The Math of Physics

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This page is to provide links and information for Orbital Mechanics Math

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Formulas

Conversion:

A mile is a longer distance than a kilometer. To convert miles to kilometers, multiply the miles value by 1.61. To convert kilometers to miles, divide the kilometers value by 1.61.

Miles = kilometers / 1.61
Kilometers = miles × 1.61

Now, here is a list of some important formulas:

Pi = 3.141592654….., this is the ratio of the circle's diameter to a circle's circumference (circumference / diameter)

Radius of circle = r
Volume of sphere = 4/3 Pi × r^3
Area of sphere = 4 Pi × r^2
Volume of cylinder =Pi × r^2 × height
Area of circle = Pi × r^2
Circumference of circle = Pi × d or 2Pi × r
Discharge = length × width × velocity = x^3/ sec.
Velocity = distance / time
Acceleration = distance / time^2

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Calculating Launch Azimuth

The formula to calculate a launch heading is :

Launch Azimuth = arcsin (cos (desired_orbital_inclination) / cos (launch_ latitude))

To be more elegant you can calculate that with the DGIV inboard calculator in Obiter:

Example:Take the scenario « Earth/Landed KCS departure ISS » Open the Orbit MFD set ISS as target SET Projection to EQU (click on FRM button) and notice the target « INC » value=51.56° Now open the surface MFD and note your latitude = 28.591° N

Open the calculator with « D8 » key, go on bottom panel and click with mouse on the following button:

« 28.591 COS M1 » (now you have the cos of latitude stored in memory 1)

« 51.56 COS / MR1 ENT ASIN » (you made cos of inclinaison, and divided this result by memory 1 then asin)

Result= 45.075° Engage PRO903SPEC45 or PRO904SPEC45 to launch with this heading.

Wait ! Usually ascent for ISS is 42° so why is there such a difference? Because the real calculation is a bit more complicated: you have to take in account hte earth's rotation, final orbital speed, etc. etc. This said if you launch with such heading you'll be off by only 3° Relative Inc, a 28 second align plane burn. So even slightly off this calculation can be useful for you in many situations. It's very more satisfying to calculate that yourself than simply blindly read what others tell you to do.

If you want to launch southbound simply do 180°-45.075° = 134.925° (« 180-45.075 ENT » on calculator)

IMPORTANT: you cannot launch if the target INC is lower than your Latitude. For MIR there is no solution

as it's INC is at 26.96° (lower than your latitude wich is 28.59°) this is why we launch at 90° and align plane later.

TIPS: You can find the launch azimuth formula and the DGIV calculator way of doing it in the check-list MFD « 12 - flight computer operation »

Accurate calculation:
If you want to be precise, you'll have to use the formula below. Is it worth a 28 second burn? Maybe! There is some beauty in a precise trajectory, but such complexity may be useless sometimes. You decide.

The complete formula is as follows:

Rough Launch Azimuth = arcsin (cos (desired_orbital_inclination) / cos (launch_ latitude))

Real Launch Azimuth = atan(tan(Rough Launch Azimuth) - SpeedOnEarth/(FinalOrbitVelocity* cos(Rough

Launch Azimuth)))

SpeedOnEarth = 407.9 can be found on Orbit MFD left parameter VEL

FinalOrbitVelocity = 7699.0 Your target's speed. (target ISS see right VEL parameter)

Now calculate the Rough Launch Azimuth. You must not have the result in Degree but in radian so do the following:

« 28.591 COS M1 51.56 COS / MR1 ENT SPEC ASIN M2 » (« Spec » will pass the calculator in Radian)

Rough Launch Azimuth = 0.7867 (Result stored in M2)

Now type:

« MR2 COS * 7699.0 ENT M1 407.9 / MR1 ENT M1 » (Result stored in M1)

And continue:

« MR2 TAN – MR1 ENT SPEC ATAN »

Final Accurate Launch Azimuth = 42.8491°

So, the complete DGIV calculator formula is as follow:

YourLat = Actual latitude

DesInc = Wanted inclinaison

FinalSpeed = Target Orbital speed

Actual Speed = Actual speed (due to planet rotation)

« YourLat COS M1 DesInc COS / MR1 ENT SPEC ASIN M2 MR2 COS * FinalSpeed ENT M1 ActualSpeed / MR1 ENT M1 MR2 TAN – MR1 ENT SPEC ATAN »

If you want this formula instead of the rough one in the check-list MFD simply edit the file       « sound/DeltaGliderIV/check_list.txt » find the place were is the old formula and replace by the above.

Take care that the horizontal place is very sparse as you'll need to cut the line several times.

Excercice:

Take the scenario « Mission /Fly me around the moon » and try to reach Luna OB1, you'll have to apply such calculation: (try it by yourself with above formula without watching below)

« 41.118 COS M1 91.48 COS / MR1 ENT SPEC ASIN M2 MR2 COS * 1480 ENT M1 3.49 / MR1 ENT M1 MR2 T AN – MR1 ENT SPEC ATAN »

Result= -2.0997°

So, as -2° is not possible we'll simply do 360° (equal to 0°) minus -2° =358° open map MFD wait that Luna OB1's trajectory pass over you and launch PRO905SPEC358

Finally , if you want to know more about this see the OrbiterWiki page:

http://www.orbiterwiki.org/wiki/Launch_Azimuth

 Above calulations was copied from DGIV Documentation by

Daniel Polli 2003-2007 http://orbiter.dansteph.com

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