Exercise: Hohmann Transfer

Note: To do this exercise you will need a valid license of STK Professional Edition and STK/Astrogator.

The Hohmann Transfer is, in terms of the velocity change (ΔV) required, the most efficient two-burn method of transferring between two circular, coplanar orbits. As shown in the illustration, a Hohmann Transfer uses an elliptical transfer orbit with its periapsis at the inner orbit and its apoapsis at the outer orbit.

In the following exercise the purpose is to transfer a satellite from a low-Earth parking orbit with a radius of 6700 km to an outer circular orbit with a radius of 42,238 km.

Reference: This exercise and the one that follows are based on Example 3-6-1 in Hale, Francis J., Introduction to Space Flight, Englewood Cliffs, N.J.: Prentice-Hall (1994), pp. 43-44.

Note: The values used here for the radii of the inner and outer orbits are for illustration purposes only. For further practice after completing this exercise, try substituting different values, such as a radius of 42,164.197 km (geosynchronous) for the outer orbit.

Setup

  1. Create a scenario and a satellite.
  2. On the Orbit page of the satellite's Basic properties, select the Astrogator propagator. You may need to expand the properties window to see all of the controls.

Constructing the MCS

To design a Hohmann transfer from a 6700 km parking orbit to a 42,238 km outer orbit, you will use the following MCS segments:

Let's take it a step at a time.

Define the Initial State

  1. The default MCS that appears when you display the satellite's Orbit page probably already begins with an Initial State segment. If not, insert one at the beginning of the MCS.
  2. Name the segment 'Inner Orbit'.
  3. Select Keplerian as the Coordinate Type and change Semi-major Axis to Radius of Periapsis with a value of 6700 km. All other elements should be set to zero.
  4. Click the Fuel Tank tab, and set Maximum Fuel Mass to 5000 kg.
  5. Set the Fuel Mass to 5000 kg.

Propagate the Parking Orbit

  1. If the second segment of the MCS is not already a Propagate segment, insert one in that position.
  2. Select Earth Point Mass as the Propagator.
  3. If the color currently assigned to the Propagate segment won't show up well on the selected 3D Graphics window background (or you just don't like it), select a different color.
  4. Set the Duration (Trip value) to 2 hours (7200 sec), more than enough to have the satellite orbit one complete pass.

Maneuver into the Transfer Ellipse

  1. Insert, as the third segment of the MCS, a Maneuver.
  2. Name the segment 'DV1'.
  3. The Maneuver Type should be set to Impulsive by default. If it is not, select that option.
  4. On the Attitude tab, select Thrust Vector for Attitude Control.
  5. Select Cartesian as the vector type.
  6. Select VNC(Earth) Thrust Axes.
  7. Set the X (Velocity) component to 2.421 km/sec. This will apply all the ΔV in the thrust direction.
  8. Click the Engine tab and select the Update Mass Based on Fuel Usage checkbox.

Propagate the Transfer Orbit to Apogee

  1. Insert, as the fourth segment of the MCS, another Propagate segment.
  2. Name the segment 'Transfer Ellipse' and select a color that will distinguish it from the first Propagate segment.
  3. Select Earth Point Mass as the Propagator.
  4. Insert an Apoapsis Stopping Condition and remove Duration.

Maneuver into the Outer Orbit

  1. Insert, as the fifth segment of the MCS, another Maneuver.
  2. Name the segment 'DV2'.
  3. Again, the Maneuver Type should be set to Impulsive.
  4. On the Attitude page, select Thrust Vector for Attitude Control.
  5. Select Cartesian as the vector type.
  6. Select VNC(Earth) Thrust Axes.
  7. Set the X (Velocity) component to 1.465 km/sec.
  8. On the Engine page, select the Update Mass Based on Fuel Usage checkbox.

Propagate the Outer Orbit

  1. Insert, as the final segment of the MCS, a Propagate segment.
  2. Name the segment 'Outer Orbit' and select a color that will distinguish it from the other two Propagate segments.
  3. Select Earth Point Mass as the Propagator.
  4. Set the Duration to 24 hours (86400 sec), so that the satellite will make a complete orbit pass (and one will be drawn in the 3D Graphics window).

The MCS tree should appear as follows when you are finished:

Running and Analyzing the MCS

Run the MCS. Select the 3D Graphics window, and adjust the view so that you can see the entire ephemeris by orienting it from the north or south pole of the Earth and then zooming out. The ephemeris displayed in the window should be similar to the illustration (except that the inner orbit is closer to the central body). The parking orbit, transfer trajectory and outer orbit should be clearly differentiated by the colors you selected for the three Propagate segments.

Note: If the Propagate segments do not display in the selected colors, open the MCS Options window and make certain that the Draw Trajectory in 3D as it is Calculated and Use Trajectory Segment Colors options are selected.

If you highlight the last Propagate segment ('Outer Orbit') and create a Summary report, you'll find that the semimajor axis of the outer orbit is very close to the desired value of 42,238 km. Also check the final Fuel Mass. As you'll recall, in defining the Initial State, you set Fuel Mass to 5000 kg, and, in setting up the two Impulsive Maneuvers, you opted to have mass decremented on the basis of fuel usage. If you highlight the second Propagate segment (after the first ΔV but before the second), you'll find a value for Fuel Mass between the initial and final values.

Suggestion: Save the scenario to compare the results of this exercise with those of the Fast Transfer exercise.