Escape velocity and Dark Matter

[Zman]

I was wondering how verifiable is the escape velocity equation.
V=SQRT(2GM/r)

How accurately has this equation been verified?

It does not take into account dark matter.

(Related ramblings)
The escape velocity from the earth’s surface is about 11.2 km/s

But there is also air resistance. How much extra velocity (above 11.2 km/s) is required to escape Earth’s gravity?

I am not after a specific value, I would just like to know if the energy requirement increase is significant or not.

How much leeway in terms of fuel does NASA give when firing rockets into space?

 

[sylas]

I was wondering how verifiable is the escape velocity equation.
V=SQRT(2GM/r)

How accurately has this equation been verified?

It's a consequence of Newtonian physics, and so it has been verified to the extent that Newtonian physics has been verified... pretty darn accurate.

It does not take into account dark matter.

The M refers to matter, whether it be dark or baryonic or anything else. A more correct statement is that it does not make any distinction between what kind of matter is involved. Nor should it.

(Related ramblings)
The escape velocity from the earth’s surface is about 11.2 km/s

But there is also air resistance. How much extra velocity (above 11.2 km/s) is required to escape Earth’s gravity?

That's an odd question... air resistance will depend on the shape of the projectile, so there's no one answer. And it is not relevant anyway. We put things into space not by giving them an escape velocity at the surface, but by using a rocket that keeps powering up all the way to orbit or beyond. The escape velocity would be more relevant if you had a gun or other sudden impulse applied at the surface of an airless planet.

I am not after a specific value, I would just like to know if the energy requirement increase is significant or not.

I don't think it is all that significant. I don't think air resistance makes all that much difference for a rocket. It would make some difference, of course; but the rocket is shaped to keep the air resistance small, and it soon gets up into pretty thin atmosphere.

But don't quote me on that. I can't given you a specific number... so it's good you didn't ask for one.

How much leeway in terms of fuel does NASA give when firing rockets into space?

Not much, I think. Something like the shuttle is placed into orbit mainly by boosters that are discarded. It carries more fuel for maneuvers in orbit; but the amount of fuel to get into orbit doesn't change enough to make "leeway" something to worry about.

Cheers -- sylas

 

[Zman]

Thanks Sylas

I was also thinking about MOND but maybe you have answered that question already.

With escape velocity, it can be derived using Potential Energy at the Earth’s surface with reference to infinity and equating it to kinetic energy.

But surely we have never tested dropping a body from infinity to the surface of a large mass?

This doesn’t detract from all the confirmation we have regarding Newtonian physics.
But when we talk about potentially large distances, do we not get general relativity issues that could alter the escape velocity equation?

Cheers Zman

 

[sylas]

This doesn’t detract from all the confirmation we have regarding Newtonian physics.
But when we talk about potentially large distances, do we not get general relativity issues that could alter the escape velocity equation?

Differences are negligible. Same for MOND. The Newtonian approximation is plenty accurate enough, particularly if you are concerned with air resistance or other non-gravity related factors. Those factors are much more significant than the tiny differences between GR, MOND and Newton applied to escape velocity from the Earth.

I am not sure how to do "escape velocity" in GR. It will depend on co-ordinates, I guess. There should be a suitable equation for a lightweight particle measured by a stationary local observer, I would think. I don't know what it is, or if it is different from the Newtonian equation.

Cheers -- sylas

 

[Zman]

Thanks again

I am changing the goal posts slightly here as I think that my thoughts on this issue are a bit clearer.

We know that velocity curves of stars are greater than expected at the edge of the galaxy. We also know that there must be a bigger gravity than Newton’s to keep the stars from flying out.

If a small mass was dropped from such a location towards the centre of the galaxy a slight increase in gravity would be compounded as a large increase in velocity.

The Newtonian escape velocity equation would surely give too small a result.

Cheers Zman

 

[sylas]

We know that velocity curves of stars are greater than expected at the edge of the galaxy. We also know that there must be a bigger gravity than Newton’s to keep the stars from flying out.

If a small mass was dropped from such a location towards the centre of the galaxy a slight increase in gravity would be compounded as a large increase in velocity.

The Newtonian escape velocity equation would surely give too small a result.

Actually, Newton gravity is just fine... as long as there is more matter in a galaxy than we can see in the form of stars and dust. That is, a galaxy might have a lot of "dark matter", in which case the equations all continue to work as usual.

There are other reasons to suspect that this is the case; but it's still worth looking at other possibilities. MOND was a possible alternative, although that doesn't work well for explaining gravitational lensing seen in deep space.

Cheers -- sylas

 

[rcgldr]

I was wondering how verifiable is the escape velocity equation.
V=SQRT(2GM/r). How accurately has this equation been verified?

Math based on this equation has been used to successfully sent satellites to other planets, as well as put a few men on the moon and return them. Apollo mission involved exceeding escape velocity to reduce the time it took to get into lunar orbit (which required using thrust to slow it down again). On the return trips, Apollo again exceeded escape velocity, but was aimed to hit Earth atmosphere at the correct angle to slow it down without burning it up.

From the Earth's surface, because of the atmoshpere, it's unlikely to achieve escape velocity because just about any real object with mass would burn up before achieving the required speed.