Weirder Than Normal--NORMAL? NOW THAT'S WEIRD!

Now that chapter 18 is posted, here's an opportunity for me to post something that hopefully helps folks make a little sense of the technobabble used. (Hope this works) All work was done by Ryuu.

If you're having trouble following what Pardus told Trevar, hopefully, this can help make it better understood....

G° 0.0000000000667428 m^3/s^2*kg
G' 0.0002394893181576 m^4/s^2*kg G'=G°*[r(e)]^1
G" 859.34563 m^5/s^2*kg G"=G°*[r(e)]^2
G'" 3,083,540,081.95494 m^6/s^2*kg G'"=G°*[r(e)]^3
G"" 11,064,488,030,754,200 m^7/s^2*kg G""=G°*[r(e)]^4
G'"" 39,702,060,660,449,400,000,000 m^8/s^2*kg G'""=G°*[r(e)]^5

In the Equestrian system, each of the above values and their dimensions of the Gravitational Constant exists for all masses at the range of distances listed below. G° is the UGC that we're familiar with, called G-original. G' is referred as G-prime, G" is G-double prime, etc. The actual values of each G is shown in this color above. This color above shows how the actual value is derived, where r(e) is the radius of Equus.

G° 0-1m
G' 1-53.5m
G" 53.5-2,862.25m
G'" 2,862.25-153,130.375m
G"" 153,130.375-8,192,475.0625m
G'"" 8,192,475.0625-438,297,415.84375m
G"" 438,297,415.84375-23,448,911,747.640625m
G'" 23,448,911,747.640625-1,254,516,778,498.7734375m
G" 1,254,516,778,498.7734375-67,116,647,649,684.37890625m
G' 67,116,647,649,684.37890625-3,590,740,649,258,114.271484375m
G° 3,590,740,649,258,114.271484375-∞m

G° (m^3/(s^2*kg)) V°o ~ (G°[M+m]/r)^.5 a°=G°M/r^2
G' (m^4/(s^2*kg)) V'o ~ (G'[M+m]/r^2)^.5 a'=G'M/r^3
G" (m^5/(s^2*kg)) V"o ~ (G"[M+m]/r^3)^.5 a"=G"M/r^4
G'" (m^6/(s^2*kg)) V'"o ~ (G'"[M+m]/r^4)^.5 a'"=G'"M/r^5
G"" (m^7/(s^2*kg)) V""o ~ (G""[M+m]/r^5)^.5 a""=G""M/r^6
G'"" (m^8/(s^2*kg)) V'""o ~ (G'""[M+m]/r^6)^.5 a'""=G'""M/r^7

For each type of G, there is a corresponding Circular Orbital Velocity and Acceleration. Plus, the dimensions of G changes from level to level.

For Equus, the following stats for Mass, Radius, Surface Gravity (with deriving formula), Density, Rotation, and distances for the G""/G'"" discontinuity and Geosynchronous orbits:

M(e) 1,894,891,630,000,000,000,000,000 kg
r(e) 3,588,242 m
a(e) 9.82257490148331 m/s^2 a=G""M/r^6
D(e) 9,791.53075482489 kg/m^3
Sidereal Rotation 0.0000114079458624521 °/s
G"" discontinuity boundary 8,192,475 m 68,292.8074134566 s for point mass in circular orbit
G'""discontinuity boundary 8,192,475 m 103,190.9635418960 s for point mass in circular orbit
Geosynch Orbit'"" 34,259,299.09 m 31,556,952 s for point mass in circular orbit

The following is values for Minimum:
distance(m-s) 680,148,665.19
force(m-s) 755,100,111.05
ForceRatio(m-s):(m-e) 267,147,083,831.44
ForceRatio(m-s):(s-e) 9,714,143,826.41

The following is values for Perpendicular alignment:
distance(m-s) 894,348,136.11
force(m-s) 146,079,070.67
ForceRatio(m-s):(m-e) 267,902,223,768.76
ForceRatio(m-s):(s-e) 8,960,234,561.03

The following is values for Maximum:
distance(m-s) 1,066,355,930.43
force(m-s) 50,841,049.05
ForceRatio(m-s):(m-e) 267,953,024,991.54
ForceRatio(m-s):(s-e) 9,009,884,764.41

For Luna, the following stats for Mass, Radius, Surface Gravity (with deriving formula), Density, Distance from Equus, Force that exists between Luna and Equus (with deriving formula), Natural Velocity of orbit (with deriving formula), Natural Period of orbit (with breakdown for days/years--see what Ryuu did?), and Apparent Diameter as viewed from Equus:

M(m) 35,654,301,600,758,500,000,000 kg
r(m) 1,850,559 m
a(m) 9.82257490148330 m/s^2 a=G""M/r^6
D(m) 1,343.11862212012 kg/m^3
R(e-m) 193,103,632.62 m
F'""(e-m) 267,902,183,942.49 m*kg/s^2 f'""=G'""M(e)M(m)/r^7
V'""o(m) 0.0384481336043122 m/s v'""=√(G'""(M(e)+M(m))/r^6)
Period(m) 31,556,952,015.57 s 365242.5002 d 1000 y
app.Dia(m) 65.8875145242359°

For Sun, the following stats for Mass, Radius, Surface Gravity (with deriving formula), Density, Distance from Equus, Force that exists between Sun and Equus (with deriving formula), Natural Velocity of orbit (with deriving formula), Natural Period of orbit (with breakdown of days/years--again, Ryuu pulled a sneaky. Guess the significance of that value!), and Apparent Diameter as viewed from Equus:

M(s) 189,489,163,000,000,000,000,000 kg
r(s) 4,169,040 m
a(s) 0.39930001707818 m/s^2 a=G""M/r^6
D(s) 624.29173040620 kg/m^3
R(e-s) 873,252,297.81 m
F(e-s) 8,959,043,715.36 m*kg/s^2 f=G""M(s)M(m)/r^6
Vo(s) 0.0067391457676225 m/s v=√(G""(M(s)+M(m))/r^5)
Period(s) 814,169,361,560.02 s 9423256.5 d 25800 y
app.Dia(s) 32.8244081265183°

System Barry Center, in Meters & Miles:
87,325,229.78 ±3,633,440.06i m (Along the E-S vector)
27,302.88881 ±2,257.71498i mi (Along the E-S vector)

The test masses Pardus used to measure and determine the discontinuity boundaries:

Test Masses:
M(a) 10,000kg

M(b) 1,000 kg
R(a-b) 0.50 m
F(a-b) 0.00 m*kg/s^2 f=G°M(a)M(b)/r^2
Vo(b) 0.0012117514596649 m/s v=√(G°(M(a)+M(b))/r^1)
Period(b) 2,592.60 s 0.720168001 h

M(b) 1,000 kg
R(a-b) 1.00 m
F(a-b) 0.00 m*kg/s^2 f=G°M(a)M(b)/r^2
Vo(b) 0.0008568376742417 m/s v=√(G°(M(a)+M(b))/r^1)
Period(b) 7,332.99 s 2.036942707 h

M(b) 1,000 kg
R(a-b) 1.00 m
F'(a-b) 2,394.89 m*kg/s^2 f'=G'M(a)M(b)/r^3
V'o(b) 1.6230780941573900 m/s v'=√(G'(M(a)+M(b))/r^2)
Period(b) 3.87 s 0.001075321 h

M(b) 1,000 kg
R(a-b) 53.50 m
F'(a-b) 0.02 m*kg/s^2 f'=G'M(a)M(b)/r^3
V'o(b) 0.0303379083020074 m/s v'=√(G'(M(a)+M(b))/r^2)
Period(b) 11,080.21 s 3.077836285 h

M(b) 1,000 kg
R(a-b) 53.50 m
F"(a-b) 1,048.95 m*kg/s^2 f"=G"M(a)M(b)/r^4
V"o(b) 7.8568706980882200 m/s v"=√(G"(M(a)+M(b))/r^3)
Period(b) 42.78 s 0.011884517 h

M(b) 1,000 kg
R(a-b) 2,862.25 m
F"(a-b) 0.00 m*kg/s^2 f"=G"M(a)M(b)/r^4
V"o(b) 0.0200779298679086 m/s v"=√(G"(M(a)+M(b))/r^3)
Period(b) 895,712.22s 10.5876149 d

M(b) 1,000 kg
R(a-b) 2,862.25 m
F'"(a-b) 0.16 m*kg/s^2 f'"=G'"M(a)M(b)/r^5
V'"o(b) 0.7108959065855590 m/s v'"=√(G'"(M(a)+M(b))/r^4)
Period(b) 25,297.72 s 7.027145051 h

The rest of these test cases are included only for completeness of the spreadsheet Ryuu made. These distances were not tested in the story:

M(b) 1,000 kg
R(a-b) 153,130.38 m
F'"(a-b) 0.00 m*kg/s^2 f'"=G'"M(a)M(b)/r^5
V'"o(b) 0.0002483696066331 m/s v'"=√(G'"(M(a)+M(b))/r^4)
Period(b) 3,873,849,684.45 s 1076069.357 h

M(b) 1,000 kg
R(a-b) 153,130.38 m
F""(a-b) 0.00 m*kg/s^2 f""=G""M(a)M(b)/r^6
V""o(b) 0.0012022883505579 m/s v""=√(G""(M(a)+M(b))/r^5)
Period(b) 800,262,700.57 s 222295.1946 h

M(b) 1,000 kg
R(a-b) 8,192,475.06 m
F""(a-b) 0.00 m*kg/s^2 f""=G""M(a)M(b)/r^6
V""o(b) 0.0000000574280651 m/s v""=√(G""(M(a)+M(b))/r^5)
Period(b) 896,335,943,971,047.00 s 2.48982E+11 h

M(b) 1,000 kg
R(a-b) 8,192,475.06 m
F'""(a-b) 0.00 m*kg/s^2 f'""=G'""M(a)M(b)/r^7
V'""o(b) 0.0000000380064654 m/s v'""=√(G'""(M(a)+M(b))/r^6)
Period(b) 1,354,370,587,749,210.00 s 3.76214E+11 h

-->Edit-->
Here's a link to Ryuu's blog post on the updated math to what's posted above.

Comments

[ChasingResonance]

Hey again! I had a couple quick questions.

Before beginning though, how is it determined what the gravitational constant of each object?

1. In that first section, why is G-pentaprime the maximum number used?

G° 0.0000000000667428 m^3/s^2*kg
G' 0.0002394893181576 m^4/s^2*kg G'=G°*[r(e)]^1
G" 859.34563 m^5/s^2*kg G"=G°*[r(e)]^2
G'" 3,083,540,081.95494 m^6/s^2*kg G'"=G°*[r(e)]^3
G"" 11,064,488,030,754,200 m^7/s^2*kg G""=G°*[r(e)]^4
G'"" 39,702,060,660,449,400,000,000 m^8/s^2*kg G'""=G°*[r(e)]^5

2. What are these numbers supposed to be?

G° 0-1m
G' 1-53.5m
G" 53.5-2,862.25m
G'" 2,862.25-153,130.375m
G"" 153,130.375-8,192,475.0625m
G'"" 8,192,475.0625-438,297,415.84375m
G"" 438,297,415.84375-23,448,911,747.640625m
G'" 23,448,911,747.640625-1,254,516,778,498.7734375m
G" 1,254,516,778,498.7734375-67,116,647,649,684.37890625m
G' 67,116,647,649,684.37890625-3,590,740,649,258,114.271484375m
G° 3,590,740,649,258,114.271484375-∞m

They aren't the same as the previous set, and the Gravitational Constant increases, then decreases, while the value next to them constantly increases, each one starting where the other ended. Why aren't the G-prime values the same?

3. Where did he get this formula?

a°=G°M/r^2

4. Why is G-tetraprime and G-Pentaprime the ones used for this section?

For Equus, the following stats for Mass, Radius, Surface Gravity (with deriving formula), Density, Rotation, and distances for the G""/G'"" discontinuity and Geosynchronous orbits:

M(e) 1,894,891,630,000,000,000,000,000 kg
r(e) 3,588,242 m
a(e) 9.82257490148331 m/s^2 a=G""M/r^6
D(e) 9,791.53075482489 kg/m^3
Sidereal Rotation 0.0000114079458624521 °/s
G"" discontinuity boundary 8,192,475 m 68,292.8074134566 s for point mass in circular orbit
G'""discontinuity boundary 8,192,475 m 103,190.9635418960 s for point mass in circular orbit
Geosynch Orbit'"" 34,259,299.09 m 31,556,952 s for point mass in circular orbit

5.

r(e) 3,588,242 m

Wait, Equus is bigger than Earth?

6.

M(e) 1,894,891,630,000,000,000,000,000 kg

But about 1/3 the mass?

7.

a(e) 9.82257490148331 m/s^2 a=G""M/r^6

This would indicate that the Gravitational constant of Equus is G-tetraprime, correct?

8.

Sidereal Rotation 0.0000114079458624521 °/s
G"" discontinuity boundary 8,192,475 m 68,292.8074134566 s for point mass in circular orbit
G'""discontinuity boundary 8,192,475 m 103,190.9635418960 s for point mass in circular orbit
Geosynch Orbit'"" 34,259,299.09 m 31,556,952 s for point mass in circular orbit

What were the formulas for each of these?

9.

The following is values for Minimum:
distance(m-s) 680,148,665.19
force(m-s) 755,100,111.05
ForceRatio(m-s):(m-e) 267,147,083,831.44
ForceRatio(m-s):(s-e) 9,714,143,826.41

Several questions on this one. Minimum distance for what? Force for what? What does (m-s) stand for, and the ones following it? I'm assuming that the first is the minimum distance between the sun and the moon, with m standing for 'moon', s for 'sun', and e for 'Equus'.

10. In this section

M(m) 35,654,301,600,758,500,000,000 kg
r(m) 1,850,559 m
a(m) 9.82257490148330 m/s^2 a=G""M/r^6
D(m) 1,343.11862212012 kg/m^3
R(e-m) 193,103,632.62 m
F'""(e-m) 267,902,183,942.49 m*kg/s^2 f'""=G'""M(e)M(m)/r^7
V'""o(m) 0.0384481336043122 m/s v'""=√(G'""(M(e)+M(m))/r^6)
Period(m) 31,556,952,015.57 s 365242.5002 d 1000 y
app.Dia(m) 65.8875145242359°

What does

R(e-m) 193,103,632.62 m

represent? It wasn't listed in the short description above it. Also, why are tetra and penta primes used again, instead of G-original?

11.

25800 y

that's a platonic year.

?.

The test masses Pardus used to measure and determine the discontinuity boundaries:

Is that the Mohorovičić discontinuity?

?. If so, how were these numbers determined?

M(b) 1,000 kg
R(a-b) 0.50 m
F(a-b) 0.00 m*kg/s^2 f=G°M(a)M(b)/r^2
Vo(b) 0.0012117514596649 m/s v=√(G°(M(a)+M(b))/r^1)
Period(b) 2,592.60 s 0.720168001 h

5.

[ChasingResonance]

Just re-read chapter 18. Disregard # 1, 4, 7, & the second half of 10. In addition though, is any explanation offered as to why G-tetraprime is the value for Equus?

[Kevin Lee]

2985887 Hi ChasingResonance!
I'm glad you were able to get some of your questions answered, but I should go ahead to reply to those in case anyone else is having issues--they're all good questions. Any confusion, I'm afraid, is my fault, since the blog post was mostly a cut&paste from my Word document, which was cut&pasted from the Excell that Ryuu sent me. We're both glad that someone's checking our work and happy that we put together a story that got someone interested in digging into it.

Hey again! I had a couple quick questions.
Before beginning though, how is it determined what the gravitational constant of each object?
Perhaps a little history is in order before jumping into the numbers.

As I mentioned in the—I’m not sure what we call that page (our “cover page” perhaps?) that has the long description for our stories, it was inspired by ZoidbergIsBestPony's "DisQord Continuum" series here as well as a fanfiction currently being posted by Ryuu at AO3 "Tunnel-Snake & Fire-Lizard".

In the TNG episode, “Daja Q”, the crew faced the problem of how to restore a moon back to its normal orbit at the same time they had to deal with Q, who had lost his powers when he pissed off all the rest of the Continuum. Seeking advice, Geordi asked Q what he would do to fix it, and Q replied, “Change the gravitational constant of the universe.”

When I first came up with the idea for this story, I remembered that quote. Since it was established by MLP canon that the moon & sun have to be moved by the princesses, and given I was basing this story off of the “DisQord Continuum”, the Enterprise found that gravity wasn’t working like it should at all in that system. Of course, when Q showed up on the ship, there wasn’t any reason to dig into why—Picard just assumed that Q was messing with stuff, as usual. But, of course the real reason is it was the ponies—who were all the evolved pets of the Continuum after Q’s people had ascended.

But when I tried to change just the value of G°, I couldn’t get anything to work right. At one point, I even considered changing the size of Equus from the same as Earth and made it the size of the Moon! I was able to find a mass/radius value that gave 1 Standard Gravity for an object that size! But I still couldn’t get the planet’s surface gravity to match Earth’s while at the same time put the sun & moon in orbits that would require manual manipulations and let them be large enough to look bigger than the sun & moon did here on Earth.

When I approached Ryuu for permission to use his characters, he was at first reluctant. But after seeing the Discord episodes and reading the “DisQord Continuum” stories, he was sold and said he was willing to help!
I mentioned the problems I was coming across, he looked into the problem and suggested changing the dimensions of G as well as its value—“after all, we are talking about Q, here!” he said. He pointed out that G° was actually a derived constant that can’t be directly measured, it really doesn’t matter what the dimensions were, so long as the final formulae for Acceleration and Force came out correct.

He also pointed out a new problem I had, which was given the size of my Equus model, the core would have to be nearly solid osmium with just a thin crust (and almost no mantel). We played with that for a bit and settled on the current diameter of this story, which played a big role with how we figured out the different gravity constants.

Our first attempt was to change the formula for gravity from a=GM/r^2, which is Newton’s original inverse square law, to a=GM/r^3. This gave us G’ and the results instantly started looking promising, but were still way too short. So Ryuu started playing around and continued increasing the distance dimensions and the values started to get better and better, but still were unsatisfactory in regards to the angular size of the moon, but G-tetraprime was looking best at that point. But just for shits and giggles, he ran it through 3 more iterations. G-pentaprime finally gave us fantastic results for the sun and moon orbits, as well as their angular size when viewed from the planet’s surface. G-hexaprime and -septaprime, however, were giving us really weird numbers—and we later found out why—but we decided to not bother with trying to make them work when G’”” was doing so great.

Way earlier, we had settled on giving the moon a 1000yr natural orbit under these new gravity constants, and the Platonic year for the sun, so that with each version of G, we only needed to adjust their distances from Equus to what we needed.

Okay, now we’re at the point that it’s time to answer your questions….

1. In that first section, why is G-pentaprime the maximum number used?

G° 0.0000000000667428 m^3/s^2*kg
G' 0.0002394893181576 m^4/s^2*kg G'=G°*[r(e)]^1
G" 859.34563 m^5/s^2*kg G"=G°*[r(e)]^2
G'" 3,083,540,081.95494 m^6/s^2*kg G'"=G°*[r(e)]^3
G"" 11,064,488,030,754,200 m^7/s^2*kg G""=G°*[r(e)]^4
G'"" 39,702,060,660,449,400,000,000 m^8/s^2*kg G'""=G°*[r(e)]^5

As I mentioned above, we started getting some really weird results with G””” (hexaprime) and higher. Not long after I had started writing the story, Ryuu was still going through, making sure he hadn’t missed something with his math. He had created a 30-some Excel spreadsheet, with each sheet dedicated to a specific value of G, filled with formulae that he only needed to make a few changes and it would auto-populate the results for what we needed. But since each page had a different dimension of G, he frequently had to go through and ensure each page had the proper correcting factors in each of the formulae.

It was during one of his rechecks that he discovered a near disaster for us. G’”” was perfect for the system’s celestial mechanics, but when he happened to plug in the values for the test masses Pardus was going to use, he discovered that with G’””, the 1kilo masses orbiting at a distance of 1meter, they would do so at more than 2C!!

(Ryuu told me that when he realized that, his face literally looked like that smilie—LOL—and he says he still hasn’t bothered trying to figure out what the Schwarzschild radius would be for them at that level!)

But in a way, it was quite fortuitous, because we still had yet to develop the transition from the Equus system to the rest of the universe….

So, in regards to the values above, each of the levels of G, from G° to G’””, have their numerical values and dimensions defined on the left hand column with the source of how they were derived on the right hand column. As you can see, each G-prime is increasing by the value of the Radius of Equus raised to the appropriate power.

2. What are these numbers supposed to be?

G° 0-1m
G' 1-53.5m
G" 53.5-2,862.25m
G'" 2,862.25-153,130.375m
G"" 153,130.375-8,192,475.0625m
G'"" 8,192,475.0625-438,297,415.84375m
G"" 438,297,415.84375-23,448,911,747.640625m
G'" 23,448,911,747.640625-1,254,516,778,498.7734375m
G" 1,254,516,778,498.7734375-67,116,647,649,684.37890625m
G' 67,116,647,649,684.37890625-3,590,740,649,258,114.271484375m
G° 3,590,740,649,258,114.271484375-∞m

They aren't the same as the previous set, and the Gravitational Constant increases, then decreases, while the value next to them constantly increases, each one starting where the other ended. Why aren't the G-prime values the same?
So…here, we’re back to talking about the Q (and their Q-like former pets)…“redefining gravity,” as Geordi put it…

Also, it’s time to mention how the transition from the Equus system to the normal universe.

Ryuu tried to resolve the problem that every stone at Pinkies’ family rock farm would be a black hole. He knew it was obviously a logarithmic problem, only it had to deal with a base other than either the Natural Log or Decimal Log. It was through trial and error that he settled on 53.5 as his Log base.

Remember, gravity never stops. Normal universal gravity, you have an asymptotic curve to measure gravitational acceleration. For any point mass orbiting a at 1 meter distant from a test mass, say a 1kg ball of iron it would experience an acceleration of 0.0000000000667428m/s^2 toward the test mass. (funny how that number appears if you plug in 1s everywhere ;) )

Now, if you compress that 1kg mass to 1.4837554589428308*10^-27meters, you’d have a black hole. But your point mass at 1meter distant, would still see the same acceleration. However, if you start approaching that 1kg black hole, your acceleration asymptotically rises toward C the closer you get to it. Likewise, the further you get away from it, the closer your acceleration gets to 0—you will never actually ever see 0 acceleration until you were infinitely far away…

Of course, it’s impossible to measure that, since there are so many other objects in the universe that would interfere with your test.

But in the Equus system, you would experience several different gravitational constants, determined only by how far away you are from the objects you're measuring against.

From a distance ranging from 0 to 1meter, a theoretical point object orbiting the test mass would be in the influence of normal gravity. But from 1meter to 53.5meters, the constant of gravity would go to G’, and so on until the maximum G’”” between distances of 8,192,475.0625meters to 438,297,415.84375meters. Beyond there, the constants would decrease until you again are back to the normal gravitational constant beyond 3,590,740,649,258,114.271484375meters. (note the Decimal Point! :D )

Ryuu explained that his initial attempt was to use the Gaussian curve for the transitions, but he was unable to develop the formula needed for values between integer powers. He said he was sorely tempted to seek help from some folks in the Physics Forum, but we were lacking time, plus he was worried that no matter how carefully he worded it, some troll would jump up just to argue semantics and drag his question into a flame war—unfortunately, they evidently seem to do that a lot over there.

So the numbers above are not the values of G° or the G-primes. They are the ranges of distances for which you would experience the various levels of gravitational constants from every object in the Equus system, no matter its size or mass.

Which leaves us with a quite a few gravitational discontinuities to worry about….

3. Where did he get this formula?

a°=G°M/r^2

Okay, so that’s from the original Newtonian Law of Gravitation (http://en.wikipedia.org/wiki/Surface_gravity
).

Last night, I had passed your message to Ryuu and he said you really had him worried that he missed correcting something before sending the Excel to me…but that’s the correct formula for G°!

Also, it’s important to remember that like the series of G-primes, the formulae for Acceleration and Circular Orbital Velocity will also change as the distance between masses change!

4. Why is G-tetraprime and G-Pentaprime the ones used for this section?

For Equus, the following stats for Mass, Radius, Surface Gravity (with deriving formula), Density, Rotation, and distances for the G""/G'"" discontinuity and Geosynchronous orbits:
M(e) 1,894,891,630,000,000,000,000,000 kg
r(e) 3,588,242 m
a(e) 9.82257490148331 m/s^2 a=G""M/r^6
D(e) 9,791.53075482489 kg/m^3
Sidereal Rotation 0.0000114079458624521 °/s
G"" discontinuity boundary 8,192,475 m 68,292.8074134566 s for point mass in circular orbit
G'""discontinuity boundary 8,192,475 m 103,190.9635418960 s for point mass in circular orbit
Geosynch Orbit'"" 34,259,299.09 m 31,556,952 s for point mass in circular orbit

The radius of Equus is 3,588,242meters, which puts it solidly in the range for G”” (153,130.375meters to 8,192,475.0625meters).

The distances between the three major bodies, Equus, Sun and Moon are all solidly within the G’”” domain.

The problem with the different gravity constants results in a discontinuity at an orbital radius of 8,192,475.0625m. If an object was large enough, it would suffer shear forces that could tear it apart if it straddled that distance for any length of time.

The 68,292.8s and 103,191s are the orbital periods for theoretical point masses in circular orbits at that distance from the center of Equus.

It’s something that would be seen at ALL the discontinuity boundaries, but this one between the inner G”” and G’”” transition line is especially problematic because we’re looking at the whole mass of the planet acting on the body in orbit where a substantial orbital speed is involved. The other transition boundaries would either be offset by the smaller masses that would likely be involved or the much slower orbital velocities further out in the system.

5.

r(e) 3,588,242 m

Wait, Equus is bigger than Earth?
Ryuu converted all kilometers to meters, so that’s actualy 3,588km, which is only a little bigger than the radius of Mars. He said it was so that he wouldn’t accidently mess up something by three orders of magnitude (or more).

6.

M(e) 1,894,891,630,000,000,000,000,000 kg

But about 1/3 the mass?
Yes, and this sets the planet’s density to a bit under twice that of Earth’s—much better than an entire moon-sized ball of osmium!!

7.

a(e) 9.82257490148331 m/s^2 a=G""M/r^6

This would indicate that the Gravitational constant of Equus is G-tetraprime, correct?
That is correct.

The funny thing is, though, on the surface of my model, it turned out to be 1 Standard Gravity no matter which constant was used!

8.

Sidereal Rotation 0.0000114079458624521 °/s
G"" discontinuity boundary 8,192,475 m 68,292.8074134566 s for point mass in circular orbit
G'""discontinuity boundary 8,192,475 m 103,190.9635418960 s for point mass in circular orbit
Geosynch Orbit'"" 34,259,299.09 m 31,556,952 s for point mass in circular orbit

What were the formulas for each of these?
Okay, Sidereal Rotation…that would be where Equus measures a “year”. It’s the angular rotational speed of the planet if it took one of our years to rotate, and you’d only be able to determine that from watching the stars from the surface of the planet! They would see it taking 1year for one of their constellations to appear to go around and return to the same point in the sky—after all, you certainly can’t rely on the sun and moon to mark your calendar, not with the princesses wrestling over them every thousand years or so!!

And especially when they hand their shit off to Twilight!!—how would anyone measure that sunrise period just before she and Tirek started blowing up everything in sight???

9.

The following is values for Minimum:
distance(m-s) 680,148,665.19
force(m-s) 755,100,111.05
ForceRatio(m-s):(m-e) 267,147,083,831.44
ForceRatio(m-s):(s-e) 9,714,143,826.41

Several questions on this one. Minimum distance for what? Force for what? What does (m-s) stand for, and the ones following it? I'm assuming that the first is the minimum distance between the sun and the moon, with m standing for 'moon', s for 'sun', and e for 'Equus'.
Yes, sorry! Any confusion you are experiencing is entirely my fault here!

There were three important measurements needed to be made: all the forces between Equus, the moon, and sun, depending upon their orientation. These numbers are the ratios of the respective forces between sun, moon and Equus. Minimum was referring to when the sun and moon were on the same side of Equus in their obits, Perpendicular for when the sun and moon were at right angles with respect to the planet and Maximum was for when the sun and moon are on opposite sides of Equus.

You are correct, the markings regarding “m-s” are between moon-and-sun, “m-e” are between moon-and-Equus, and “s-e” are for between sun-and-Equus.

Just remember, in this model, the sun is actually just another small moon of the planet, but Celestia had opened a regulated Gateway to another universe in this other moon’s core, and the resulting radiation percolating through its mass is causing the surface to be white-hot, as well as being nearly vaporized!

10. In this section

M(m) 35,654,301,600,758,500,000,000 kg
r(m) 1,850,559 m
a(m) 9.82257490148330 m/s^2 a=G""M/r^6
D(m) 1,343.11862212012 kg/m^3
R(e-m) 193,103,632.62 m
F'""(e-m) 267,902,183,942.49 m*kg/s^2 f'""=G'""M(e)M(m)/r^7
V'""o(m) 0.0384481336043122 m/s v'""=√(G'""(M(e)+M(m))/r^6)
Period(m) 31,556,952,015.57 s 365242.5002 d 1000 y
app.Dia(m) 65.8875145242359°

What does

R(e-m) 193,103,632.62 m

represent? It wasn't listed in the short description above it. Also, why are tetra and penta primes used again, instead of G-original?
So this is dealing with Luna’s moon:

r(m) is the moon’s radius, which is solidly in the G”” zone. Despite being so much smaller than Equus (&much less Earth), when you plug & chug the numbers through the G-petaprime calculations, you actually end up with a surface gravity also of nearly 1 Standard Gravity!

And the R(e-m) is the distance between the center of Equus and the moon, again, in meters. For that distance, the appropriate Grav constant would be G-pentaprime. For whatever reason, Ryuu used “R” for both, but he differentiated between them by using the small letter for the planetary body radius and the capital for the distance between planetary bodies.

11.

25800 y

that's a platonic year.
Yes. Ryuu figured why not give the sun a natural orbital period that had significance when the moon had a millennial period.

The test masses Pardus used to measure and determine the discontinuity boundaries:
Is that the Mohorovičić discontinuity?
Heh! Cute
Although Pardus could claim title for it, being the first from Earth to discover it, but he’d most likely give the credit to Sir Issac Newcolt.

?. If so, how were these numbers determined?

M(b) 1,000 kg
R(a-b) 0.50 m
F(a-b) 0.00 m*kg/s^2 f=G°M(a)M(b)/r^2
Vo(b) 0.0012117514596649 m/s v=√(G°(M(a)+M(b))/r^1)
Period(b) 2,592.60 s 0.720168001 h

And this series was representing the tests that Pardus did when he took those stones from the ground into orbit with him, so he could try to figure out the system’s gravity. One test mass—Labled “A” was 10,000kg; the other—labeled “B” was only 1,000kg.
From the story, it was implied he was able to determine the gravity discontinuities were based on powers of 53.5meters.

Since this first sample is putting them at .5m apart, we have the circular orbital velocity, V°o ~ (G°[M+m]/r)^.5
V°o ~ ((6.67428*10^-11)(10,000+1,000)/(0.5))^.5
V°o ~ ((6.67428*10^-11)(11,000)/(0.5))^.5
V°o ~ (0.0000014683416)^.5
V°o ~ 0.00121175m/s

Period would be 2πr/V°o
2*3.14159*0.5/0.00121175 = 2592.6s, or 0.72hours

All the others are plug & chug as well, with attention being made for the different gravitational constants.

So I hope that resolves any confusion you may have had. Again, I do appologize for any problems I caused.

[ChasingResonance]

2988764
Alright, response time!
1.

he discovered that with G’””, the 1kilo masses orbiting at a distance of 1meter, they would do so at more than 2C!!

Twice the speed of light. Holy fuck.

2.

Last night, I had passed your message to Ryuu and he said you really had him worried that he missed correcting something before sending the Excel to me…but that’s the correct formula for G°!

That was my mistake. I was looking for the standard formula for acceleration, not the Gravitation Theory. I see that the formula would increase exponent degree based on the G value, as the radius would be changing due to the length going from 4 to 8 power.

3.

(Ryuu told me that when he realized that, his face literally looked like that smilie—LOL—and he says he still hasn’t bothered trying to figure out what the Schwarzschild radius would be for them at that level!)

What? It's just (2GM)/c^2 . . .

4.

Geosynch Orbit'"" 34,259,299.09 m 31,556,952 s for point mass in circular orbit

I'm assuming that the formula used for that was (μ(P/2pi)^2)^1/3, where μ is the gravitational constant times the mass, and P is the orbital perid, so what was the value for P? I know that it is the sidereal rotation, but did you input it as degrees per second, or something else?

5.

The radius of Equus is 3,588,242meters, which puts it solidly in the range for G”” (153,130.375meters to 8,192,475.0625meters).
The distances between the three major bodies, Equus, Sun and Moon are all solidly within the G’”” domain.

I had gotten confused and started thinking that the G value was from the surface, not the center of the planet, so I was extra confused when they were experiencing G'''' on the surface. Thanks for clearing that up.

6.

Wait, Equus is bigger than Earth?

Not sure where I even got that one. Earth's radius is something like 6.7k kilometers, right?

7.

much better than an entire moon-sized ball of osmium!!

Very, very true.

8.

The following is values for Minimum:
distance(m-s) 680,148,665.19
force(m-s) 755,100,111.05
ForceRatio(m-s):(m-e) 267,147,083,831.44
ForceRatio(m-s):(s-e) 9,714,143,826.41

The following is values for Perpendicular alignment:
distance(m-s) 894,348,136.11
force(m-s) 146,079,070.67
ForceRatio(m-s):(m-e) 267,902,223,768.76
ForceRatio(m-s):(s-e) 8,960,234,561.03

The following is values for Maximum:
distance(m-s) 1,066,355,930.43
force(m-s) 50,841,049.05
ForceRatio(m-s):(m-e) 267,953,024,991.54
ForceRatio(m-s):(s-e) 9,009,884,764.41

I see now why you needed these values. This helps explain why they weren't getting totally fucked by the Grav. Con being different.

9.

Okay, Sidereal Rotation…that would be where Equus measures a “year”. It’s the angular rotational speed of the planet if it took one of our years to rotate, and you’d only be able to determine that from watching the stars from the surface of the planet! They would see it taking 1year for one of their constellations to appear to go around and return to the same point in the sky—after all, you certainly can’t rely on the sun and moon to mark your calendar, not with the princesses wrestling over them every thousand years or so!!

So why exactly does it still rotate? It isn't orbiting a sun; it's the other way around, in fact. However, the sidereal rotation, if I understand correctly, would mean that the larger the planet, the larger that value would have to be, so that it could still rotate in one earth year.

10.

And especially when they hand their shit off to Twilight!!—how would anyone measure that sunrise period just before she and Tirek started blowing up everything in sight???

Yeah, that might cause a bit more of a problem.

11.

Vo(b) 0.0012117514596649 m/s

I feel that that is rather slow. I'm not sure if I can compare it to earth's orbital velocity, as this is a circular, whereas Earth has a slightly elliptical one.

12.
What did you use to calculate all of these? Did you work these out with any type of simulator? Either way, I'm very impressed. I've learned more from this discussion than my physics teacher. In one year.

Also, tell Ryuu I said hi and thanks. I'll probably be asking for some more explanations in the near future. I want to learn as much as possible. Should I or do I need to? Absolutely not. I will never use this. However, I still like to learn a bunch. I showed one of my buddies some of this, and he stopped after about 20 seconds of reading.
In any case, I'll await your response, and my brain to store some more info. Thanks,

-ChasingResonance

[Kevin Lee]

2988895

1.Twice the speed of light. Holy fuck.

Ooops...my mistake. I took another look at his work and found it would actually be 664.6C! That's like warp 8.72 in TOS scale!

2.That was my mistake. I was looking for the standard formula for acceleration, not the Gravitation Theory. I see that the formula would increase exponent degree based on the G value, as the radius would be changing due to the length going from 4 to 8 power.

Yeah, I had the same problem wrapping my head about it, also. Since the dimensions for G are made up from Mass, Time, and Distance, you really couldn't change it for either Mass or Time without really warping the universe. Distance and absolute value are the only options that one could alter for changing it. (of course, with Q, we shouldn't put it past him....and if he did, shouldn't Discord be better known as Qthulhu? )--don't peek if you don't want to melt your brain

3.What? It's just (2GM)/c^2 . . .

Well, that would come to: 883,490meters for a 1kg mass, if I'm doing that right...???
edited after completing #4-->Considering what I just got with the formula below, yeah, I can see why he didn't bother. It's not taking into account the altered value & dimensions of G-pentaprime!

4.Geosynch Orbit'"" 34,259,299.09 m 31,556,952 s for point mass in circular orbit
I'm assuming that the formula used for that was (μ(P/2pi)^2)^1/3, where μ is the gravitational constant times the mass, and P is the orbital perid, so what was the value for P? I know that it is the sidereal rotation, but did you input it as degrees per second, or something else?

Only problem is that formula takes certain shortcuts where G is concerned and is still using normal Grav Constant. That formula gives me the orbital radius of 123,806,257,643,110,000,000meters!--That CAN'T be right!

He used the following matrix:
V°o ~ (G°[M+m]/r)^.5 G° (m^3/(s^2*kg)) a°=G°M/r^2
V'o ~ (G'[M+m]/r^2)^.5 G' (m^4/(s^2*kg)) a'=G'M/r^3
V"o ~ (G"[M+m]/r^3)^.5 G" (m^5/(s^2*kg)) a"=G"M/r^4
V'"o ~ (G'"[M+m]/r^4)^.5 G'" (m^6/(s^2*kg)) a'"=G'"M/r^5
V""o ~ (G""[M+m]/r^5)^.5 G"" (m^7/(s^2*kg)) a""=G""M/r^6
V'""o ~ (G'""[M+m]/r^6)^.5 G'"" (m^8/(s^2*kg)) a'""=G'""M/r^7

Whereas, the spreadsheet Ryuu devised calculates the Circumference of the orbit would be 2πr, but we need to find r. Since we already know that P= 2πr/Vo = 1yr or 31,556,952s,
and Vo'"" is √(G'""[M+m]/r^6), we can reduce m to theoretical point mass to eliminate it, so we're left with √(G'""M)/r^3 after taking out the r^6 from within the Square Root.
Putting them both together works out to P = 2πr/(√(G'""M)/r^3). Now we can solve for r.
P = 2πr/(√(G'""M)/r^3)
P = 2πr^4/(√(G'""M))
r^4 = (P√(G'""M))/2π
that gives us 34,259,299meters, which corresponds to what Ryuu has in the spreadsheet.

5. No problem.
6. Ditto
7.Yes, very, very true.

8.

I see now why you needed these values. This helps explain why they weren't getting totally fucked by the Grav. Con being different.

Plus, with the Grav Constant at other values, the forces that the princesses have to apply to move the sun and moon are much, much less! On the order of only a few hundred to a few thousand times the force that the ISS experiences in orbit about the Earth! (Moslestia is getting off easy! She only has to use 400x the force to shift the sun, while Luna has to use 34,800x to move the moon--that bitch!! )

One other thing that needs to be considered, but I haven't ever seen touched in the show or in many of the stories I've seen, the princesses must actually be moving both the sun AND the moon throughout all their watches--after all, in the morning, the sun is always ready to rise in the east while Celestia goes straight to bed after dropping the sun in the west, and the same factor exists for the moon always being ready for Luna to lift it up in the evening...I'm planning on having this mentioned, so spoilers

9.

So why exactly does it still rotate? It isn't orbiting a sun; it's the other way around, in fact. However, the sidereal rotation, if I understand correctly, would mean that the larger the planet, the larger that value would have to be, so that it could still rotate in one earth year.

There's really no reason it wouldn't rotate...in fact, to say it doesn't rotate at all would make it truly and oddball planet. Almost everything in the universe rotates, even if a bit slow. And size of the object really has no factor other than the amount of force needed to change it. Look at Venus and Mars. Mars is much smaller than the Earth, yet it has a day nearly equal to ours, while Venus is much closer to our size and mass, yet not only rotates extremely slow, but is retrograde at that! Give that randomness, there really no reason that Jupiter should have such a huge angular momentum, but it has ~60% of the angular momentum in the entire Solar System!

But setting Equus to a 1year rotation solves two things: it sets a reason for Equestria to even have a "year", since the show already demonstrates that the princesses are lousy timekeepers, and it answers to the universe that almost everything in it rotates! (even if very, very slowly). And unless one was watching very, very carefully, it could easily be wrongly assumed to not rotate at all.

10.

Yeah, that might cause a bit more of a problem.

Indeed!
11.

Vo(b) 0.0012117514596649 m/s
I feel that that is rather slow. I'm not sure if I can compare it to earth's orbital velocity, as this is a circular, whereas Earth has a slightly elliptical one.

It seems rather fast for me, that such small rocks would orbit each other within 3/4 of an hour? But that is what the numbers suggest it should do. I suppose it could be tested, if we made the expense to launch a set of 10-metric ton and 1-metric ton rocks into orbit. I think it'd have to be done in orbit since any air and suspension mechanism on Earth would likely interfere with the testing.

12. Ryuu apparently just put all the basic gravity formulae in an Excel spreadsheet. He must've been searching all through the Internet for all those formulae. There are about 30pages of it, though, each built about the idea of the different dimensions and values of the Grav Constants.
Once he made the one for G-original, it was just a matter of copy, paste & correct for all the various changes needed to the Grav Constants, Circular Orbit Velocity, Acceleration, and Force on each page.
I can understand why your friend would give up on it...I think Ryuu must eat, drink and shit math! This stuff is pure theory, and really wouldn't have any real-world applications...unless somebody ever gets around to building a set of misaligned warp coils

But we do appreciate your interest. We certainly hope you'll enjoy the rest of the story as it gets posted.

Cheers,
Kevin

[Muranuse]

Looks at first couple of lines.
Uhuh... makes senses so far...
End of post
BRAIN! You get back here!
NO! This is too much for meeee-hehehe...