Kepler's 3rd Law: Constant Orbital Velocity Around Black Holes?

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In summary, Kepler's 3rd law is a consequence of Newtonian Mechanics and does not hold in cases of strong gravity or velocities close to the speed of light. This can be seen in the example of an object orbiting a black hole, where the mass is not constant and there is a radius within which stable orbits are not possible. While Kepler's laws work well in most realistic cases, there is no general rule for the relativistic case and each situation must be evaluated individually. In this case, the object could potentially be torn apart by gravitational forces before any calculations can be made.
  • #1
Suraj M
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While proving Kepler's 3rd law we get the equation
$$ \frac{ΔA}{Δt} = \frac{2L}{m}$$
we say L and m are constant,so aerial velocity is constant!
But consider a body going around a black hole really quickly, then the mass would not be constant, right?
So is Kepler's 3rd law violated or am i missing something?
 
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Kepler's laws are a consequence of Newtonian Mechanics. They don't hold in the case of strong gravity or velocities close to the speed of light. For example as you get close to a black hole, there is a radius (clued the ISCO, for Innermost Stable Circular Orbit) within which there are no possible stable orbits. This is still outside of the event horizon.
 
  • #3
Oh Okay, is there some defined relation that works in all realistic cases, relating Time period and radius?
 
  • #4
Kepler's laws work really well in almost all realistic cases. I suspect it is very difficult to find an astronomical case where the velocity is so high or the gravity so strong that Kepler's laws don't apply. In the relativistic case (velocities close to the speed of light, or very strong gravity), I don't think there is any general rule like Kepler's laws. You have to work out each case.
 
  • #5
phyzguy said:
In the relativistic case (velocities close to the speed of light, or very strong gravity)
I guess the object would just get torn apart due to the gravitational force, before we calculate anything!
 
  • #6
Suraj M said:
I guess the object would just get torn apart due to the gravitational force, before we calculate anything!
The object could be small which would make tidal forces negligible, but its trajectory could still be highly non-Newtonian.
 

Related to Kepler's 3rd Law: Constant Orbital Velocity Around Black Holes?

1. What is Kepler's 3rd Law?

Kepler's 3rd Law, also known as the Harmonic Law, states that the square of a planet's orbital period is directly proportional to the cube of its semi-major axis. In simpler terms, it relates the time it takes for a planet to orbit its star to the distance between them.

2. How does Kepler's 3rd Law apply to black holes?

Kepler's 3rd Law also applies to objects orbiting black holes. However, for objects orbiting very close to the black hole, the orbital period and semi-major axis must be recalculated using the Schwarzschild radius, which takes into account the intense gravitational pull of the black hole.

3. Why is orbital velocity constant around black holes?

The intense gravitational pull of a black hole causes objects in orbit to travel at incredibly high speeds. However, due to the conservation of angular momentum, the orbital velocity remains constant as the object gets closer to the black hole. This means that as the object's distance from the black hole decreases, its speed increases to maintain a constant orbital velocity.

4. How does the mass of the black hole affect the orbital velocity?

The mass of the black hole directly influences the orbital velocity of objects around it. The greater the mass of the black hole, the stronger its gravitational pull and the higher the orbital velocity of objects orbiting it. This is why objects orbiting supermassive black holes, like those at the center of galaxies, have much higher orbital velocities compared to objects orbiting smaller black holes.

5. Can Kepler's 3rd Law be applied to objects orbiting other celestial bodies?

Yes, Kepler's 3rd Law can be applied to objects orbiting other celestial bodies, such as stars, planets, and moons. It is a fundamental law of planetary motion and has been used to accurately predict the orbits of objects in our solar system and beyond. However, as mentioned before, the calculations may need to be adjusted for extreme cases, such as objects orbiting black holes.

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