# List of Mathematical Equations

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This thread needs more Vis Viva.
Agreed.

It's in there!

Yo Dawg, I heard you like spoilers.

So I put more spoilers, inside my spoilers.

Yeah, I was considering putting yet another set of spoilers inside one of the spoiler but I talked myself out of it

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Semi major axis is the average of the apoapsis and periapsis.

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You don't need the speed of a circular orbit. It's handled by the Vis-Viva equation.

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You might want the equations for an ellipse, as well as certain objects that make up ellipses.

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For what it's worth, I recently wound up putting together a Google Doc on Calculating Orbital Position as a function of Time on Elliptical and Hyperbolic orbits.

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You shouldn't really have units in a list of equations.

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Semi major axis is the average of the apoapsis and periapsis.

+ half the radia of the M1 and M2, if the apoaspis is not quoted as a radius. In KSP the are both missing Kerbins radius.

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You shouldn't really have units in a list of equations.

Hmmm...

General terms, maybe?

Like:

force equals mass times acceleration.

distance per unit time

Things like that?

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Hmmm...

General terms, maybe?

Like:

force equals mass times acceleration.

distance per unit time

Things like that?

Very unnecessary if the equation is known to the point that it is remotely useful, but yeah saying the dimensions would be slightly reasonable, unlike saying units.

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+ half the radia of the M1 and M2, if the apoaspis is not quoted as a radius. In KSP the are both missing Kerbins radius.

You mean half the diameter?

It would then be (listed apoapsis + planetary radius) + (listed periapsis + planetary radius) then divided by 2.

That's only if Ap and Pe are above sea level.

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......-M...../ T - MG - KV2 \

Y1 = Ã¢â‚¬â€ Ln|Ã¢â‚¬â€Ã¢â‚¬â€Ã¢â‚¬â€Ã¢â‚¬â€Ã¢â‚¬â€Ã¢â‚¬â€Ã¢â‚¬â€Ã¢â‚¬â€|

.......2K....\.....T - MG....../

Y = Height your rocket reaches

G = 9.81 m/s2

M = Mass of your rocket

V = Velocity of the rocket

K = Wind resistance forces

T = Motor thrust

All very interesting, but this one here is pretty huge.

That equation makes absolutely no sense. You're measuring distance in time squared per distance. Adding a force to a force distance squared per time squared. Taking a logarithm of a dimensional quantity.

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[TD=align: center]Joules

(J)[/TD]

[TD]

To determine the maximum potential energy of matter.

[/TD]

[/TR]

[/TABLE]

[TABLE=class: grid, width: 1500]

[TR]

[TD]Name[/TD]

[TD]Equation[/TD]

[TD]Where[/TD]

[TD]Measured in:[/TD]

[TD]Purpose[/TD]

[/TR]

[TR]

[TD=align: center]

Mass-Energy Equivalence (

[/TD]

[TD=align: center]http://latex.codecogs.com/gif.latex?mc%5E%7B2%7D[/TD]

[TD]

m = Mass (g)

c = Speed of light (m/s)

[/TD]

E=McÃ‚Â² is the energy an object has in a reference frame where its momentum is 0, it is the minimum possible energy of an object.

F=ma is not valid when just the mass and acceleration are known, mass must also be constant, and velocity <<c. It's also just known as the force not "potential force"

s=ut+(1/2)atÃ‚Â² and vÃ‚Â²=uÃ‚Â²+2as are only valid for constant acceleration and u<<c

Edited by BlueCosmology

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Here's one that I don't think has come up yet: The Synodic Period Formula.

It is the time it takes for two cycles of differing periods to synchronize again.

If we're talking rotating and/or orbiting bodies, and all bodies are orbiting/rotating in the same direction, then...

It's good for:

• Mean Solar Day (Use the Planet's Sidereal Orbital Period and the Sidereal Day Length)
• Mean Time Between Full Moon Phases (Use the Planet's Sidereal Orbital Period and its Moon's Sidereal Orbital Period)
• Mean Time Between Hohmann Transfer Windows (Use the Sidereal Orbital Period of the Origin and Destination objects, both must be orbiting the same body)

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That equation makes absolutely no sense. You're measuring distance in time squared per distance. Adding a force to a force distance squared per time squared. Taking a logarithm of a dimensional quantity.

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E=McÃ‚Â² is the energy an object has in a reference frame where its momentum is 0, it is the minimum possible energy of an object.

F=ma is not valid when just the mass and acceleration are known, mass must also be constant, and velocity <<c. It's also just known as the force not "potential force"

s=ut+(1/2)atÃ‚Â² and vÃ‚Â²=uÃ‚Â²+2as are only valid for constant acceleration and u<<c

Moment of Force = Mass(moment) * acceleration. This is useful if you have one of the stretchy-man kerbal rocket that can't fly strait.

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You mean half the diameter?

It would then be (listed apoapsis + planetary radius) + (listed periapsis + planetary radius) then divided by 2.

That's only if Ap and Pe are above sea level.

You would still add the radius if they are below sea level. Pe is negative if it's below sea level when measured from sea level, adding the planetary radius will give the correct Pe radius.

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there's a problem with your semi major axis equation, in the apoiler it states:

$Ap$ = Apoapis of orbit (m) - Measured in altitude from surface

$Pe$ = Periapsis of orbit (m) - Measured in altitude from surface

Both should be measured from the center of the body, not from surface. A sphere is the same as a point mass!

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Proposal 1/2v^2-GM/r=C

Where:

no thrust is added

r is the distance from the center of the body

You use this to figure out how fast you'll be going at a given point on an escape trajectory (the vis viva equation can actually do this according to wikipedia but I don't know how to calculate the semi major axis of a hyperbola, this is what the nearest physics textbook coughed up)

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uh, a better way to write that might be:

1/2 V2^2+GM/r1=1/2V2^2+GM/r2

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Proposals (probably useful for those who creates new planetary systems in-game, or wonder whether an asteroid should broke up or not) :

Tidal force (source) :

1) F = 2GMmR/(d^3) (for equator wrt orbital plane, away from center of satellite body)

2) F = GMmR/(d^3) (for poles wrt orbital plane, to the center of the satellite)

where

M, m = mass of main object & secondary

R = radius of secondary

d = distance between main & secondary, center to center

G = gravitational constant

--------

Roche Limit (source) :

d Ã¢â€°Ë† 2.44*R*(ÃŽÂ¡/ÃÂ)^(1/3)

where

d = the distance of roche limit from main body

R = radius of main body

P = density of main body

ÃÂ = density of satellite

Coefficient vary from 1.26 (only considering tidal forces) to 2.455 (fully liquid body, considering centrifugal force and the possibility of triaxial ellipsoid). 2.44 is as given by roche himself.

--------

Minimum radius of satellite limit whether you can apply the roche limit (reference, other than my personal files)

R Ã¢â€°Ë† (ÃŽÂ¶/GÃÂ)^(1/2)

where

ÃŽÂ¶ = tensile strength

G = gravitational constant

ÃÂ = density of satellite

A body with radius of less than the calculated R will not break at roche limit.

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Thanks YNM! I added the first two equations and will likely add the other at a later date.

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Here are a few more for aerodynamics. The drag is already on there, but I've edited some variables for the conformity and added a description.

[TABLE=class: grid, width: 1500]

[TR]

[TD]Name[/TD]

[TD]Equation[/TD]

[TD]Where[/TD]

[TD]Measured in:[/TD]

[TD]Purpose[/TD]

[/TR]

[TR]

[TD]Drag Equation (FD)[/TD]

[TD=align: center]$\frac{1}{2}\rho v^2 C_D A$[/TD]

[TD]

ÃÂ = Mass density of the penetrated fluid

v = Flow velocity relative to the object

CD = Drag coefficient

A = Cross-section area

[/TD]

[TD=align: center]Force[/TD]

[TD=align: center]Drag generated by a body moving through fluid/air at sub-sonic or low supersonic speeds.[/TD]

[/TR]

[TR]

[TD]Lift Equation (FL)[/TD]

[TD=align: center]$\frac{1}{2}\rho v^2 C_L A$[/TD]

[TD]

ÃÂ = Mass density of the penetrated fluid

v = Flow velocity relative to the object

CL = Lift coefficient

A = Planform area

[/TD]

[TD=align: center]Force[/TD]

[TD=align: center]Lift generated by an airfoil, such as airplane's wing.[/TD]

[/TR]

[TR]

[TD]Lift Coefficient (CL)[/TD]

[TD=align: center]$2 \pi \alpha$[/TD]

[TD]

ÃŽÂ± = Angle of attack (in radians)

[/TD]

[TD=align: center]Dimensionless[/TD]

[TD=align: center]Lift coefficient for a thin, symmetrical airfoil. Valid only for small angle of attack.[/TD]

[/TR]

[/Table]

P.S. I would advise replacing "Measured in:" column with "Dimensions:" column. Units depend on your system, but a length in equation will always be a length, and so on. So that's a useful bit of information

Edited by K^2

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