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Author Topic: How does one calculate the escape velocity starting at the center of a planet?  (Read 14618 times)

Offline jeffreyH

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BTW I am looking for a better site for evidence of g underground. That one was pseudoscience at its worst.
 

Offline jeffreyH

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Before this thread is transferred or I move posts into new theories I have this:

http://earthdynamics.org/papers-ED/2010/2010-Steinberger-etal-Icarus.pdf

This discusses investigations of gravity anomalies with respect to density variations so is not new theory material but I will be looking into this further.
 

Offline alancalverd

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Have you a link for the wikipedia reference. I can't find it?

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

The equation I quoted is about 2/3 of the way down the article
 

Offline jeffreyH

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Have you a link for the wikipedia reference. I can't find it?

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

The equation I quoted is about 2/3 of the way down the article

I'd already sorted the Ve calculations I was more interested in a plot of Ve from the centre of gravity. I am transferring posts on this subject to here if anyone is interested in carrying on the debate.

http://www.thenakedscientists.com/forum/index.php?topic=50260.0
 

Offline jeffreyH

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Re: The similarities between a model of gravity and the atom
« Reply #54 on: 08/02/2014 03:25:39 »


If we consider a spherical mass as being made up of a gradient of internal spheres going down to the core we can treat each one as a separate gravitational source working out from the centre. At the very centre all gravitation cancels. At any point P out from the radius we will have a sphere at which P will be coincident with a spherical surface. We could treat our source of gravitation as being concentrated within each of these spheres as we move outwards and calculate Ve using this method. I will show the problems with this approach over the following posts.
« Last Edit: 08/02/2014 03:32:09 by jeffreyH »
 

Offline jeffreyH

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Re: The similarities between a model of gravity and the atom
« Reply #55 on: 08/02/2014 03:34:51 »
The image below as posted previously by Cliffordk shows the first problem with the internal spheres method. These images firstly only take into consideration the gravitation radiating outwards. Gravitation radiates in all directions even internally. This multi-directional radiation needs to be taken into account when considering gravitational cancellation.
« Last Edit: 08/02/2014 03:43:54 by jeffreyH »
 

Offline CliffordK

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I went ahead and moved this discussion to New Theories.
Your new topic appears to be related to atoms and not planets, so I didn't merge the two topics.

As RD had mentioned quite a while ago, if the core is denser than the crust, then the acceleration due to gravity does increase slightly as one descends through the crust, or deeper into the ocean.  We currently do not have any technology that would allow drilling into the mantle, or core of the planet. 

See:
https://en.wikipedia.org/wiki/Gravity_of_Earth#Depth
 

Offline jeffreyH

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I went ahead and moved this discussion to New Theories.
Your new topic appears to be related to atoms and not planets, so I didn't merge the two topics.

As RD had mentioned quite a while ago, if the core is denser than the crust, then the acceleration due to gravity does increase slightly as one descends through the crust, or deeper into the ocean.  We currently do not have any technology that would allow drilling into the mantle, or core of the planet. 

See:
https://en.wikipedia.org/wiki/Gravity_of_Earth#Depth

Sorry forget the other thread and we'll stick with this one. Well there must be another way other than drilling down that far. If I'm wrong then I'd rather know that sooner than later.

I'm not sure I agree with the method of gravitational cancellation used in the wikipedia entry. I'll have to put together an illustration of exactly what I mean by that. I can see how it makes sense with the internal spheres idea but I think it is slightly different than that.
« Last Edit: 08/02/2014 07:05:15 by jeffreyH »
 

Offline alancalverd

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I'd already sorted the Ve calculations I was more interested in a plot of Ve from the centre of gravity.

Just take the square root of the g/r plot. The PREM graph (see reply #14) is probably the best estimate for this planet. The shell integrals and presumed densities are all pretty much as I recall from undergraduate days.
« Last Edit: 08/02/2014 09:41:50 by alancalverd »
 

Offline jeffreyH

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Let's go back a bit. Schwarzschild radius rs=2Gm/c^2. This is what initially puzzled me. Where did the 2 come from. From Jim Baggott's book "Gravity" I got the answer. Look at Clifford's cubes example. Contributions criss-cross everywhere not only outwards from the centre. This gives a contribution of the form 1 + 1/2 + 1/4 + 1/8 + 1/16. the limit of this sequence, or the more complex exact sequence, has a limit towards 2. This gives our multiplier of 2. While gravity kernels will upset this uniformity I am ignoring them in my model as they can be added later and will only complicate the model. As this derivation results in a uniform internal field distribution we can treat it as smooth and continuous. Any shell theorem will thus fail to describe the effects.

As an additional note before I finish this post. I will set out the method of proof in later posts.
« Last Edit: 08/02/2014 20:10:24 by jeffreyH »
 

Offline jeffreyH

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I have an adjustment that needs to be made to the model which I hadn't taken into account. I need a distance function to be applied after the cancellation function. This should still describe a curve but should describe a smooth trend towards zero g at the centre. It is something I should have taken into account but hey this is a new theory so it is a work in progress. Once this is completed it should make more sense.
 

Offline jeffreyH

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Houston we have a problem (minor issue). At the surface the plot doesn't match so I have to address this. There was a distance adjustment issue which I should have spotted. This now better resembles RDs data from PREM.
« Last Edit: 08/02/2014 22:22:38 by jeffreyH »
 

Offline jeffreyH

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I have graphed the data for the earth compressed to a radius of just over 6 metres.
 

Offline jeffreyH

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I will be updating this post to explain the diagrams below. The first things of note are the spherical domes described by s and c in figure 2. These are important aspects of the model. They visually describe gravitational cancellation methodology.

In figure 1 we see the situation where our mass is at the centre of gravity where all forces cancel. The vector direction of travel is shown by V. The mass marked as s as can be thought of as subtractive as its gravitation is in the direction of travel and can be cancelled by f = a - s. In figure 2 our mass is moving towards the surface and s, still needing to be cancelled, no longer equals a. We now need to cancel the equivalent force. Inverting this spherical dome gives c, where all mass distances are equivalent and so all forces equalize. Cancelling this way gives us a new portion of the mass at a. a now contains the only mass that can be considered as acting against the momentum of our moving mass. However a can now be thought of as having a different centre of gravity to be used in our Ve calculation. If this is not adjusted correctly then the plot of internal gravitation will not meet with the inverse square plot at the surface. The line L is the demarcation of the sum of positive gravitational vectors in s and negative gravitational vectors in a. Figure 3 shows that L is coincident with the spherical surface indicating that no cancellations exist as s is now equivalent with free space.
« Last Edit: 09/02/2014 14:02:37 by jeffreyH »
 

Offline jeffreyH

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I now need to determine a calculus function to use in the adjustment to the original centre of gravity. This may take a while. I will also have to adjust for mass distances and volumes for force distribution. This final plot will differ from those illustrated so far. I am hoping this final version of the model can be modified to introduce gravity kernels so that the PREM data can be reproduced.
« Last Edit: 09/02/2014 14:36:09 by jeffreyH »
 

Offline jeffreyH

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Here is an example of the problem from another viewpoint. See the attached figure of a hoop-shaped mass with lobes. For vector V0 the gravitational forces from both lobes are the same magnitude and in the same direction. In the case on V1 the two lobes oppose each other in an inverse square relationship. Due to the vector direction the force exerted by the right lobe will decrease with respect to distance travelled along V1. For the right lobe these forces will increase. At a stationary point at the centre of gravity all forces cancel.
 

Offline jeffreyH

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The figure below shows the partial shells that need to be analyzed to obtain the overall gravitational strength at each distance from the mass moving along vector V. The sphere s1 can be considered to be in free space as all gravitation has been mathematically cancelled in this region. The sphere s2 touches the centre point of the spherical dome removed via cancellation and can also be considered to be in free space. These can be ignored. All outer spheres from this point up to a coincidence with the outer surface of the whole mass contain the portions of the mass to be considered as gravitation contributors. All shells have their centres coincident with the centre of the mass at vector V.
The calculation of a modified centre of gravity and the gravitational strength contributions are non-trivial functions.
« Last Edit: 09/02/2014 22:23:07 by jeffreyH »
 

Offline jeffreyH

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Attached is the plot with centre of gravity corrected so the two parts of the plot meet at the surface. I will be sorting the calculus to derive a comparison plot to determine any differences between that and this simplified model.
« Last Edit: 09/02/2014 23:11:46 by jeffreyH »
 

Offline jeffreyH

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Calculating for time dilation there is a 0.7 ns difference between the core and surface of the earth. This can be ignored and is unlikely to be verified experimentally. This results in a difference of 60.48 microseconds a day.
 

Offline jeffreyH

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This model may be used to study gravitational collapse, at least in the early stages. By adjusting the value for the radius to the surface increases in escape velocity can be studied. As ordinary matter cannot travel faster than light then, when the escape velocity at the surface (event horizon) reaches c, the internal velocity should still decrease to zero at the centre. This may run counter to accepted theory but is indicated by the model.
 

Offline jeffreyH

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The calculation of Ve from the centre of a mass in terms of gravitational cancellation can be reduced to a very simple formula for an idealized sphere with uniform density. Because the mass is moving away from the centre of gravity the shape of the mass that is left after removing those portions cancelled out is perfectly symmetrical. Thus only a small wedge-shaped slice that includes the centre of gravity need be considered for purpose of integration. This can then be multiplied by a factor based on the mass size to produce the total gravitational force operating on the mass moving away from the centre. While this simplified version of the calculations is perfectly suited to the situation where a mass is moving directly away from the centre of gravity in a straight line trajectory it will fail to model more complex situations. However, as a starting point, it can be adapted over time to model those more complex situations. The main objective of developing this model is to determine the exact nature of the curve for Ve away from the centre of gravity and whether it can be validated experimentally. A secondary objective is to determine if this model can be developed to model a mass under gravitational collapse and simplify some of the procedures currently used for this type of model.
« Last Edit: 20/02/2014 03:59:21 by jeffreyH »
 

Offline jeffreyH

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I have saved this PDF http://www-gpsg.mit.edu/12.201_12.501/BOOK/chapter2.pdf

In section 2.2 I have found that I should be able to use zonal spherical harmonics with assumptions of latitude perpendicular to the direction of Ve. From what I have read so far the contributions should be greater nearer the surface than at the centre of gravity as the distance from the non-cancelled mass declines as the escaping mass nears the surface. Calculations nearer the centre of gravity may as well consider the mass to be spherical and uniform and ignore mass distribution as it can be considered to be a greater distance away than the non-cancelled mass when approaching the surface.
« Last Edit: 22/02/2014 22:14:42 by jeffreyH »
 

Offline jeffreyH

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As a quick estimation the following can be used to find the volume of a spherical segment with two bases. This ignores curvature with respect to the mass moving away from the centre of gravity but can provide an indication of the resulting curve profile.

V = 1/6pi * h(3a^2 + 3b^2 + h^2)

I will post results once done.
« Last Edit: 23/02/2014 07:22:26 by jeffreyH »
 

Offline jeffreyH

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Just as a reference I have found a PDF of a 1939 paper by Einstein online.

http://www.cscamm.umd.edu/tiglio/GR2012/Syllabus_files/EinsteinSchwarzschild.pdf

Of particular interest is the statement on page numbered 928.

"Equation (14) represents a complicated relation between the particle density
n and the function a representing the gravitational field. The limiting case,
however, in which the gravitating particles are concentrated within an infinitely
thin spherical shell, between r = ro- A and r = ro, is comparatively simple.
Of course, this case could only be realized if the individual particles had the
rest-volume zero, which cannot be the case. This idealization, however, still is
of interest as a limiting case for the radial distribution of the particles"
« Last Edit: 25/02/2014 02:20:12 by jeffreyH »
 


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