An Introduction to Orbital Mechanics

Arshia – Year 10 Student

Editor’s note: Talented Year 10 student Arshia successfully submitted this insightful piece of work to the school Science Magazine essay competition. Year 12 student Aashmi, a Chief Editor of the publication, notes, “On behalf of the GSAL Science Magazine, I’d like to congratulate you as a winner in the recent essay competition. Thank you so much for your brilliant entry – the editorial team thoroughly enjoyed reading it and were very impressed by the talent shown by your work.” CPD

Since humans emerged, we have explored; it’s in our DNA. We explore to feed our curiosity, to gain wealth or simply to be the first one there. Over the hundreds of thousands of years that humans have been around, we have explored nearly every corner of our planet. We have mapped the surface, explored the caves, and visited the highest and lowest points on the surface and under the ocean. There isn’t much more on this planet we haven’t found, so what comes next? Over the next few decades, we will see one of the most important periods in history, where humans will leave this planet to continue exploring. Since we will explore new frontiers, there will be a demand for people, meaning there will be more jobs in space. It is important that today’s young people know more about the subject to ensure our progress in space exploration goes smoothly.

Circular and elliptical orbits:

An orbit occurs when an object falls towards a celestial body but because it has such a high horizontal velocity, the object’s trajectory curves with the curvature of the celestial body. There are four types of orbits but the most important ones are circular and elliptical. There are a few terms regarding elliptical orbits that you need to know:

  • Perigee = When the object on its orbit is at the closest point to the body it’s orbiting.
  • Apogee = When the object on its orbit is at the furthest point from the body it’s orbiting.
  • Semi-major axis = The mean value of the perigee and apogee, it is given the symbol ‘a’.

For circular orbits we don’t use apogee and perigee since the object is equidistant from the body at all points in the orbit.

Intuitively, the closer to the body the object is orbiting, the faster the orbital velocity is because the orbital circumference is smaller and the force of gravity is higher, so it needs to be moving faster to stay in orbit. We can calculate orbital velocity using the equations below:

Elliptical (left); Circular (right).

Where G is the Universal Gravitational Constant (6.67 x 10-11) and M is the mass of the body you are orbiting in kg, r is the current distance from the body’s centre of gravity and a is the semi major axis.

As you can see, for a circular orbit, a = r, because the perigee equals the apogee so therefore, the equation can be simplified.

Orbital manouevering:

Orbital manouevering occurs when a spacecraft changes its velocity. However, this doesn’t mean the speed needs to change since velocity is a vector quantity (a quantity with both direction and magnitude). This allows the craft to change the shape or inclination of its orbit.

Orbital inclination is the angle at which a satellites orbit is offset from the celestial body’s equatorial plane. A spacecraft may need to change its orbital inclination to reach a target orbit which was not possible to achieve directly from launch without orbital manouevering or for many other reasons. The equation to calculate how much you need to accelerate to change your orbital inclination to a given angle is below:

Where i is the angle and v is the velocity.

But why would you want to change your orbital inclination? This could be done for a number of reasons. For example, if a spacecraft A wanted to encounter spacecraft B, it would need to raise its orbits apogee to spacecraft B’s orbit’s semi major axis so it can encounter spacecraft B. Here is a diagram of this:

To do this it needs to accelerate at the burn point as that will raise the apogee to the encounter point. Spacecraft A’s orbital velocity would be 288 m/s but it would need to accelerate at the burn point so that its orbit would change into the transfer orbit. To do this, we just need to calculate what the orbital velocity would be at the burn point in the transfer orbit and accelerate to that velocity. This would mean spacecraft A would need to accelerate to 325 m/s (37 m/s increase in velocity). Once it reaches the apogee, it would have a lower velocity as it is further away from the planet meaning the gravitational potential energy would have increased, therefore, the kinetic energy must decrease.

This means Spacecraft A (186m/s) is moving slower than spacecraft B (218m/s) which means that their relative speed is 32 m/s. However, if the spacecraft collide at that speed, both spacecraft will be destroyed so spacecraft A needs to accelerate to match the velocity of spacecraft B. This acceleration is called a circularisation burn as it changes our elliptical transfer orbit into the circular target orbit. If we add up how much we accelerated in total, we get the number 69 which is our Delta v requirement.

Delta V is the maximum amount that a spacecraft’s velocity can change with the amount of fuel it currently has. This means if a spacecraft had a delta V of 100 m/s, it could only change its velocity by 100m/s. We can calculate delta V using the rocket equation below:

Where ve is the Isp of the engines (efficiency) multiplied by the acceleration due to gravity, m0 is the total mass of the craft and mf is the mass of the craft without any propellant.

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