Time dilation magnitude in orbit around a black hole

[robartinc]

Hello,

I'm working on a hypothetical situation involving a planetary body orbiting a black hole (similar to the scenario in Interstellar, but for different reasons), trying to balance tidal forces with orbital distance and time dilation.

First, I'm interested in the effect of gravitational forces on tidal heating, so I calculated that based on an equation derived in an astrobiology class:

The intent is that the effects of tidal forces would be enough to power massive generators via geothermal energy without being enough to melt or rip apart the planet.
Given a black hole of stellar mass, planetary body the size of our moon, 1000AU orbit, 0.5% orbital variation, and 5% efficiency* converting tidal force into heat, I get a temperature of about 600K - high but potentially manageable (depending especially on if the body has an atmosphere, and how the heat is converted into usable energy).
*still looking for data on the calculated efficiency of tidal heating, eg. on Jupiter's moons; this is just an estimate.

Second, I want to ensure that the planet at this orbital distance would not have a significant time dilation due to those gravitational effects. Assuming a static black hole for now, but a moving observer (i.e. on the orbiting planetary body itself), I used the equation,

from https://www.physicsforums.com/threads/calculating-gravitational-time-dilation.385822/
At 1000AU, I expect they would not be very large, but my calculations are showing a time dilation factor of less than two parts per hundred billion (10^11). When I alter the parameters to see what would be 'significant' on the order of one part in a thousand, it would require the planet orbit at only 2km, when the Schwarzschild radius is 3000km, and photon sphere is 4400km.
This seems counter-intuitive to me, that the planet would have to be so close to 'experience' (relative to an outside observer) time dilation on an appreciable magnitude. Again, my goal is to have the planet orbit outside this region, but I want to be sure my calculations are based on a reasonably accurate representation of the situation.

I've skimmed through these threads as well:
https://www.physicsforums.com/threa...-body-in-orbit-around-kerr-black-hole.781691/
https://www.physicsforums.com/threads/distance-from-black-hole-to-experience-time-dilation.805105/
but I still don't think I have what I need to make this work yet.

Any suggestions for getting a handle on the time dilation especially would be greatly appreciated!

 

[PeterDonis]

an equation derived in an astrobiology class

Can you give a reference to an online derivation? Or at least to a textbook or other source where this comes from?

Also, what kind of tidal heating are you imagining? Is the planet supposed to be rotating fast, and the tidal effect of the hole slows its rotation down? You haven't sqrtmentioned the planet's rotation period.

a black hole of stellar mass, planetary body the size of our moon, 1000AU orbit

Why such a far orbit? 1000 AU is about 10^{11} times the Schwarzschild radius of the hole, so relativistic effects are negligible, including time dilation. Not only that, but tidal effects are negligible--tidal effects go like the inverse cube of the distance, so the hole's tidal effect on the planet at 1000 AU will be about a billion times smaller than the Sun's tidal effect on Earth. See further comments below.

I get a temperature of about 600K

This can't possibly be right. The Sun's tidal effect on Earth doesn't heat it measurably, and as above, the hole's effect on a planet at 1000 AU will be about a billion times smaller than the Sun's effect on the Earth.

When I alter the parameters to see what would be 'significant' on the order of one part in a thousand, it would require the planet orbit at only 2km, when the Schwarzschild radius is 3000km

You need to check your math. The Schwarzschild radius of a black hole with the mass of the Sun is about 3 km, not 3000 km. For a time dilation factor of one part in a thousand, you would want an orbit at roughly 500 times the Schwarzschild radius.

 

[Janus]

I have to agree that 1000 AU tidal heating would be insignificant. Since you do not give a rotation rate in your formula but do give the distance variation due to the eccentricity of the orbit, I assume you are dealing with tidal heating like that which Io experiences as it movie in and out from Jupiter.

The formula I found for tidal heating is:

q= \frac{63 p n^5 r^4 e^2}{38 u Q}

where p is the density of the body
n is the mean orbital motion (2 pi divided by the orbital period)
r is the radius of the body
e is the eccentricity of the orbit
u is the shear modulus ( a measure of the elasticity of the body)
Q is the dissipation factor ( the rate of energy loss of a oscillating system)

The last two are bit difficult to put numbers to, but we can do some comparison using Io as a baseline. We can compare the tidal heating of Io now to what it would get placed 1000 AU from the Sun with an eccentricity of 0.025
Plugging in Io's present values, we get an answer of 1493.5/(uQ). changing n to the needed value for orbiting at 1000 AU from the Sun and e to 0.025, we get and answer of 5.38e-30/(uQ). Meaning Io presently receives 2.78e32 times more tidal heating than it would if we moved it to your suggested orbit.