A satellite in orbit

Q/A Excerpt

Mithun Mohan: How does a satellite or a spacecraft gets placed in orbit?

Nilesh: A satellite is continuously falling towards the Earth. The reason why it does not come down is because it has an exactly the right amount of horizontal/tangential velocity. Giving it that kind of velocity is the rockets job. Lower than this, the satellite will fall back on Earth. Higher = it will fly away and won’t stay in an orbit. You can use Newton’s gravitation law to calculate how much velocity is required for an object to stay in orbit.

In low Earth orbit (called LEO, height between 150 and 2000 km from Earth’s surface, speed required to stay in orbit is around 7.8 km / second. Escape velocity (at the surface) is around 11.2 km / second.

Explanation

Placing a satellite in the orbit is more like throwing it with the right speed in the right direction. While for you kids, it may seem a little too vague of an explanation, but this is what is really happening when a rocket is launched.

A little thought experiment – Newton’s cannonball

Imagine that you are standing on the top of a very high mountain with a cannon. That cannon is there to teach you science and not to help you fight the Demogorgon! For the sake of our experiment, let’s turn off gravity and any air resistance. When you fire the cannon, the cannonball would leave the earth and continue to go in a straight-line forever — without gravity to pull it towards Earth. Here is a visual guide to that event. The case is represented by a straight red line C.

Newton's Cannonball Illustration
Newton’s cannonball — A classical thought experiment to understand gravity and orbits.

But in reality, when you fire a cannon from the top of that mountain, the cannonball (a projectile) would just follow a parabolic trajectory and hit the bottom of the mountain, like the green line A in the figure above. This is because the Earth’s gravity pulls the slow moving cannonball downwards.

Now for the sake of the experiment again, let us upgrade our cannon into a super cannon that has a very big explosive charge. When you fire it, the cannonball would escape the cannon at a very high velocity. The ball would simply miss the ground and keep going on and on in a trajectory similar to that of the blue one B in the figure. This trajectory is called as the orbit. As long as air resistance and other forces, don’t slow down the cannonball, it will orbit the Earth perpetually.

Why would the cannonball orbit?

As I mentioned earlier, you fired the cannonball with a very high velocity. So this very high velocity means, the cannonball would have left the atmosphere significantly where there is no air resistance. And due to inertia, it stays moving with that speed forever. But there is also gravity still acting on the ball, that pulls it downwards towards the Earth’s centre. But as the Earth is round, the fast moving ball just misses the ground and continue trying to fall towards the Earth forever. We call this ‘continuous falling without hitting the Earth’ as orbiting.

In case, if the initial velocity is way too much — greater than 11 km/s (escape velocity) on Earth — then the ball would simply escape Earth’s sphere of influence and go away into space.

Therefore, when the cannonball has the right velocity (i.e. lesser than the escape velocity), it balances out gravity and gets into a stable orbit around the Earth. Also, looking at the dynamics of the circular motion of the ball, we find that an inward force is exerted on the cannonball that draws it towards the centre of the Earth. This essential force that keeps the ball in the orbit is known as the centripetal force, which is equal to the force of gravity.

Cannonball and a Rocket

Now let’s get back to rockets and satellites! In a sense, the rocket is analogous to the cannonball in our thought experiment. Here is a step-by-step breakdown of how a satellite gets to the orbit and stays there.

During the launch, the rocket that carries the satellite is usually placed at a certain launch pad on the Earth — close to the equator for most of the launches. In India, it would be Sriharikota, which is 13° north of the equator. When the rocket goes boom, it soars into the clouds vertically with one mission — put the little satellite in its nose into the orbit around the Earth.

For most cases, the orbit would be designated at an altitude of 400 km to 2000 km — called as the Low Earth Orbit (LEO). So to reach such heights, rockets are launched vertically for that required initial ascent to that heights. It really wouldn’t make sense to launch a rocket completely horizontal like the cannonball in our experiment. The horizontal boost usually comes when the rocket reaches a certain altitude around the designated one. Ground control slowly pitches the rocket towards the east and once it reaches the altitude, the final boost sets the orbital velocity for the satellite to stay in its orbit — just like a cannonball that is launched with engines to manoeuvre.

Orbital Velocity

How can we know the required speed for stable and closed orbits? For simplicity, orbital velocity of an object orbiting a massive object in a circular orbit can be found with the help of Newton’s gravitational law. As we have seen earlier, the force that keeps the satellite drawn towards the centre of the Earth is the centripetal force.

Mathematically, the centripetal force is defined by Newton’s second law of motion as:

Fc = ms × ac

Where Fc is the centripetal force, ms is the mass of the satellite, and ac is the centripetal acceleration.

ac = v2/r (how?)

Where v is the velocity of the satellite and r is the radius of the orbit from the centre of the Earth. This makes the first equation as,

Fc = ms × v2/r

We have already learnt that the centripetal force is equal to the gravitational force. Newton’s law of gravitation is written as,

Fg = (G × me × ms)/r

Where Fg is the gravitational force, G is the universal gravitational constant, and me is the mass of the Earth.

Equating Fg and Fc and solving them for v, we get,

v = √((G × me)/r)

This is how you can calculate the orbital velocity of any satellites in the LEO, as most of them have a circular orbit.

Will satellites stay in the orbit forever?

In reality, even the satellites orbiting in LEO still experience a little drag from the very thin atmosphere leading to orbital decay. So they do lose orbital speed, but they eventually gain it back by periodically firing their onboard thrusters.

The International Space Station is one good example for this. It weighs around 420 kg and is really big in size — 72 m × 102m. With an orbital speed of 7.6 km/s, the ISS experiences drag due to the thin atmosphere (10−8 Pa) at its altitude of roughly 400 km. So to maintain orbit, the station periodically fires its thrusters to boost it back. To avoid drag, it also orients its solar panels parallel to the ground whenever it reaches the Earth’s shadow. As the atmosphere is too thin, the solar activity plays an important role in the drag experienced by the station.

For further understanding the dynamics of orbits in a practical way with loads of fun, try out Kerbal Space Program. You can launch your own satellites in your own launch vehicles and play around with orbits.

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Karthikeyan KCPosts | Website

Karthik is an engineer, gamer, author, and a techie who wanders the realms of the interwebs, exploring and learning. He is the creator of the science encyclopedia Swyde. He loves communicating science online and sparking web applications at Mango Education.

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