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n n sin(1/n)1 0.8414712 0.958851...10 0.998334...100 0.999983
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.
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 [Links inactive - To make links active and clickable, login or click here to register]
Snap ?http://en.wikipedia.org/wiki/Gravity_of_Earth#Depth [Links inactive - To make links active and clickable, login or click here to register]
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).
Quote from: JP on 06/02/2014 00:07:37What 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.
Quote from: jeffreyH on 06/02/2014 00:36:49Quote from: JP on 06/02/2014 00:07:37What 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.
Quote from: JP on 06/02/2014 01:09:53Quote from: jeffreyH on 06/02/2014 00:36:49Quote from: JP on 06/02/2014 00:07:37What 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.
Woo Hoo and there is the Earth's Scwarzschild gravity well.
Ok, Jeffrey. But if you're going to be publishing plots of your own calculations without telling us what you're doing, could you please keep it to the New Theories section? This part of the forum is for open discussion of the science behind "mainstream" ideas in physics, such as Einstein's field equations or Newtonian gravity.
The latter graph seems to show the escape velocity decreasing when you reach 10m from the center of the Earth.Assuming a spherically symmetric Earth, the gravitational force would be the attraction of a 10m radius sphere (possibly with a significant content of gold, osmium or other dense materials). You would be effectively weightless (if you weren't instantly crushed by thousands of kilometers of iron and rock).This 10m radius sphere does not have enough density to form a black hole.I think the graphed reduction of escape velocity near the center of the Earth is a real effect - I think it might be some sort of underflow error.
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!