Difference between revisions of "Tutorial: Basic Orbiting (Math)"

Revision as of 21:47, 28 September 2012

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In the basic orbiting tutorial, you were introduced to the concept of orbiting, and basic orbit stabilization, as well as an orbital table to help you along. Now, what if you want an orbit that isn't on that table? What if you want to have an orbit with a specific period? That's where these formulae come in.

Orbital Speed

The relation between orbital speed and acceleration is given by the formula:

${\displaystyle a={\frac {v^{2}}{r}},}$

where a is the acceleration due to gravity, v is the horizontal speed, and r is the radius of orbit.

Of course, gravity varies depending on your distance from the planet, so we also need the following formula to determine a based on your altitude:

${\displaystyle a=g\left({\frac {R}{R+h}}\right)^{2},}$

where g is the acceleration due to gravity at sea level (9.807 m/s²), R is the radius of Kerbin (600 km), and h is the altitude of your orbit.

Note: Since the radius of your orbit r is equal to the radius of Kerbin R plus your altitude h, we can substitute R + h for r.

Now we have two expressions for a, so we set them on opposite sides of an equation and simplify:

${\displaystyle {\begin{aligned}g\left({\frac {R}{(R+h)}}\right)^{2}&={\frac {v^{2}}{r}}\\g\left({\frac {R}{(R+h)}}\right)^{2}&={\frac {v^{2}}{R+h}}\\g{\frac {R^{2}}{(R+h)^{2}}}&={\frac {v^{2}}{R+h}}\\g{\frac {R^{2}}{R+h}}&=v^{2}\\v^{2}&=g{\frac {R^{2}}{R+h}}\\v&={\sqrt {g{\frac {R^{2}}{R+h}}}}\\v&=R{\sqrt {\frac {g}{R+h}}}\\\end{aligned}}}$

Finally, substituting known values for g and R,

${\displaystyle v=600\ 000\ \mathrm {m} {\sqrt {\frac {9.807\ \mathrm {m} /\mathrm {s} ^{2}}{600\ 000\ \mathrm {m} +h}}}}$

Orbital Period

From the basic mechanics formula:

${\displaystyle d=vt}$

We know v from the above, and d is simply the circumference of a circle with a radius equal to your orbital altitude plus the radius of Kerbin:

${\displaystyle t=2\pi {\frac {600\ 000\ \mathrm {m} +h}{v}}}$