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

[Ryuu]

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.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 formula F° = (G°*M₁*M₂)/d², and we know the value for the Gravitational Constant (G°) = 6.67428 *10^-11 m³/s²*kg. Additionally, the formula for surface gravity is A° = G°*M_p/r_p², and the formula for circular orbits is V°_o ~ √(G°[M_p+m]/d) .I tried an initial change of Newton's formula to F' = (G'*M₁*M₂)/d³ and a change of the Gravitational Constant (G') = 2.394893181576 *10^-4 m⁴/s²*kg as 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 3588242—which happens to be the radius of the planet in meters. The value of 53.5 with 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 be A' = G'*M_p/r_p³, and the formula for circular orbits is V'o ~ √(G'[M_p+m]/d²) respectively.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 m⁸/s²*kg. As a result, finding the value of Force becomes F'"" = (G'""*M₁*M₂)/d⁷, the Surface Gravity became A'"" = G'""*M_p/r_p⁷, and Velocity of Circular Orbits were now found by V'""_o ~ √(G'""[M_p+m]/d⁶). The orbits of this hypothetical planet's moons were now quite satisfactory.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 of 0-1m, the operating value of G is our normal G°. From 1m to 53.5m, however, the value of G is G'. And from 53.5m to (53.5)²m --- (2862.25), the value of G is G", and so on until the final values of G'"" are in play from distances of 53.5⁴m to 53.5⁵m. Thereafter, the values of G reduce until back to normal beyond 53.5⁹m in accordance to the following table:

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 of 3588242m):

G°___0.0000000000667428________6.67428*10^-11______m³/s²*kg
G'___0.0002394893181576______________G'=G°*r_p¹_____m⁴/s²*kg
G"___859.34563_________________________G"=G°*r_p²_____m⁵/s²*kg
G'"__3,083,540,081.95494_______________G'"=G°*r_p³_____m⁶/s²*kg
G""__11,064,488,030,754,200____________G""=G°*r_p⁴____m⁷/s²*kg
G'""_39,702,060,660,449,400,000,000___G'""=G°*r_p⁵____m⁸/s²*kg

Along with the corresponding Surface accelerations:

a° = G°M_p/r_p²
a' = G'M_p/r_p³
a" = G"M_p/r_p⁴
a'" = G'"M_p/r_p⁵
a"" = G""M_p/r_p⁶
a'""= G'""M_p/r_p⁷

And Circular Orbit Velocities:

V°_o ~ √(G°[M_p+m]/d)
V'_o ~ √(G'[M_p+m]/d²)
V"_o ~ √(G"[M_p+m]/d³)
V'"_o ~ √(G'"[M_p+m]/d⁴)
V""_o ~ √(G""[M_p+m]/d⁵)
V'""_o ~ √(G'""[M_p+m]/d⁶)

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

The boundaries between where G'" and G"" and G"" and G'"" 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')^.25 = 0.0104233322m^4.25/s²*kg covering distances between 53.5^.125m to 53.5^.375m, a (G')^.5 = 0.45365637 m^4.5/s²*kg covering distances between 53.5^.375m to 53.5^.625m, a (G')^.75 = 19.744559154m^4.75/s²*kg covering distances between 53.5^.625m to 53.5^.875m, that would exist between G' and G", 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 the Altered G relies on the ratio of the (logarithm of the planet's radius) vs the measured distances (in the logarithm of base 53.5) times G°.

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 the Altered G would 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

 

[mfb]

A 1/d³ gravitational force law (or even higher powers in the denominator) does not allow stable orbits. The moon would either crash or escape permanently quickly (within a few orbits).
The change would also mess around with several fundamental concepts of physics.

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 be A' = G'*M_p/r_p³, and the formula for circular orbits is V'o ~ √(G'[M_p+m]/d²) respectively.

No, those formulas would look completely different with a different distance law. Treating the planet as point-mass (which those formulas do) works with our 1/d² force law only.

You are asking an x-y-question here: Why don't you post the setting and ask for possible solutions for this setting? There might be an easier solution.

 

[Ryuu]

Thanks for your reply, MFB.

To be honest, I don't know if the changes would necessarily cause the concept of "point-mass" to no longer have validity. Certainly, the equations for Force of Gravity would have to be changed to account for the higher distance dimension in an Altered G constant, while Acceleration and the equations for orbits are all derived from the Force law. Also, I looked into the other affects, and only saw perhaps a handful of physics concepts that would've been affected--the Gravity Coupling Constant being the only significant and noteworthy effect--which was how and where I found the biggest problem for the G'"". Most of our other concepts in physics, deal with the electromagnetic forces, and have very little interaction with gravity.

And, as I discovered on my own, the forces would be more extreme when dealing with smaller distances--astronomically, objects would continue to orbit each other serenely for millions of years as opposed to tens of years...but everyday things like people and pencils and pebbles would all become black holes Thus, the requirement for the stratification. But I'm not happy with it, as there's no smooth progression between the needed changes of the ALTGs.

But since this is Star Trek physics, we're talking about, leeways have to be made for it--especially when dealing with physics-stretching abilities from creatures like Q. Of course, we know that Q would most likely have developed a smoothed-out equation for self-adjusting ALTGs. As chaotic as Q is, he does like doing things with style.

And knowing how this place instantly shuts down any discussion that might even hint at physics warping topic unless in the SciFi section--even then, it's not safe--that's why I started here. So where is this section you're suggesting I take this to?

Thanks,
Ryuu.

 

[mfb]

To be honest, I don't know if the changes would necessarily cause the concept of "point-mass" to no longer have validity.

Don't worry, I know the equations. And you saw the effect with the 1/r^7 force yourself: a small object nearby would lead to more gravitational attraction than you calculated for the whole earth, including the floor which is closer and more massive than the person nearby. You cannot reduce objects to point-masses unless you have the usual 1/r^2 law.

Also, I looked into the other affects, and only saw perhaps a handful of physics concepts that would've been affected--the Gravity Coupling Constant being the only significant and noteworthy effect--which was how and where I found the biggest problem for the G'"". Most of our other concepts in physics, deal with the electromagnetic forces, and have very little interaction with gravity.

It would break general relativity. It would change the masses of objects in Planck units, which might break quantum field theory (all of particle physics) (but we don't know in the absence of a unification of both) and therefore all of physics.
A 1/r^4 force (or stronger) everywhere starting from astronomical scales would lead to notable effects of gravity on the scale of nuclei. A transition between different power laws depending on distance to avoid that (distance measured in which frame?) would break special relativity and therefore both general relativity and quantum field theory at the same time.

So where is this section you're suggesting I take this to?

Here. I suggest you start with the problem instead of problems with a solution attempt.
X-Y-questions