Derivation of Escape Velocity Inconsistency

Hey, I'm a first year Astrophysics student, and in revising the Schwarzchild radius, I wanted to derive it and so I started by deriving the escape velocity from first principles, then rearrange to get the Schwarzchild Radius. Child's play, really.

However, depending from where you come from, either from energy or rotational motion, you end up being a factor of $$\sqrt{2}$$ out:

First Derivation, from Newtonian Mechanics

I am ignoring vector notation for speed

$$F = \frac{GMm}{r^{2}} = ma$$

$$a = \frac{GM}{r^{2}}$$

However,

$$a = \frac{v^{2}}{r}$$

Therefore,

$$\frac{v^{2}}{r} = \frac{GM}{r^{2}}$$

Rearranging for v gives

$$v = \sqrt{\frac{GM}{r}}$$

Second Derivation, from Energy

We can assume that by units,

$$E = \frac{1}{2}mv^{2} = \frac{GMm}{r}$$

$$E = \frac{1}{2}v^{2} = \frac{GM}{r}$$

$$E = v^{2} = \frac{2GM}{r}$$

$$v = \sqrt{\frac{2GM}{r}}$$

Are my mathematics skills not up to scratch, or is it that for two equally valid derivations, we get two equally valid equations, that mean the same thing, but are not equal to one another?

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I suspect that in your first derivation, you are not calculating the escape velocity, but instead the velocity needed to keep the test particle in circular orbit of radius r. The 3rd equation, thats the bad one:

$$a = \frac{v^{2}}{r}$$

This is the centrifugal acceleration of a particle moving with speed v in a circular orbit of radius r - its not related to the escape velocity.

Further, I see a more fundamental problem with your approach. You are using Newtonian Mechanics, but what you aim at is the Schwarzschild radius, which is a concept based on the General Theory of Relativity. So you have to use the equations of General Relativity right from the start (Though I have no clue how to do this, sorry...)

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ah so the first derivation was for orbital velocity, whereas the second was for escape velocity... I hate it when people don't label things correctly, or when I cannot read.

Janus
Staff Emeritus
Gold Member
Further, I see a more fundamental problem with your approach. You are using Newtonian Mechanics, but what you aim at is the Schwarzschild radius, which is a concept based on the General Theory of Relativity. So you have to use the equations of General Relativity right from the start (Though I have no clue how to do this, sorry...)

Actually, you can derive the Schwarzschild radius from Newtonian escape velocity by simply setting the escape velocity to c.

Astronuc
Staff Emeritus
In the first equation, one had a balance of force (acceleration) between gravitational force/acceleration and centripetal force/acceleration.

In the energy equation, one simply equates the change in gravitational potential energy required to go from r to infinity with the change in kinetic energy at r and assuming zero kinetic energy at infinity, i.e. one coasts to a stop at infinity.

So escape velocity is the minimum velocity to escape without additional thrust after leaving from r.

Actually, you can derive the Schwarzschild radius from Newtonian escape velocity by simply setting the escape velocity to c.

Oooopsie... of course you are right and I was wrong, checked it at some other sources. Though I think its really strange - I would have thought that, if there is one place in the universe where newtonian mechanics is not applicable anymore, then this would be near a black hole ! Strange ! Anyway, thanks a lot for the correction.

Kurdt
Staff Emeritus
Gold Member
There is another way to derive the escape velocity involving calculus and Newton's second law.

rock.freak667
Homework Helper
There is another way to derive the escape velocity involving calculus and Newton's second law.

Really? How does one do it using that? Have only done it using kinetic energy and the gravitational potential

Kurdt
Staff Emeritus
Gold Member
Really? How does one do it using that? Have only done it using kinetic energy and the gravitational potential

Try it yourself

$$F = m\frac{dv}{dt} = -\frac{GMm}{r^2}$$

rock.freak667
Homework Helper
Try it yourself

$$F = m\frac{dv}{dt} = -\frac{GMm}{r^2}$$

hm...cancel out one of the m's and then say that $\frac{dv}{dt}=v \frac{dv}{dr}$,then integrate both sides and I'll get the same beginning as if I start with ke=gravitational pe. That's the correct way to do it?

Kurdt
Staff Emeritus
Gold Member
hm...cancel out one of the m's and then say that $\frac{dv}{dt}=v \frac{dv}{dr}$,then integrate both sides and I'll get the same beginning as if I start with ke=gravitational pe. That's the correct way to do it?

Yeah pretty much.

still doesn't change the fact that one is a root 2 factor out compared to the other though..

Deriving the Schwarzchild radius isn't the problem, regardless of method. I am just curious as to which escape velocity to use.

I am assuming from Newtonian Mechanics, that the velocity there is actually orbital velocity (the velocity it takes to stay in orbit)

By Conservation of Energy, it is the velocity needed to escape a body of mass M, with a radius of r.

So the Newtonian is actually irrelevant, as it appears that because it does not contain that vital '2', and also from the equations used, it does not match up the logic.

Kurdt
Staff Emeritus