In any annular orbit, the centripetal force appropriate to advance the apogee is provided by the gravitational force on the satellite. To account the geostationary apogee altitude, one begins with this equivalence, and uses the actuality that the alternate aeon is one sidereal day.
\mathbf{F}_\text{c} = \mathbf{F}_\text{g}
By Newton's additional law of motion, we can alter the armament F with the accumulation m of the article assorted by the dispatch acquainted by the article due to that force:
m \mathbf{a}_\text{c} = m \mathbf{g}
We agenda that the accumulation of the accessory m appears on both abandon — geostationary apogee is absolute of the accumulation of the satellite.[4] So artful the distance simplifies into artful the point area the magnitudes of the centripetal dispatch appropriate for alternate motion and the gravitational dispatch provided by Earth's force are equal.
The centripetal acceleration's consequence is:
|\mathbf{a}_\text{c}| = \omega^2 r
where ω is the angular speed, and r is the alternate ambit as abstinent from the Earth's centermost of mass.
The consequence of the gravitational dispatch is:
|\mathbf{g}| = \frac{G M}{r^2}
where M is the accumulation of Earth, 5.9736 × 1024 kg, and G is the gravitational constant, 6.67428 ± 0.00067 × 10−11 m3 kg−1 s−2.
Equating the two accelerations gives:
r^3 = \frac{G M}{\omega^2} \to r = \sqrt[3]{\frac{G M}{\omega^2}}
The artefact GM is accepted with abundant greater attention than either agency alone; it is accepted as the geocentric gravitational connected μ = 398,600.4418 ± 0.0008 km3 s−2:
r = \sqrt[3]{\frac\mu{\omega^2}}
The angular dispatch ω is begin by adding the bend travelled in one anarchy (360° = 2π rad) by the alternate aeon (the time it takes to accomplish one abounding revolution: one sidereal day, or 86,164.09054 seconds).[5] This gives:
\omega \approx \frac{2 \mathrm\pi~\mathrm{rad}} {86\,164~\mathrm{s}} \approx 7.2921 \times 10^{-5}~\mathrm{rad} / \mathrm{s}
The consistent alternate ambit is 42,164 kilometres (26,199 mi). Adding the Earth's close radius, 6,378 kilometres (3,963 mi), gives the distance of 35,786 kilometres (22,236 mi).
Orbital dispatch (how fast the accessory is affective through space) is affected by adding the angular dispatch by the alternate radius:
v = \omega r \approx 3.0746~\mathrm{km}/\mathrm{s} \approx 11\,068~\mathrm{km}/\mathrm{h} \approx 6877.8~\mathrm{mph}\text{.}
Now, by the aforementioned formula, let us acquisition the geostationary apogee of an article in affiliation to Mars (an areostationary orbit). The geocentric gravitational connected GM (which is μ) for Mars has the amount of 42,828 km3s-2, and the accepted rotational aeon (T) of Mars is 88,642.66 seconds. Since ω = 2π/T, application the blueprint above, the amount of ω is begin to be approx 7.088218×10-5 s-1. Thus, r3 = 8.5243×1012 km3, whose cube basis of is 20,427 km; adding the close ambit of Mars (3396.2 km) we accept 17,031 km.
\mathbf{F}_\text{c} = \mathbf{F}_\text{g}
By Newton's additional law of motion, we can alter the armament F with the accumulation m of the article assorted by the dispatch acquainted by the article due to that force:
m \mathbf{a}_\text{c} = m \mathbf{g}
We agenda that the accumulation of the accessory m appears on both abandon — geostationary apogee is absolute of the accumulation of the satellite.[4] So artful the distance simplifies into artful the point area the magnitudes of the centripetal dispatch appropriate for alternate motion and the gravitational dispatch provided by Earth's force are equal.
The centripetal acceleration's consequence is:
|\mathbf{a}_\text{c}| = \omega^2 r
where ω is the angular speed, and r is the alternate ambit as abstinent from the Earth's centermost of mass.
The consequence of the gravitational dispatch is:
|\mathbf{g}| = \frac{G M}{r^2}
where M is the accumulation of Earth, 5.9736 × 1024 kg, and G is the gravitational constant, 6.67428 ± 0.00067 × 10−11 m3 kg−1 s−2.
Equating the two accelerations gives:
r^3 = \frac{G M}{\omega^2} \to r = \sqrt[3]{\frac{G M}{\omega^2}}
The artefact GM is accepted with abundant greater attention than either agency alone; it is accepted as the geocentric gravitational connected μ = 398,600.4418 ± 0.0008 km3 s−2:
r = \sqrt[3]{\frac\mu{\omega^2}}
The angular dispatch ω is begin by adding the bend travelled in one anarchy (360° = 2π rad) by the alternate aeon (the time it takes to accomplish one abounding revolution: one sidereal day, or 86,164.09054 seconds).[5] This gives:
\omega \approx \frac{2 \mathrm\pi~\mathrm{rad}} {86\,164~\mathrm{s}} \approx 7.2921 \times 10^{-5}~\mathrm{rad} / \mathrm{s}
The consistent alternate ambit is 42,164 kilometres (26,199 mi). Adding the Earth's close radius, 6,378 kilometres (3,963 mi), gives the distance of 35,786 kilometres (22,236 mi).
Orbital dispatch (how fast the accessory is affective through space) is affected by adding the angular dispatch by the alternate radius:
v = \omega r \approx 3.0746~\mathrm{km}/\mathrm{s} \approx 11\,068~\mathrm{km}/\mathrm{h} \approx 6877.8~\mathrm{mph}\text{.}
Now, by the aforementioned formula, let us acquisition the geostationary apogee of an article in affiliation to Mars (an areostationary orbit). The geocentric gravitational connected GM (which is μ) for Mars has the amount of 42,828 km3s-2, and the accepted rotational aeon (T) of Mars is 88,642.66 seconds. Since ω = 2π/T, application the blueprint above, the amount of ω is begin to be approx 7.088218×10-5 s-1. Thus, r3 = 8.5243×1012 km3, whose cube basis of is 20,427 km; adding the close ambit of Mars (3396.2 km) we accept 17,031 km.
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