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I'm trying to write a hard sci-fi novel with good accuracy, but when it comes to astronomy, I'm a total amateur.

Here's my fictional planet that orbit a black hole information:

Star Mass = 10000 Solar Masses Planet Mass = 1 Earth Mass

The planet orbit is around 23 AU from the black hole. I did some calculations and it seems it had orbital velocity of 621 km/s. (compared to Earth: 30km/s).

But would it be harder for a rocket escape from the planet or otherwise?

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No, the planet's orbital speed doesn't affect the escape velocity. That's just a function of the planet's mass and radius. However, it does affect the speed you need to escape from your stellar system.

There's a simple relationship between the speed of a circular orbit and escape velocity. The circular orbit speed is given by $$v^2 = GM/r$$ where $G$ is the gravitational constant, $M$ is the mass, $r$ is the orbit radius. But I assume you already know that, since you correctly calculated your planet's orbital speed.

The escape velocity is given by $$v^2 = 2GM/r$$ To calculate the escape speed from the planet, use the planet's radius for $r$. Of course, that ignores air resistance.

But once you've escaped from the planet, if you want to escape from your stellar system, you'll need ~878.3 km/s, relative to the black hole, which is $\sqrt2$ times the planet's orbital speed. That's not going to be easy!

To calculate the orbital speed and escape velocity close to the black hole, we really should use general relativity, but those numbers should be ok at 23 AU. FWIW, the Schwarzschild radius of your black hole is 29532.5 km.

By the way, we only know the value of $G$ to 4 or 5 significant figures. Instead, we normally use the gravitational parameter, which we can measure to much greater precision. That Wikipedia article has a table of values for our Solar System, but JPL has a better one. I have links & info here: https://astronomy.stackexchange.com/a/48616/16685

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  • $\begingroup$ Are you claiming tidal forces do not affect escape velocity? $\endgroup$
    – TimRias
    Commented Oct 8, 2022 at 22:29
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    $\begingroup$ @Tim No, but the tidal force from that black hole is relatively small at 23 AU, so I just ignored it. ;) Consider $10000/23^3\approx 0.822$, and the tidal force from the Sun on Earth is (mostly) negligible. But see leapsecond.com/hsn2006 $\endgroup$
    – PM 2Ring
    Commented Oct 8, 2022 at 22:53
  • $\begingroup$ So, any civillization that live within black hole system would have really hard time escaping their "star system"?? i meant 800+km delta-v is huge, and impossible to achieve with conventional rockets... $\endgroup$
    – Totally Amateur
    Commented Oct 10, 2022 at 14:22
  • $\begingroup$ Probably they need to do research something like nuclear rockets before they could go interstellar... $\endgroup$
    – Totally Amateur
    Commented Oct 10, 2022 at 14:24
  • $\begingroup$ @TotallyAmateur Yes, it would be very hard to leave! But see en.wikipedia.org/wiki/Gravity_assist $\endgroup$
    – PM 2Ring
    Commented Oct 10, 2022 at 15:00

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