Basics of Orbital Dynamics – Jaden Weiss

Basics of Orbital Dynamics

Chances are that you have seen a rocket lift off, either on television or via a video. Most people have also seen the iconic Apollo 11 footage from the moon, where Neil Armstrong famously said “One small step for man, one giant leap for mankind.”

So, how does a spacecraft get from point A to point B?

4 Forces on Rockets

Thrust

Perhaps the most obvious force acting on a rocket is thrust. Fuel and oxygen react, generating hot gases. As these hot gases exit the base of the engine, an equal and opposite force pushes up on the rocket to conserve momentum.

Gravity (Weight)

If it were not for gravity, the rocket would keep rising forever after the engines shut down. When the rocket first ascends, the engines are fighting gravity, which slows the ascent.

The space above the rocket is not empty — it is filled with tons of air. As the rocket ascends, it must push air out of the way. This transfers momentum from the rocket to the air, slowing the rocket down. In the process, friction on the surface of the rocket generates heat. Drag is also used to slow the crew capsule/return vehicle to safe speeds without expending fuel.

If you have only seen a rocket immediately after liftoff, it may be unclear why lift is involved. As the rocket burns straight up after liftoff, there is no initial lift. However, later in the ascent the rocket tips over to gather horizontal velocity for orbit. When the rocket is moving at an angle to the air, the outer shell acts as a poorly shaped wing surface, generating a force perpendicular to the rocket in an upward direction. Fins on rockets also generate lift in adjustable directions to help steer the rocket.

Ascent

When the clamps holding down the rocket are first released, the only two forces acting on the rocket are thrust and weight (gravity). The balance between the force of gravity and the thrust of the engine is commonly expressed as the Thrust-to-Weight Ratio, or TWR. In order for the rocket to rise, the TWR must be greater than 1, meaning that the thrust generated exceeds the force of gravity on the rocket.

As the rocket gains speed, lift and drag forces increase in intensity and the problem of tilt angle comes into action. The more the rocket tips over, the more it acts like an airplane — the rocket gains velocity more easily because it does not have to fight as much of gravity. However, if the rocket is tilted over more then it gains altitude slower and spends more time in the atmosphere. Lower in the atmosphere, drag is stronger and the thrust is weaker due to opposing pressure from the air.

As the rocket rises higher into the atmosphere, the speed increases and the atmosphere begins thinning out, as there is less air above compressing it. The atmospheric forces initially increase, until an important point in the ascent called Max-Q. At Max-Q, the atmospheric forces reach their most intense point and aerodynamic stress on the structure of the rocket reaches its maximum value. At this point, fuel flow to the engines is reduced to minimizes forces on the structure of the rocket and stop it from being torn apart. During Max-Q, the rocket accelerates less and is carried into the upper atmosphere by its momentum. As the atmosphere thins out, the engines can be slowly throttled back up, holding these forces roughly constant. Eventually, the atmosphere is thin enough that drag and lift drop off despite the increasing speed of the rocket, and Max-Q is passed. This is considered a milestone in a spaceflight, as the spacecraft has (hopefully) survived its worst forces.

Circularization

As the rocket exits on the atmosphere, it is on a suborbital trajectory — if the engines cut out right then, the rocket would fall back into the atmosphere and crash into the ground (or the escape system would activate and the crew capsule would be saved by parachutes).

In order to stay in space, a maneuver called circularization is preformed. The goal of circularization is to reach a point where the spacecraft is neither moving towards nor away from the surface, but is rather carried around the planet by its tangential (horizontal) velocity. This accomplished by burning near-horizontally. As the spacecraft speeds up, gravity changes the speed less and changes the direction of movement more. This continues until the change in speed produced by gravity is close enough to 0.

Orbit

Once in orbit, all movement is made by preforming small burns to tweak the velocity. These changes in velocity change the shape of the orbit. However, since the craft is moving in a circle, a different system of directions is needed than on earth, where we can use cardinal directions as well as up and down.

Directions in Orbit

Prograde & Retrograde — One of the easiest directions to define is prograde — the direction that the craft is moving. Retrograde is the opposite of prograde — it is the direction that the craft is moving away from.

Radial In/Out — Another easy direction to reference with is the direction towards or away from the planet. Radial In and Radial Out are just this — in is towards the planet, and out is away from the planet.

Normal/Anti-Normal — Take your right hand and curl your fingers into a fist. Stick your thumb up. Align the curl of your fingers to the direction of the orbit, and your thumb will point normal to the orbit. If you repeat this with your left hand, your thumb will point anti-normal. These directions are useful for adjusting tilted orbits.

Elliptical Orbits

Spacecraft typically travel in a type of orbit called an Elliptical Orbit — an orbit forming the shape of an ellipse. There are two very important points in an elliptical orbit — the Apoapsis & the Periapsis. The Apoapsis is the highest point of the orbit, and is the point on the orbit where the craft travels the slowest. The Periapsis is the lowest point of the orbit, and is the point on the orbit where the craft travels the fastest. As the craft rises from the periapsis to the apoapsis, gravity slows it down until it no longer has enough momentum to keep rising. As the craft falls from the apoapsis to the periapsis, gravity speeds it up until it is moving so fast that its momentum pulls it back up to the apoapsis.

Imagine that the craft burned retrograde at the apoapsis. The craft would slow down, and thus fall faster. It would then have to fall further to regain its momentum to return to the apoapsis. Therefore, this would lower the periapsis without changing the apoapsis. If it slowed down enough, the periapsis would fall within the atmosphere, and drag will slow it down there. Slowing down at the periapis would mean that it would not rise as far before it started to fall again — this would raise the periapsis. These passes through the atmosphere would continue slowing the craft down, pulling it into a spiral that would eventually crash into the surface (although the heat and drag force would destroy it unless the craft was designed to tolerate these extremes).

Normal burns are used to correct orbital inclination. This is useful when attempting to rendezvous with another spacecraft, where some rendezvous maneuvers assume that the target craft is on the same orbital plane.

Gravity Assists

Now imagine that the moon is in the general direction of the apoapsis, and the craft burns prograde at the apoapsis. It picks up enough speed that the apoapsis intersects the orbit of the moon. As the craft rises, it gets close to the moon and is pulled into an orbit passing close to the surface. If the craft did nothing (warning: several approximations), it would be thrown past the moon, returning to an earth orbit with the same starting speed but in a different direction and at a different point. This is the concept of gravity assists — manuvers which allow slingshots past the moon (or other planets) to change orbits and gather speed.

Alternately, the craft could burn retrograde at the point closest to the moon’s surface instead of continuing. If it slows down enough, the flyby will become an elliptical orbit around the moon. This burn injecting into a lunar orbit is called an injection burn.

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