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How do scientists place satellites into orbit? How do they calculate the gravity acting on the spacecraft in order to generate exact opposite force by means of Kinetic energy to keep the satellite stable?

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    $\begingroup$ Scientists don't, Engineers do. $\endgroup$ – ja72 Feb 24 '14 at 13:53
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Do you think anyone calculated the earth's speed to stay in orbit around the sun? As long as the speed is in the correct range the satellite will stay in orbit.

For a satellite around the earth, the minimum speed is about 7 km/s. This is tangential speed, i.e. speed parallel to the earth's surface. Anything below 7 km/s, and the satellite will fall back.

If the speed is above 11.2 km/s, the gravity of the earth is insufficient to hold the satellite back, and it will escape the earth - but it needs more speed still (in the right direction) to escape the sun as well.

Between 7 and 11.2 km/s, the satellite will be in some orbit. For example, at an altitude of about 35700 km, a speed of 3.1 km/s is sufficient to keep the satellite in geostationary orbit. At the distance of the moon, about 1 km/s is sufficient. Wikipedia has a good starting article.

All these figures assume a circular orbit. In elliptical orbits the speed varies depending on the position in the orbit. See Kepler's laws. He developed them for planets, but they apply to satellites as well.

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First of all it is a bit strange to say that scientists place satellites into orbit. Since a rocket does all the work, which in turn is build by engineers. But you might say that the people who control the rocket/satellite can be called scientists.

I am not an expert on the planning of trajectories of satellites. However I do suspect that the trajectories of satellites who are put in orbit around Earth will use something like an Hohmann transfer or for higher orbits even a bi-elliptic transfer.

But there are also more complex orbits. For example the Earth is not spherical, but more like an oblate spheroid, which causes inclined orbits to have nodal precession. Even gravitational anomalies can make low orbits unstable over time, which is especially noticeable in Lunar orbit.
You also have to take into account orbital decay due to atmospheric drag, since space around Earth is not a complete vacuum, but this is more about maintaining orbit.

And there are also satellites who's orbit is hugely affected by multiple celestial bodies. These orbits are often near a Lagrangian point. For example the Gaia spacecraft which orbits around Sun–Earth L2 Lagrangian point.

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At a very basic level for the computation of a circular orbit it is just enough to equate the centripetal and the gravitational force:

$$F_g=F_c$$ $$G \frac{mM}{r^2} = m \frac{v^2}{r}$$ where $G$ is the gravitational constant, $m$ is the mass of the satellite, $M$ is the mass of the Earth, $v$ is the satellites tangential velocity and $r$ is the altitude of the satellite with respect to Earth's centre. We then obtain:

$$v=\sqrt{\frac{GM}r}$$

If the velocity does not strictly meet this condition, it does not mean that the satellite will fall, it will just follow an elliptical orbit where the altitude is not constant as a lot of celestial bodies do.

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