Material Falling Onto Neutron Stars

[RyanJ]

Hi all.

I've been trying to calculate how much energy would be released from dropping things onto the surface of a neutron star from certain heights. I'm using...

[math]\Delta E = GMm (\frac{1}{r_2} - \frac{1}{r_1})[/math]

To calculate the potential energy that could be released. Where G is the gravitational constant, M is the mass of the neutron star, m is the mass of the falling object, r₂ is the radius of the neutron star plus the height and r₁ is the radius of the neutron star. Phew!

Would this be the correct formulation to use here? I don't know if I would need to take relativistic effects into account here (and I wouldn't know where to start with that, so advice would also be welcome). Are there any other factors I need to consider here? What proportion of that potential energy would realistically be released from such a collision?

I'm trying to get back into practising using mathematics and physics in practice, I'm feeling seriously rusty. I also recall hearing something along the lines of "dropping a marshmallow onto the surface of the neutron star releases as much energy as an atomic bomb" and I wanted to put that claim to the test.

I think this is the correct forum, but if not I would appreciate it if the thread could be moved.

Thanks!

[Romao Mota]

The overall strategy of the equation is correct, but the gravitational potential energy is a negative value and delta E should be positive, so, the equation needs to be multiplied by (-1), or invert the positions of r₁ and r₂.

or

When r₂>>>r₁, the influence of r2 can be neglected and delta E is exactly equivalent to the absolute value of the potential energy on the surface of the neutron star, then:

 

Edited March 14 by Romao Mota
latex problem

[swansont]

12 hours ago, RyanJ said:

I don't know if I would need to take relativistic effects into account here (and I wouldn't know where to start with that, so advice would also be welcome)

If the energy from the result approaches or exceeds a few percent of mc^2, that an indication that you need to use a relativistic treatment

[RyanJ]

10 hours ago, swansont said:

If the energy from the result approaches or exceeds a few percent of mc^2, that an indication that you need to use a relativistic treatment

What would that look like? Would that require something like this:

1. Calculate the acceleration due to gravity (g) using [math]g = \frac{GM}{r^2}[/math]
2. Calculate the time taken to fall over the distance h under the influence of g using [math] t = \sqrt{\frac{2h}{g}}[/math]
3. Calculate the terminal velocity of the object after falling for time t using [math] v = v_0+gt[/math]
4. Calculate the kinetic energy at the end of the fall using the relativistic kinetic energy equation [math]k_e = m_0 × c^2 × (\sqrt{1 - \frac{v^2}{c^2}} - 1)[/math]

While this seems correct outwardly, I can see some issues. It assumes that the acceleration is uniform over the time of the fall and it doesn't factor in that there is an upper speed limit due to relativistic effects. I don't know how to account for the latter, for small heights it's a non-issue but for larger ones, the speed quickly exceeded what is actually possible - implying my model and method is ultimately faulty.

Edited March 15 by RyanJ

[KJW]

Let [math]M[/math] be the mass of the non-rotating spherical mass (neutron star but assumed to be non-rotating), [math]R[/math] be the radius of the spherical mass, and [math]h[/math] be the height above the ground at radius [math]R[/math] from which the object of mass [math]m[/math] (measured at height [math]h[/math]) is dropped. Assuming that the collision with the ground is completely non-elastic, the energy [math]E[/math] (also measured at height [math]h[/math]) released is:

 

[math]E = \left(1 - \sqrt{\dfrac{g_{tt}(R)}{g_{tt}(R+h)}}\right) m c^2[/math]

 

where [math]g_{tt}(R)[/math] and [math]g_{tt}(R+h)[/math] are the [math]tt[/math]-components of the Schwarzschild metric at [math]R[/math] and [math]R+h[/math] respectively. Thus:

 

[math]E = \left(1 - \sqrt{\dfrac{1 - \dfrac{2 G M}{c^2 R}}{1 - \dfrac{2 G M}{c^2 (R+h)}}}\right) m c^2[/math]

 

Note that [math]\sqrt{\dfrac{g_{tt}(R)}{g_{tt}(R+h)}} = \sqrt{\dfrac{1 - \dfrac{2 G M}{c^2 R}}{1 - \dfrac{2 G M}{c^2 (R+h)}}}[/math] is the ratio of the mass of an object at [math]R[/math] to the mass of the same object at [math]R+h[/math], the object being at rest at both heights.

 

 

Edited March 15 by KJW

[swansont]

1 hour ago, RyanJ said:

What would that look like? Would that require something like this:

1. Calculate the acceleration due to gravity (g) using g=GMr2
2. Calculate the time taken to fall over the distance h under the influence of g using t=2hg−−√
3. Calculate the terminal velocity of the object after falling for time t using v=v0+gt
4. Calculate the kinetic energy at the end of the fall using the relativistic kinetic energy equation ke=m0×c2×(1−v2c2−−−−−√−1)

While this seems correct outwardly, I can see some issues. It assumes that the acceleration is uniform over the time of the fall and it doesn't factor in that there is an upper speed limit due to relativistic effects. I don't know how to account for the latter, for small heights it's a non-issue but for larger ones, the speed quickly exceeded what is actually possible - implying my model and method is ultimately faulty.

You can’t use the Newtonian kinematics equations if the motion is relativistic.

There is no terminal velocity - terminal velocity requires an opposing, speed-dependent force.

[RyanJ]

On 3/15/2024 at 11:48 AM, KJW said:

Let M be the mass of the non-rotating spherical mass (neutron star but assumed to be non-rotating) ...

Thank you. I need to take a bit of time to digest the implications of the equation you've presented so I can get a good understanding of what it is doing and why.