Orbital Mechanics

Question 1

Mention the three Laws of Kepler:

Answer 1

1. Orbit of each planet is an eclipse with the sun in one of the foci.
2. A line from the planet to the sun travels equal areas during equal intervals of time.
3. The square of the orbital period (T) is equal to the cube of the semi-major axis (a) of the orbit: T² = a³

Question 2

Write down the Energy Equation:

Answer 2

• Specific energy for elliptical orbits is always negative.
• Increasing sp. energy (i.e Ԑ ~ 0) the orbit becomes parabolic.
• A parabolic trajectory has the minimum energy for escaping Earth.
• Further increasing the sp. energy will lead to a hyperbolic trajectory i.e going interplanetary.

Question 3

What are the six parameters used to define an orbit?

Answer 3

1. Eccentricity (e): Ratio of minor to major axis dimensions
2. Semimajor axis “a”: half of the major axis
3. Inclination (i): Angle between the plane of reference vs movement
4. Argument of periapsis “ω”: Angle from ascending node to perigee in S/C movement direction.
5. Longitude of ascending node “Ω”: Angle from vernal equinox to ascending node.
6. True anomaly “θ”: Angle S/C Position-periapsis in S/C movement direction

Question 4

Small dV is applied tangentially in flight direction to my circular orbit?
S/C thrusts in flight direction i.e tangentially to its orbit at the apogee? (elliptical)

Answer 4

• If we are in circular orbit ⇒ Hofmann transfer happens
• If we are in elliptical orbit ⇒ Perigee radious increases hence “a” increases

Question 5

Small dV is applied in the perigee, what happens?

We keep increasing dV, what happens?

Answer 5

The perigee keeps constant and the ellipse is bigger. Apogee radius increases.

If we still increase dV, perigee will remain constant up to the point which V_s/c > V_e so the energy Ɛ ~ 0 then the S/C will be out of the body of influence of the planet thus ⇒ parabolic orbit.

Question 6

Small dV is applied against flight direction in perigee?

Answer 6

If the dV is enough the velocity reduces the apogee as much as the perigee size ⇒ circular orbit, if not, just a smaller eclipse.

Question 7

Apply thrust normally to the orbital plane?

Answer 7

S/C changes its orbit inclination angle with respect to the reference plane.

Question 8

How to calculate Hohmann Transfer?

Answer 8

1. Obtain semi-major axis (a)
2. Calcular circular velocities per each orbit (source and destiny)
3. Calculate the periapsis and apoapsis velocites
4. The sum of the velocity differences between the circular and elliptical velocities at the periapsis and apoapsis is equal to the total dV for this maneuver

Question 9

What does a one tangent burn mean and how do you calculate it?

Answer 9

One tangent burn means the transfer path is tangent to only one orbit but not the second.

It is calculated as follows: (SIMILAR TO HOHMANN TRANSFER UNTIL STEP#3)

1. Find flight path angle (Φ) from e= 1- (r_a/a)
2. Calculate dV_b = sqrt(V_cb²+V_eb² - 2V_cV_ecos(Φ))
3. Calculate dV_a = (V_c - V_e)
4. Sum it up to get the required dV

Question 10

What is plane change and when is the best moment to apply it?

Answer 10

Plane change is the change of direction of the velocity vector

The best moment to apply is when the S/C velocity is minimum, hence at apogee

It requires dV component normal to the orbital plane and to the initial velocity vector

Question 11

What is the Bi-elliptical transfer and what’s the difference vs Hohnmann Transfer?

Answer 11

The bi-elliptical transfer (also known as three-impulse transfer) consists of two transfer ellipses. The transfer from the first to the second transfer occurs at the apoapsis of both ellipses.

In general, the Hohmann transfer is the most energy-efficient transfer orbit BUT if r₂ >> r₁ then bi-elliptical transfer is chosen.

Question 12

What is orbit determination?

Answer 12

Determining an orbit on knowledge of velocity, distance (r), and a flight path angle

Question 13

What is Low Thrust Maneuver?

Answer 13

When enough thrust is not available: 1mN > Thrust (e.g EP). It consists of providing low but continuous thrust over a long period of time.

Less efficient than Hohmann transfer

Question 14

What type of orbit is needed for interplanetary travel?

Answer 14

Hyperbolic Orbit

Question 15

What is the sphere of influence?

Answer 15

• Imaginary sphere around a body where its gravitational force is larger than any other body
• Used to simplify body problem when we calculate orbit trajectories
• Neglects all the influence of other celestial bodies

Question 16

What is a Hill sphere?

Answer 16

It has a physical meaning, it is based on Lagrange points where centrifugal & gravitational forces are balanced.