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Author Topic: How does one calculate the escape velocity starting at the center of a planet?  (Read 14604 times)

jeffreyH

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How does one calculate the escape velocity starting at the center of a planet?
« on: 05/02/2014 10:05:05 »
I am looking for an equation that will always result in a value of 1 above a certain value. So if we have a range between 0-n, n will be the value above which the result will always be 1. Am I asking too much?
« Last Edit: 07/02/2014 23:10:58 by CliffordK »

RD

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Re: An impossible equation?
« Reply #1 on: 05/02/2014 10:41:18 »
A limit of a sequence could converge to 1 as n increases ...
e.g.   n sin(1/n)

Quote
n      n sin(1/n)
1       0.841471
2       0.958851
...
10      0.998334
...
100    0.999983
http://en.wikipedia.org/wiki/Limit_of_a_sequence

n=1000000 ,    n sin(1/n)=0.999999999999833

[ the formula uses sin in radians not degrees ]
« Last Edit: 05/02/2014 11:00:57 by RD »

CliffordK

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Re: An impossible equation?
« Reply #2 on: 05/02/2014 11:03:22 »
All of your limit equations approach 1, but never quite equal 1.  Is that what you want?

Did you say your equation has to equal anything other than 1?

Y = 1x

One could also have an equation like:

Y = 1 + |x| - x

If you want to shift the point where it becomes 1, then you just add in another factor:
Y = 1 + |x-n| - (x-n)

jeffreyH

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Re: An impossible equation?
« Reply #3 on: 05/02/2014 11:16:54 »
I don't think what I am asking is possible. At least not without horrendous mathematics. I have worked round the issue. The jpeg attached shows what I was doing. The tipping point is at the spherical surface. The figures used are rough estimates of values without spherical components. I am working on that part of the calculation. So don't take these as correct.
« Last Edit: 05/02/2014 11:18:50 by jeffreyH »

evan_au

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Re: An impossible equation?
« Reply #4 on: 05/02/2014 11:17:20 »
Software notation gives us a very compact notation for describing a function like this.

Depending on the computer language, it could be as simple as "(n>N)", where:
• n is an integer variable (or it could be a floating-point variable)
• N is an integer constant
• This expression returns 0 if n<=N
• This expression returns 1 if n>N

jeffreyH

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Re: An impossible equation?
« Reply #5 on: 05/02/2014 11:30:44 »
Internally this function bears a remarkable similarity to the atomic asymtotic freedom principle. The plateau effect nearing the surface is also interesting. If we take this as an analogy of the atom then electron orbitals would congregate around this plateau.
« Last Edit: 05/02/2014 11:33:16 by jeffreyH »

CliffordK

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Re: An impossible equation?
« Reply #6 on: 05/02/2014 12:01:39 »
I think Evan is right, some kind of if-then-else statement would be easiest.

My absolute value equation above will give you a nice linear progression that you could use, but it would just make all of your equations unnecessarily complicated.

As far as your escape velocity calculation, I think there is a problem.
Say you start 1000 miles below the surface of the Earth.  The gravity at the starting point is less than at the surface, but to actually escape from the planet, one has to travel up the hole to the surface, and continue through the atmosphere and into space.  Thus, starting at the bottom of the hole, the escape velocity must be greater than it would be starting at the surface.

jeffreyH

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Re: An impossible equation?
« Reply #7 on: 05/02/2014 12:26:46 »
I think Evan is right, some kind of if-then-else statement would be easiest.

My absolute value equation above will give you a nice linear progression that you could use, but it would just make all of your equations unnecessarily complicated.

As far as your escape velocity calculation, I think there is a problem.
Say you start 1000 miles below the surface of the Earth.  The gravity at the starting point is less than at the surface, but to actually escape from the planet, one has to travel up the hole to the surface, and continue through the atmosphere and into space.  Thus, starting at the bottom of the hole, the escape velocity must be greater than it would be starting at the surface.

If gravity cancels at the centre then movement upwards is easiest surely and increases in an exponential manner from there. You have to take into account the partial gravitational cancellations all the way up to the surface. This is still a work in progress and once I have the proper spherical terms the profile may look rather different so any conclusions will have to wait. I have no idea how to test this but if true it would show that gravitation has the same properties as some of the atomic mechanisms. Of course I may also be talking rubbish.

jeffreyH

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Re: An impossible equation?
« Reply #8 on: 05/02/2014 12:38:30 »
Take point P0 to be at the centre of gravitation. Take point P1 to be n diatance from Po. As long as n < r then it should get harder to move from P0 the nearer we get to distance n. If n > r then this situation is revered and assumes an inverse square formula. So to get to P1 from P0 will be easier and require less energy than moving from P1 to Pn. Again Pn must reside at a distance less than r from the centre.

jeffreyH

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Re: An impossible equation?
« Reply #9 on: 05/02/2014 12:42:47 »
I think Evan is right, some kind of if-then-else statement would be easiest.

My absolute value equation above will give you a nice linear progression that you could use, but it would just make all of your equations unnecessarily complicated.

As far as your escape velocity calculation, I think there is a problem.
Say you start 1000 miles below the surface of the Earth.  The gravity at the starting point is less than at the surface, but to actually escape from the planet, one has to travel up the hole to the surface, and continue through the atmosphere and into space.  Thus, starting at the bottom of the hole, the escape velocity must be greater than it would be starting at the surface.

Sorry I see your point. This is an escape velocity gradient. This is not saying that if you start at point P within the mass there will be a constant escape velocity. Internally this is an increasing vector function requiring more input of energy the further out one travels. This makes it HARDER to escape from an internal gravity well and not easier.

jeffreyH

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Re: An impossible equation?
« Reply #10 on: 05/02/2014 12:54:14 »
Just out of interest the function I have worked out as 1 - f(n) where n approaches zero.

JP

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Re: An impossible equation?
« Reply #11 on: 05/02/2014 13:55:25 »
The function you're looking for is the Heaviside step function H(x).  Or to get the effect you want, 1-H(x).

http://en.wikipedia.org/wiki/Heaviside_step_function

jeffreyH

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Re: An impossible equation?
« Reply #12 on: 05/02/2014 16:33:27 »
The function you're looking for is the Heaviside step function H(x).  Or to get the effect you want, 1-H(x).

http://en.wikipedia.org/wiki/Heaviside_step_function

I may yet need this so thanks for the link.

jeffreyH

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The similarities between a model of gravity and the atom
« Reply #13 on: 05/02/2014 18:53:55 »
I will be posting some results here in advance of posting a paper to a uk.arxiv.org.

RD

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Re: An impossible equation?
« Reply #14 on: 05/02/2014 19:46:59 »
« Last Edit: 05/02/2014 19:51:05 by RD »

jeffreyH

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Re: An impossible equation?
« Reply #15 on: 05/02/2014 21:07:54 »
Snap ?

http://en.wikipedia.org/wiki/Gravity_of_Earth#Depth

The plateau profile is more rounded on the wiki graph as I haven't added the spherical geometry calculations to mine. As the wiki graph relates to density then my original assertion that mass-energy density affects gravitational feedback must have some credence.

JP

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Re: An impossible equation?
« Reply #16 on: 06/02/2014 00:07:37 »
What do you mean by gravitational feedback?  The effects due to nonlinearity of Einstein's field equations is going to be negligible (and is ignored in these graphs).

jeffreyH

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Re: An impossible equation?
« Reply #17 on: 06/02/2014 00:36:49 »
What do you mean by gravitational feedback?  The effects due to nonlinearity of Einstein's field equations is going to be negligible (and is ignored in these graphs).

I am not saying they are not negligible.

jeffreyH

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Re: An impossible equation?
« Reply #18 on: 06/02/2014 00:38:12 »
A new graph but with the spherical geometry. This plot doesn't start from the centre of gravity. It is off centre so I need to correct that. Very unhelpfully Excel seems to continue the line.

JP

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Re: An impossible equation?
« Reply #19 on: 06/02/2014 01:09:53 »
What do you mean by gravitational feedback?  The effects due to nonlinearity of Einstein's field equations is going to be negligible (and is ignored in these graphs).

I am not saying they are not negligible.

Ah, but they are.  Unless you're planning on replacing general relativity with your own theory.

jeffreyH

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Re: An impossible equation?
« Reply #20 on: 06/02/2014 02:25:13 »
What do you mean by gravitational feedback?  The effects due to nonlinearity of Einstein's field equations is going to be negligible (and is ignored in these graphs).

I am not saying they are not negligible.

Ah, but they are.  Unless you're planning on replacing general relativity with your own theory.

Now that would be silly.

JP

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Re: An impossible equation?
« Reply #21 on: 06/02/2014 02:34:17 »
What do you mean by gravitational feedback?  The effects due to nonlinearity of Einstein's field equations is going to be negligible (and is ignored in these graphs).

I am not saying they are not negligible.

Ah, but they are.  Unless you're planning on replacing general relativity with your own theory.

Now that would be silly.

Indeed, and it would belong in the New Theories section of the forum.  :)

jeffreyH

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Re: An impossible equation?
« Reply #22 on: 06/02/2014 12:31:06 »
I have just saved a PDF of the PREM results. I am going to look at the core data to see what conclusions were reached. I did notice they mentioned one unexpected result. It would be better if my calculations were able to follow the PREM results. Oh Joy more fun mathematics.
« Last Edit: 06/02/2014 12:33:52 by jeffreyH »

jeffreyH

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Re: An impossible equation?
« Reply #23 on: 06/02/2014 17:16:19 »
There was an error in the other plots. I picked up the wrong value for the Gravitational Constant. OOPS! So here is a new plot. I have zoomed in to investigate nearer the centre of gravity but that is not shown in this plot as the number of errors due to rounding made the data questionable. The pink line on the attached plot shows the point where we reach the spherical surface. Here it translates to the usual inverse square relationship. There is a sharp drop in Ve towards the centre. My hunch is this will coincide with the Schwarzschild radius. I still have to find a method to confirm this. If I find a match with rs then this should validate the model.

jeffreyH

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Re: An impossible equation?
« Reply #24 on: 06/02/2014 17:35:36 »
This is a first attempt at a zoom with less uncertainty. The x axis shows metres from the centre of gravity.

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Re: An impossible equation?
« Reply #24 on: 06/02/2014 17:35:36 »