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Escape speed at a distance r from the center of mass of a sphere is se= sqrt (2gr) where g is the gravitational acceleration at r. At the center of a sphere both g and r are zero, and increase smoothly (if the sphere is homogeneous) to the surface. For the nonhomogeneities of the earth, see the plot of g versus r given in the Wikipedia reference, which takes account of the depth/density profile of the planet.
RD posted a couple of models for the internal gravity of the Earth. The gravity should go to zero at the middle of the body.Were you looking at theoretical singularities with all the mass concentrated at a single point? That is a completely different beast than a planet with the mass surrounding a person as one bores towards the center.Your escape velocity should always increase as you move towards the center of the body. I.E. no jagged lines as you have on some of your graphs. I threw this chart together last night. Acceleration Due to Gravity.I drew a linear line from the center of Earth to the surface, which I believe is indicative of a constant density of Earth. Current theories indicate an iron core, and greater density in the center of Earth, but for the purposes of my estimate, this was good enough. [ Invalid Attachment ] Equations: (d is the distance to the center of Earth).From center to surface:Acceleration due to gravity is: 9.8*d/(6371 km)From the surface into space, Acceleration due to gravity is:I'll try to convert my "acceleration" to a "velocity" soon.
Technically you can divide your sphere into cubes, and calculate the acceleration due to gravity to each sub-cube, or perhaps one could do it with concentric rings or shells. According to RD's Notes, the linear line represents the acceleration due to gravity if the planet or body has a constant density.Anyway, it should be good enough for a crude estimate. Otherwise one would need well defined acceleration or density functions.
Jeffrey, while I appreciate that you're discussing gravity and making calculations using your model, we have a strict policy here that we don't allow new models to be proposed and discussed outside of the New Theories sub-forum. A major reason for this is that this is mostly a science Q&A board and users who come here seeking answers (without any scientific background) will find the forum very confusing if new theories are mingled in with accepted theories. I'd like to ask that you keep this thread to asking questions about Newtonian gravity, general relativity and perhaps some questions about peer-reviewed theories of quantum gravity. Please keep posts discussing your own models and their results to a thread in the New Theories section.Thanks!
Have you a link for the wikipedia reference. I can't find it?
Quote from: jeffreyH on 07/02/2014 22:48:48Have you a link for the wikipedia reference. I can't find it?http://en.wikipedia.org/wiki/Escape_velocityThe equation I quoted is about 2/3 of the way down the article
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
I'd already sorted the Ve calculations I was more interested in a plot of Ve from the centre of gravity.