What should the mass of a planet be in-order for its escape velocity to be the speed of light? Is it even possible? What will it look like from an outside viewer? Will it even be visible in the human eye spectrum?
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2$\begingroup$ Escape velocity depends on mass and radius. If you keep adding rocks and atmosphere to a planet, it will collapse into a neutron star, and no longer be a planet. If you keep adding mass it will collapse to a black hole at about 2 solar masses. $\endgroup$– Keith McClaryDec 6, 2020 at 4:00
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2$\begingroup$ Welcome to Astronomy Stack Exchange! This is a great question, don't worry if this is closed as duplicate of a previously posted question with an answer that also answers this one. $\endgroup$– uhohDec 6, 2020 at 5:14
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$\begingroup$ How does the linked question (can you escape a black hole) relate to this (what is the mass of body with a escape vel near speed of light) This question is explictly not about black holes. $\endgroup$– James KDec 6, 2020 at 7:19
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4$\begingroup$ @JamesK Any object whose escape velocity actually reaches the speed of light is a black hole by definition. $\endgroup$– Steve LintonDec 6, 2020 at 8:07
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2$\begingroup$ I agree with JamesK that the linked question is not a suitable dupe target for this question. And as StephenG suggests, the impossibility of escaping from a black hole is because there are no paths leading away from the BH, your velocity isn't really relevant. $\endgroup$– PM 2RingDec 6, 2020 at 15:56
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2 Answers
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As @KeithMcClary says in his comment, escape velocity depends on both mass and radius. The smaller the radius for a given mass the higher the escape velocity. So if you could somehow compress the Earth until it was just a few centimeters across, its escape velocity would approach the speed of light.
At the other extreme, if you filled a space twice the diameter of Earth's orbit with copies of the Earth, without doing any compressing at all, it would already be a black hole.
If you just pile matter onto a planet and let its own gravity compress it, you need about 2 solar masses before it is close to being a black hole (at which point it is 10-20 km across).
answered Dec 6, 2020 at 8:15
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Escape velocity can be descibed by
$$v=\sqrt{\frac{2GM}{r}}\tag{1}$$
Where $v$ is the escape velocity, $r$ is the distance from the mass (in case of the planet, the minimum distance is the radius of the planet), $M$ is the mass and $G$ is Newton's Gravitational Constant.
If the planet should have the excape velocity of the speed of light, it has to be so dense that you have to consider relativistic effects. In General Relativity, the Schwarzschild Radius of a black hole (this radius is the distance at which the escape velocity equals $c$, which is exactly what you want) is described by:
$$r=\frac{2GM}{c^2}\tag{2}$$
Whch can be solved for $m$:
$$M=\frac{rc^2}{2G}\tag{3}$$
So if you have either a fixed radius of mass, you can easily calculate the missing parameter using these equations.
The problem, as mentioned before, is that for a planet (or any other body) to have an escape velocity of the speed of light, it must be extremely dense. In fact, if it is dense enough to have $v_{esc} = c$, the body is a black hole (Think about it - the event horizon of a blackhole if the distance where the escape velocity equals the speed of light, so anything beyond this horizon cannot escape as it would require a speed greater than $c$).
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2$\begingroup$ It is kind of a coincidence that the Newtonian escape velocity calculation results in the the same value as the Schwartzchild radius in GR. # $\endgroup$– James KDec 6, 2020 at 17:38
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1$\begingroup$ @JamesK Thanks for your comment. I always thought that one of these formulae is derived from the other, but it actually doesn't make sense because you have to consider relativistic effects. I will edit my answer $\endgroup$– jng224Dec 6, 2020 at 18:14
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