Geosynchronous Orbit (Math)/ru

Геосинхронная или, в частности, геостационарная орбита - это орбита, на которой период обращения равен "суткам" на этом небесном теле (обозначается как сидерический период или сидерический период обращения), таким образом, Вы постоянно остаетесь в одном и том же районе планеты. Сила тяготения и центростремительная сила также должны быть равными, что справедливо для любой круговой орбиты. Различие между геосинхронными и геостационарными орбитами в том, что у геосинхронных орбит существует орбитальный период в одни сутки, однако, геостационарные орбиты также являются экваториальными.

Формулы

So, in order to calculate the geostationary orbit around any given body, we must first create an equation with the force of gravity and the centripetal force.The force of gravity is equal to:

$F_{G}={\frac {G\cdot M_{1}\cdot M_{2}}{r^{2}}}$

where G is the gravitational constant ($6.67384\cdot 10^{-11}{\frac {m^{3}}{kg\cdot s^{2}}}$), M₁ is the mass of the body, M₂ is the mass of the satellite, and r is the distance between the center of mass of the planet and that of the satellite.

Now, because in a geostationary orbit the gravitational force is the same as the centripetal force, we can put them in opposite sides of the equation:

$F_{G}={\frac {G\cdot M_{1}\cdot M_{2}}{r^{2}}}=M_{2}\cdot {\frac {v^{2}}{r}}=F_{C}$

The masses of the satellite cancel out, so we are left with:

${\frac {G\cdot M_{1}}{r^{2}}}={\frac {v^{2}}{r}}$

So, no matter how large your satellite is, the geostationary altitude will be the same.

We know that velocity is equal to distance divided by time. In this case, the distance is the circumference of your orbit, which is 2πr, and time in this instance is your orbital period, or one day for the planetary body. So, we now have:

$v={\frac {2\cdot \pi \cdot r}{t}}$

But we are dealing with velocity squared, so we square our equation and get:

$v^{2}={\frac {4\cdot \pi ^{2}\cdot r^{2}}{t^{2}}}$

Plugging that back into the original equation, we get:

${\frac {G\cdot M_{1}}{r^{2}}}={\frac {4\cdot \pi ^{2}\cdot r^{2}}{t^{2}\cdot r}}$

The r² cancels out the r in the denominator and becomes a plain old r. Thus, we have:

${\frac {G\cdot M_{1}}{r^{2}}}={\frac {4\cdot \pi ^{2}\cdot r}{t^{2}}}$

We now multiply by r²:

$G\cdot M_{1}={\frac {4\cdot \pi ^{2}\cdot r^{3}}{t^{2}}}$

And multiply by t²/4π²:

$r^{3}={\frac {G\cdot M_{1}\cdot t^{2}}{4\cdot \pi ^{2}}}$

And by taking the cube root of that, we arrive at our answer (well, sort of):

$r={\sqrt[{3}]{\frac {G\cdot M_{1}\cdot t^{2}}{4\cdot \pi ^{2}}}}$

But remember how r is the distance from the center of mass of the planet to that of the satellite? Well, it turns out that, unlike satellites, the center of mass of a planet is rather far away from its surface:

$a={\sqrt[{3}]{\frac {G\cdot M_{1}\cdot t^{2}}{4\cdot \pi ^{2}}}}-R_{p}$

Where $a$ is the altitude from the sea level of the planet, and $R_{p}$ is the radius of the planet. This formula calculates the altitude above sea level for any given orbital period $t$. To get the height for a stationary orbit, the orbital period must be as long as the sidereal rotation period, the time of a full revolution of the planet relative to the sky.

Always remember to subtract the radius of the planet from your answer once you find altitude.

Note: You probably noticed that the gravitational constant is in three units at once, two of them to a negative power. Don't panic, it will all work out when you plug in real numbers.

Обратите внимание

• Видеоролик о выводе на геосинхронную орбиту.