List of Mathematical Equations

[Chezburgar7300]

Hello! I thought I'd start up a community effort to put a fair majority of relevant equations for space travel and physics all in the one thread. *

[TABLE=class: grid, width: 1500]

[TR]

[TD]Name[/TD]

[TD]Equation[/TD]

[TD]Where[/TD]

[TD]Measured in:[/TD]

[TD]Purpose[/TD]

[TD]Courtesy of:[/TD]

[/TR]

[TR]

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

[TD=align: center][/TD]

[TD]

ÃÂ = Mass density of the penetrated fluid

v = Flow velocity relative to the object

C_D = Drag coefficient

A = Cross-section area

[/TD]

[TD=align: center]Newtons

(N)[/TD]

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

[TD=align: center]K^2[/TD]

[/TR]

[TR]

[TD]Drag Coefficient ()[/TD]

[TD][/TD]

[TD]

= Drag force

= Mass density of fluid

= Speed of object relative to fluid

= Reference area

[/TD]

[TD=align: center]N/A[/TD]

[TD=align: center]This equation is a stub! This needs to be improved ASAP! Feel free to post the relevant details in the comments![/TD]

[TD=align: center][/TD]

[/TR]

[TR]

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

[TD][/TD]

[TD]

ÃÂ = Mass density of the penetrated fluid

v = Flow velocity relative to the object

C_L = Lift coefficient

A = Planform area

[/TD]

[TD=align: center]Newtons

(N)[/TD]

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

[TD=align: center]K^2[/TD]

[/TR]

[TR]

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

[TD][/TD]

[TD]

α = 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]

[TD=align: center]K^2[/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]

[TD]Courtesy of:[/TD]

[/TR]

[TR]

[TD]Orbital Speed of a Circular Orbit

()[/TD]

[TD=align: center][/TD]

[TD]

= Radius of orbited body

= Gravity of orbited body (

= Altitude over surface of orbited body (m)

[/TD]

[TD=align: center]Metres per second

(m.s^-1)[/TD]

[TD=align: center]To determine the speed of a spacecraft on a given circular orbit around a celestial body.[/TD]

[TD=align: center][/TD]

[/TR]

[TR]

[TD]The 'Rocket' Equation / Delta Velocity ()[/TD]

[TD=align: center][/TD]

[TD]

= Exhaust Velocity (Engine Isp (s) * Surface Gravity (m.s^-2)

= Log Base e (2.718)

= Total Mass with propellant (kg)

= Total Mass without propellant (kg)

[/TD]

[TD=align: center]Metres per second

(m.s^-1)[/TD]

[TD=align: center]To determine the maximum velocity change of a spacecraft.[/TD]

[TD=align: center]Skyler4856[/TD]

[/TR]

[TR]

[TD]Orbital Period (T)[/TD]

[TD=align: center][/TD]

[TD]

= Altitude of semi major axis

= Standard gravitational parametre (m^-3.s^-2)

[/TD]

[TD=align: center]Seconds

(s)[/TD]

[TD=align: center]To determine the length of time to completely revolve around a given orbit.[/TD]

[TD=align: center][/TD]

[/TR]

[TR]

[TD]Gravitational Parametre ()[/TD]

[TD=align: center][/TD]

[TD]

= Gravitational constant

= Mass of body (kg)

[/TD]

[TD=align: center]Metres cubed per

second squared

(m^-3.s^-2)[/TD]

[TD=align: center]This equation is a stub! This needs to be improved ASAP! Feel free to post the relevant details in the comments![/TD]

[TD=align: center][/TD]

[/TR]

[TR]

[TD]Vis-Viva Equation ()[/TD]

[TD=align: center][/TD]

[TD]

= Gravitational constant

= Mass of central body (kg)

= Distance between the two bodies (m)

= Semi-Major Axis (m)

[/TD]

[TD=align: center][/TD]

[TD=align: center]This equation is a stub! This needs to be improved ASAP! Feel free to post the relevant details in the comments![/TD]

[TD=align: center]Red Iron Crown[/TD]

[/TR]

[TR]

[TD]Semi-Major Axis ()[/TD]

[TD=align: center][/TD]

[TD]

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

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

[/TD]

[TD=align: center]Metres

(m)[/TD]

[TD=align: center]To determine the Semi-Major Axis of an orbiting body/craft[/TD]

[TD=align: center][/TD]

[/TR]

[TR]

[TD]Escape Velocity ()[/TD]

[TD=align: center][/TD]

[TD]

= Gravitational constant

= Mass of body (kg)

= Distance from the centre of the central body

[/TD]

[TD=align: center]Metres per second

(m.s^-1)[/TD]

[TD=align: center]To determine the escape velocity of a body.[/TD]

[TD=align: center][/TD]

[/TR]

[/TABLE]

[TABLE=class: grid, width: 1500, align: left]

[TR]

[TD]Name[/TD]

[TD]Equation[/TD]

[TD]Where[/TD]

[TD]Measured in:[/TD]

[TD]Purpose[/TD]

[TD]Courtesy of:[/TD]

[/TR]

[TR]

[TD]Tidal Force

()[/TD]

[TD][/TD]

[TD]

= Gravitational Constant

= Mass of Main Object

= Mass of Secondary Object

= Radius of Secondary Object

= Distance between Secondary and Primary Objects (From center to center)

[/TD]

[TD=align: center]Newtons

(N)[/TD]

[TD=align: center]N/A[/TD]

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

[/TR]

[TR]

[TD]Roche Limit ()[/TD]

[TD][/TD]

[TD]

r = Radius of Primary Body

= Density of Primary Body

= Density of Satellite Body

[/TD]

[TD=align: center]Metres

(m)[/TD]

[TD=align: center]To determine the distance at which a body held together by it's own gravity will disintegrate from the tidal forces of another body.[/TD]

[TD=align: center]YNM[/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]

[TD]Courtesy of:[/TD]

[/TR]

[TR]

[TD]Schwarzschild Radius ()[/TD]

[TD][/TD]

[TD]

= The speed of light (m.s^-1)

= Mass (kg)

= Gravitational Constant

[/TD]

[TD=align: center]Metres

(m)[/TD]

[TD=align: center]To determine the radius of a non-rotating (Schwarzschild) Black Hole, given the mass.[/TD]

[TD=align: center][/TD]

[/TR]

[TR]

[TD]Gas Temperature

()[/TD]

[TD][/TD]

[TD]

= Distance from centre of Black Hole (m)

[/TD]

[TD=align: center]Kelvins

(K)[/TD]

[TD=align: center]To determine the temperature of gas from a certain distance from the center of a Black Hole.[/TD]

[TD=align: center][/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]

[TD]Courtesy of:[/TD]

[/TR]

[TR]

[TD=align: center]

)

Mass-Energy Equivalence (

[/TD]

[TD=align: center][/TD]

[TD]

m = Mass (g)

c = Speed of light (m/s)

[/TD]

[TD=align: center]Joules

(J)[/TD]

[TD]

To determine the maximum potential energy of matter.

[/TD]

[TD][/TD]

[/TR]

[TR]

[TD]Density ()[/TD]

[TD=align: center][/TD]

[TD]

= Mass (kg)

= Volume (m³)

[/TD]

[TD=align: center]Kilograms per

cubed metre

(kg.m^-3)[/TD]

[TD=align: center]To determine the average density of an object of any three dimensional shape.[/TD]

[TD=align: center][/TD]

[/TR]

[TR]

[TD]Force ()[/TD]

[TD=align: center][/TD]

[TD]

= Mass (Kg)

= Acceleration (m.s^-2)

[/TD]

[TD=align: center]Newtons

(N)[/TD]

[TD=align: center]To determine the potential force of an object where the mass and acceleration are known.[/TD]

[TD=align: center][/TD]

[/TR]

[TR]

[TD]Newton's Theory of Gravitation ()[/TD]

[TD=align: center][/TD]

[TD]

= Gravitational Constant

= Mass of Object 1 (kg)

= Mass of Object 2 (kg)

= Distance between centres of masses

[/TD]

[TD=align: center]Newtons

(N)[/TD]

[TD=align: center]This equation is a stub! This needs to be improved ASAP! Feel free to post the relevant details in the comments![/TD]

[TD=align: center]

cicatrix[/TD]

[/TR]

[TR]

[TD]Displacement ()[/TD]

[TD=align: center][/TD]

[TD]

= Initial velocity of object (m.s^-1)

= Time (s)

= Acceleration of object (m.s^-2)

[/TD]

[TD=align: center]Metres

(m)[/TD]

[TD=align: center]To determine the displacement of a moving object from it's starting position.[/TD]

[TD=align: center][/TD]

[/TR]

[TR]

[TD]Velocity Squared ()[/TD]

[TD=align: center][/TD]

[TD]

= Initial velocity of object (m.s^-1)

= Acceleration of object (m.s^-2)

= Displacement of object (m)

[/TD]

[TD=align: center]Metres per second

squared

(m.s^-2)[/TD]

[TD=align: center]To determine the squared velocity of an object, so that it is possible to determine the square root of that.[/TD]

[TD=align: center][/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]

[TD]Courtesy of:[/TD]

[/TR]

[TR]

[TD]Thrust ()[/TD]

[TD=align: center][/TD]

[TD]

= Speed of exhaust gases relative to rocket

= Rate of change of mass with respect to time (mass flow rate of exhaust)

[/TD]

[TD=align: center]Newtons

(N)[/TD]

[TD=align: center]To determine the thrust of a rocket engine (once launched/tested).[/TD]

[TD=align: center][/TD]

[/TR]

[/TABLE]

Want to help out?

Please, by all means post relevant equations in the thread and I'll be sure to edit them in! Please state the name of the equation, the formula and the other parametres of the table apart including the purpose. A source is always welcomed!

If you have three or more equations in the one post, please post them in a spoiler to avoid clogging up the pages

If you don't have equations that fit the current categories, but are still related to space by all means post them! I'll be sure to add another category.

Have a problem with a currently listed equation? Post a revised version that is (more) accurate! Please include a source or example for verification.

This is currently not a complete resource, and will rely on the community to help the community. I will add more equations as I find or hear about them, though.

*Note: Equations have not been tested and/or verified, so they may not be accurate and/or may be missing parametres.

Edited June 29, 2015 by Chezburgar7300
Aerodynamics updates thanks to K^2! Now added a 'courtesy of:' column.

[Skyler4856]

Itd be ideal to incorporate this

[longbyte1]

I laughed when I saw density on the list.

[Bill Phil]

Would force not be a good equation to have? Pretty handy. Also a few other equations I can't recall just yet.

Oh! Kinetic energy! And Potential energy. As well as the gravitational acceleration equation.

d = 1/2*a*t^2 is important as well.

[TheDarkStar]

Don't forget this:

s''(t) = a(t)

s'(t) = v(t)

where s, v, and a are position, velocity, and acceleration as a function of t.

[Bill Phil]

Oh, and Kepler's Third Law:

the square of the orbital period is equal to the cube of the semi-major axis.

[SaturnV]

You should do this on the wiki of KSP (not to mention there is one already), this forum does not support TeX or MathML

[Shaggygoblin]

um...

missing THE most important one...

Towel =/= 42

∞ = 42

Edited March 11, 2015 by Shaggygoblin

[EzinX]

How would you build a reasonable accurate model for aerodynamic and hydrodynamic drag in realtime? It doesn't need to be accurate enough to build a real wing or boat hull, just to approximate the drag given :

A low resolution grid of the surface of the craft exposed to the air stream. As in, a series of samples over the surface with average surface slope values at those points.

And that's it, I guess. Ferram doesn't document his code, and I don't have an aerospace engineering degree.

[cicatrix]

A nice effort, yet 'where' parts should also list relative units of measurement.

In most cases it's obvious like velocity is in m/s, but here, for example:

g = Gravity of orbited body

what unit? m/s², newtons, or what?

= Standard gravitational parametre

- you should also specify that it's equal to G * m (kg) and provide G = 6.67384(80) * 10^-11

^{and where's this:}

Edited March 12, 2015 by cicatrix

[PB666]

A nice effort, yet 'where' parts should also list relative units of measurement.

In most cases it's obvious like velocity is in m/s, but here, for example:

g = Gravity of orbited body

what unit? m/s², newtons, or what?

http://upload.wikimedia.org/math/9/3/9/939974a71dda1b83cce5ab82a2d2cec1.png = Standard gravitational parametre

- you should also specify that it's equal to G * m (kg) and provide G = 6.67384(80) * 10^-11

^{and where's this:}

http://upload.wikimedia.org/math/0/f/3/0f36df929ac9d711a8ba8c5658c3bfee.png

Wiki overstates the precision of G, best not to use more than 5 digits. The most recent attempt to determine G precisely found a confidence interval wider than previous estimates.

[Xannari Ferrows]

......-M...../ T - MG - KV² \

Y₁ =  L_n|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.

Edited March 12, 2015 by Xannari Ferrows

[PB666]

I think you are missing the point.

What we need is a sticky page somewhere with equations (e.g. excel) in a format that most people and utilize such that they need only plug their parameters into

r = <Kerbin's radius>

alt = <altitude>

inc = <inclination>

etc.

And then the wonderful little program determines all the orbital parameters.

Having 60 different equations does not help if you have to spend 3 days calculate something a program can do in 5 trillionths of a second.

- - - Updated - - -

G = universal gravitational constant

g = acceleration due to gravity at earths surface

[Rune]

You don't have the rocket equation in yet? dV=Vex*ln(MR), where dV= delta-v, Vex= exhaust velocity= 9.81*isp and MR is mass ration (GLOW/OEW).

Rune. That's like, the only one I use.

[maltesh]

I wound up throwing this together a couple weekends ago whenbashing together a program to simulate the surface coverage of the Draim tetrahedral satellite configuration.

For the purposes of the above equations, θ is in radians, not degrees.

[PB666]

Elliptical Equations using any 3 dimensional spatial coordinates

If an ellipse is composed of a center © and two focal points (F1, F2)) Then

C = X₀ Y₀ Z₀ (the center of the elliptical area)

F1 = X₁ Y₁ Z₁ (the systems center of mass)

F2 = X₂ Y₂ Z₂ (an imaginary point in space that lies along the line CF₁ at an equal distance from C on the opposing side)

E_X = X_Ex Y_Ex Z_Ex (a point along the ellipse or orbit)

f = ((X₁ - X₀)² + (Y₁ - Y₀)² + (Z₁- Z₀)²)^1/2

2a = ((X₁ - X_Ex)² + (Y₁ - Y_Ex)² + (Z₁- Z_Ex)²)^1/2 + ((X₂ - X_Ex)² + (Y₂ - Y_Ex)² + (Z₂- Z_Ex)²)^1/2

2a is coincidentally the Major-axis

a = 2a/2 = semi-major axis

a² = b² + f²

b² = (1-f²/a²)^1/2

e = f/a

l = b²/a² = semi-latis rectum.

It is possible to reassign the coordinates to two dimensional geometry bases on (-a,0) (a,0) and (0,-, (0,+ since the line CF1, CF2 can be projected to -a and +a and since

((X₁ - X_Ex)² + (Y₁ - Y_Ex)² + (Z₁- Z_Ex)²)^1/2 = ((X₂ - X_Ex)² + (Y₂ - Y_Ex)² + (Z₂- Z_Ex)²)^1/2 at reassigned points (0,-, (0,+

This will simplify the determination of spatial position by converting the points back when arithmatic is complete. [Appended, it is necessary to have at least one more point other then Pe and Ap to derive a converter to the planar coordinate system since there are a circle of points of r = b from the center that can satisfy the above equity. In order find which 2 points lie in the plane of the ellipse the semi-latis rectum intercept might be convenient. The problem will arise in trying to convert the coordinates back into the original coordinate system]. If you are using a program like mechJeb the program gives an angle-to-prograde (AtP), therefore the periapsis also has an angle to prograde, the coordinates at positions rotated +/- 90^o relative the AtP_Pe or AtP_Apo will reflect the elliptical semi-latis rectum intercepts. This assumes that the physics engine does not change they way is determines SOI in the future, because it assumes that the F1 = central bodies center. These slr intercept coordinates can be related to the planar slr coordinates system to derive a conversion equation.

If central body is massive and satellite is not then

2f = Ap-Pe

therefore

e = (Ap-Pe)/Axis_major

Axis_major = Ap + Pe + r_{central body} (only if Ap and Pe are measured in altitude and not radius from CB otherwise Ap + Pe)

if a is known and the period is also known then it is possible to determine GM with great precision but not G or system mass

GM = 4pi²a³ / T_p²

Now wouldn't we rather have a computer program that handles these?

Edited March 12, 2015 by PB666

[stupid_chris]

I'm fairly sure making all the equations in the OP use tex would make it much more readable.

[cicatrix]

It would also be great if formulas were given in TeX Notation.

More info about TeX

PS. stupid_chris ninjae'd my suggestion

[Chezburgar7300]

You don't have the rocket equation in yet? dV=Vex*ln(MR), where dV= delta-v, Vex= exhaust velocity= 9.81*isp and MR is mass ration (GLOW/OEW).

Rune. That's like, the only one I use.

It's in there, under Astrodynamics. I'll add an alternate name for it.

Also, how do I use Tex? I had a quick read over the wiki page thanks to cicatrix and it sounds really helpful and easy enough to use for my brain Is it 'inbuilt' into the forum software?

Thanks for all of the equations, guys! I'm *trying* to add them in now.

Edited March 12, 2015 by Chezburgar7300

[cicatrix]

Also, how do I use Tex? I had a quick read over the wiki page thanks to cicatrix and it sounds really helpful and easy enough to use for my brain Is it 'inbuilt' into the forum software?

Thanks for all of the equations, guys! I'm adding them in now.

There are plugins for vBulletin, I think, that generate TEX, but you could use external tool like this, for example:

http://www.codecogs.com/latex/eqneditor.php?lang=en-us

and provide both image and TeX notation, for example:

F = G\frac{m_{1}m_{2}}{r^{2}}

[Voyager55]

Yo Dawg, I heard you like spoilers.

So I put more spoilers, inside my spoilers.

[Red Iron Crown]

This thread needs more Vis Viva.

[Chezburgar7300]

There are plugins for vBulletin, I think, that generate TEX, but you could use external tool like this, for example:

http://www.codecogs.com/latex/eqneditor.php?lang=en-us

and provide both image and TeX notation, for example:

http://latex.codecogs.com/gif.latex?F&space;=&space;G\frac{m_{1}m_{2}}{r^{2}}

F = G\frac{m_{1}m_{2}}{r^{2}}

That helped me out more than you know I'll progressively convert each equation over the next few days. Thanks!

[Bill Phil]

This thread needs more Vis Viva.

Agreed.

- - - Updated - - -

Using an excel spreadsheet, you could have 5 pages, one for each category. Then you have each equatioj in one column, with the variables in the same column. Plug and go.

[PB666]

So Im finding a problem in trying to extract position information from the program, I suppose some have managed to work around this. When a ship is away from the planet, the universe mores relative to the ship, the origin can be meters away from the ship, so Im guessing the scene origin is then working backwards to determine the relative positions and motions from the planets and ship. Obviously there is a game clock somewhere positioning the planets (feeding information where the planets should be at time x) as ships move in an out of SOIs, the game is inspecting the game clock, then determining where the satellites are going to be with respect to the CB of the SOI. Ad hoc calculations are going to be less useful unless one can get a frame of reference. For example I placed ship in perfect 100km orbit with miniscule inclination (0 but calculated that it was +/- 8 meters deviancy from the equitorial plane).After applying F12 'show debug pane' I then froze the game at precisely Longditude 0, however the velocity vectors were not as expected, and they did not follow the angle to prograde (235.6) either. So apparently there is a masked coordinate system that determines the velocity vectors. The Y velocity coincided close to the expected values for Y based on inclination, but the V_X and V_Z coordinates appear to follow V² = V_X² + V_Z² but not sure what the orientation of the X and Z dimensions are.

The other problem was in trying to extract the information my computer had a memory free up because there was an overrun, and I had to shut down everything and reboot. I can manage the debug info, all I need is a reference position.

If I find something different from the above I will edit this post.

I think Kerbal needs a 'force' coordinate debug button that works with the last velocity and in [escape] freeze presents X, Y and Z coordinates or the rotational equivilants.