. equal to the radius of the primary body, the neutron star.

GPE is given by

, where is the gravitational constant, the mass of the primary, the mass of the test body, the distance from the center of gravity of the system to the test body.

Change in GPE, then, is

, where is the test body’s initial distance, its distance at impact.

Taking an “average” neutron star of 1.5 solar masses and surface radius 12000 m, and for ease of calculation a test body massing 1 kg, this evaluates:



Suspecting that this is associated with a speed for which relativistic effects are slight, we can calculate impact velocity with the classical equation for kinetic energy,


getting



, close to wikipedia & freeztar’s example of

"... fall from just one meter high it would hit the surface of the neutron star at 2 thousand kilometers per second, or 4.3 million miles per hour."

which is indeed less than 1% of the speed of light.

More relevant than impact speed from 1 m above its surface, though, is impact speed from a great distance (it’s traditional in gravitational mechanics to assume “an infinite distance” to set an upper limit on such values). Recalculating with , , and .
This is high enough that we should use a relativistic calculation for kinetic energy,

, which gives an impact velocity of about 0.4 c, about 12000000 m/s.

Although a very high speed, this is still not enough to produce a dramatic mass dilation – a factor of about 1.19, (a 19% increase), so we can see that the original post’s question

Originally Posted by litespeed

1) Does this mass accelerated by gravity increase as it approaches the speed of light, thus increasing the mass and gravity of the BH?

incorrectly assumes that bodies falling toward the surfaces of compact bodies such as neutron stars or the event horizons of black holes (which requires a separate calculation, but one yielding a larger yet similar result) always reach high relativistic speeds.

It’s also necessary to realistically consider conditions around a star-mass body like a neutron star. Because our test body would not be alone, but part of a large accretion disk, it would almost certainly collide many times with similar bodies, preventing it from approaching this upper limit value of 0.4 c.

I think the answer to the title question “Can Neutron Star Become Black Hole”, is “yes, but before it became a black hole, it would cease being a neutron star. An explanation would require a separate, fairly lengthy post, which I’ll get to as time permits, and invite anyone who can spare the time now the pleasure of writing.