I was helping someone with his fanfiction where a system didn't have the same gravitational constant that exists within the rest of the universe.(adsbygoogle = window.adsbygoogle || []).push({});

There was a throw-away line given by Q on the Star Trek episode, "Deja Q", where Q was asked how would he fix the problem of a moon falling out of orbit, he replied: "Change the Gravitational Constant of the Universe.".

In the fanfiction, the system's inhabitants have a connection with the Q Continuum, so it made sense for us to play around with the formulas a bit to try and fit the situation as given.

We know the force of Gravity is given by Newton's formulaF° = (G°*M, and we know the value for the Gravitational Constant_{1}*M_{2})/d^{2}(G°) = 6.67428 *10. Additionally, the formula for surface gravity is^{-11}m^{3}/s^{2}*kgA° = G°*M, and the formula for circular orbits is_{p}/r_{p}^{2}V°._{o}~ √(G°[M_{p}+m]/d)

I tried an initial change of Newton's formula toF' = (G'*Mand a change of the Gravitational Constant_{1}*M_{2})/d^{3}(G') = 2.394893181576 *10as well. (For reasons not relevant to my question, the value of the various G's increase was settled to be in terms of factors of^{-4}m^{4}/s^{2}*kg3588242—which happens to be the radius of the planet in meters. The value of53.5with respect to d (distance) in various formulas was also settled during the negotiations. But the significance of those values will become apparent in a moment.)

When I made that change, it was found the best way to deal with the formula regarding surface gravity and circular orbits were to modify them as well to beA' = G'*M, and the formula for circular orbits is_{p}/r_{p}^{3}V'o ~ √(G'[Mrespectively._{p}+m]/d^{2})

Initial results looked promising, however, there was still too much force remaining between the planet and its moons for our liking. But by increasing the changes to the Gravitational Constant, the results were finally satisfactory by the time I settled upon a value of(G'"") = 39,702,060,660,449,400,000,000 m. As a result, finding the value of Force becomes^{8}/s^{2}*kgF'"" = (G'""*M, the Surface Gravity became_{1}*M_{2})/d^{7}A'"" = G'""*M, and Velocity of Circular Orbits were now found by_{p}/r_{p}^{7}V'"". The orbits of this hypothetical planet's moons were now quite satisfactory._{o}~ √(G'""[M_{p}+m]/d^{6})

But then, I ran into a small problem. I happened to take a look at how small everyday masses (~1kg) would react at normal small everyday distances (~1m)—and found to my horror that the resulting forces would require orbital speeds of~2C! Ooops!!

Fortunately, our efforts weren't a total loss. The best solution was decided to make a stratified series of different Constants of Gravity, the values of which were defined by how far apart the objects involved were.

For distances of0-1m, the operating value of G is our normalG°. From1m to 53.5m, however, the value of G isG'. And from53.5m to (53.5), the value of G is^{2}m --- (2862.25)G", and so on until the final values ofG'""are in play from distances of53.5. Thereafter, the values of G reduce until back to normal beyond^{4}m to 53.5^{5}m53.5in accordance to the following table:^{9}m

G°____0______________________________________1

G'____1______________________________________53.5

G"____53.5__________________________________2,862.25

G'"___2,862.25______________________________153,130.375

G""___153,130.375__________________________8,192,475.0625

G'""__8,192,475.0625_______________________438,297,415.84375

G""___438,297,415.84375___________________23,448,911,747.640625

G'"___23,448,911,747.640625_______________1,254,516,778,498.7734375

G"____1,254,516,778,498.7734375__________67,116,647,649,684.37890625

G'____67,116,647,649,684.37890625________3,590,740,649,258,114.271484375

G°____3,590,740,649,258,114.271484375___∞

The values and dimensions of the various G's, and how they were derived, are given below (with r_{p}being the value of3588242m):

G°___0.0000000000667428________6.67428*10^{-11}______m^{3}/s^{2}*kg

G'___0.0002394893181576______________G'=G°*r_{p}^{1}_____m^{4}/s^{2}*kg

G"___859.34563_________________________G"=G°*r_{p}^{2}_____m^{5}/s^{2}*kg

G'"__3,083,540,081.95494_______________G'"=G°*r_{p}^{3}_____m^{6}/s^{2}*kg

G""__11,064,488,030,754,200____________G""=G°*r_{p}^{4}____m^{7}/s^{2}*kg

G'""_39,702,060,660,449,400,000,000___G'""=G°*r_{p}^{5}____m^{8}/s^{2}*kg

Along with the corresponding Surface accelerations:

a° = G°M_{p}/r_{p}^{2}

a' = G'M_{p}/r_{p}^{3}

a" = G"M_{p}/r_{p}^{4}

a'" = G'"M_{p}/r_{p}^{5}

a"" = G""M_{p}/r_{p}^{6}

a'""= G'""M_{p}/r_{p}^{7}

And Circular Orbit Velocities:

V°_{o}~√(G°[M_{p}+m]/d)

V'_{o}~√(G'[M_{p}+m]/d^{2})

V"_{o}~√(G"[M_{p}+m]/d^{3})

V'"_{o}~√(G'"[M_{p}+m]/d^{4})

V""_{o}~√(G""[M_{p}+m]/d^{5})

V'""_{o}~√(G'""[M_{p}+m]/d^{6})

This stratification of varying Constants of Gravity solved many problems, but there are still other issues to consider.

The boundaries between whereG'"andG""andG""andG'""come into play, large objects (small moons, and such) crossing those boundaries surrounding other large objects (planets, for example) are subjected to large sheer forces which would tear a small moon or asteroid apart. Small objects might withstand such stresses, however.

I think that one way to work around that problem is to further subdivide the layers into integer roots—for example: a(G')covering distances between^{.25}= 0.0104233322m^{4.25}/s^{2}*kg53.5to^{.125}m53.5, a^{.375}m(G')covering distances between^{.5}= 0.45365637 m^{4.5}/s^{2}*kg53.5to^{.375}m53.5, a^{.625}m(G')covering distances between^{.75}= 19.744559154m^{4.75}/s^{2}*kg53.5to^{.625}m53.5, that would exist between^{.875}mG'andG", and so on between the other layers. But this only makes matters worse where it comes into the number of layers involved.

Also, I suspect the vertical stratification is a Gaussian Curve (with perhaps a strong kurtosis for the G'"" values) on a horizontal logarithmic scale, I was hoping for a more eloquent formula that would let me find the solutions for non-circular orbits and such.

So far, I can see the relevant factors for determining the smooth values of G across the system would simply derive from the distances between the objects being looked at, where the values of theAltered Grelies on the ratio of the (logarithm of the planet's radius) vs the measured distances (in the logarithm of base 53.5) timesG°.

However, I'm unable to derive a smooth formula for covering all situations in this system. Unfortunately, I'm at my limit of calculus that I know for what is needed to figure this out.

What I would like to do is develop the formula that could allow someone to easily determine what theAltered Gwould be if just given the distance between the objects, and in that way, I would be able to apply that toward finding the general formulas for motion in the system.

Any assistance would be greatly appreciated. Thanks,

Ryuu

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# Need a Little Help With Gravity--just hypothetical & theory

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