Exercise: Fast Transfer (Using Targeter)

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

Here, as in the Hohmann Transfer and targeting exercises, 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.

While a Hohmann Transfer is the most efficient two-burn maneuver to use in this situation, it is also one of the slowest. Among other things, the satellite is required to travel the entire length of the elliptical transfer orbit, including the approach to apoapsis, where its velocity is considerably slower than in the portion of the orbit near periapsis. If it is of great importance to reduce the time of flight (e.g. for a rendezvous or a planetary intercept) a maneuver such as that shown in the illustration can be used. This maneuver, known as a fast transfer, is considerably faster than a Hohmann Transfer, but, of course, it uses more fuel.

Reference: This exercise is based on Example 3-6-5 in Hale, Francis J., Introduction to Space Flight, Englewood Cliffs, N.J.: Prentice-Hall (1994), pp. 48-50.

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 fast 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 you wish, select a different color for the segment.
  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

Now, use the targeter to calculate the ΔV required to move the spacecraft from the parking orbit into the transfer orbit. The goal of the targeter will be defined in terms of the radius of apoapsis of the transfer ellipse, twice the radius of the desired final orbit.

Define a Target Sequence
  1. Insert a Target Sequence segment.
  2. Name the Target Sequence segment 'Start Transfer'.
  3. Nest a Maneuver in the Target Sequence.
  4. Name the nested Maneuver segment 'DV1'.
Select Variables
  1. Highlight the nested Maneuver and make certain that the Maneuver Type is set to Impulsive.
  2. Select Thrust Vector for Attitude Control.
  3. Select Cartesian as the vector type.
  4. Select VNC(Earth) Thrust Axes.
  5. Select the X (Velocity) component as the sole independent variable.
  6. Click the Engine tab and select the Update Mass Based on Fuel Usage checkbox.
  7. Click Results... and select Radius of Apoapsis (Keplerian Elements folder) as the only dependent variable.
Set up the Targeter
  1. Select the Target Sequence, highlight the default Profile (Differential Corrector), and open its Variables page by clicking Properties....
  2. Select the Use options under Control Parameters and Equality Constraints.
  3. Set the Desired Value for Radius of Apoapsis to 88176 km.
  4. Set Tolerance (under Equality Constraints) to 0.1 km.
  5. Display the Convergence page, increase the Maximum Iterations amount to 50 and select the Display Status option. Click OK to close the Properties window for the Profile.
  6. Set the Mode for the Profile to Iterate.
  7. Make sure the targeter is turned on (select Run active profiles in the Action field).

Propagate the Transfer Orbit to 42238 km

  1. Insert another Propagate segment after the Target Sequence.
  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 R Magnitude Stopping Condition, set the Trip value to 42238 km, and remove Duration.

Maneuver into the Outer Orbit

Here you will use the targeter to calculate the ΔV required to break out of the transfer orbit midway to apogee and enter the outer circular orbit. The goal will be to circularize the orbit, i.e., change its eccentricity to zero.

Define a Target Sequence
  1. Insert another Target Sequence segment.
  2. Name the Target Sequence segment 'Finish Transfer'.
  3. Nest a Maneuver in the Target Sequence.
  4. Name the nested Maneuver segment 'DV2'.
Select Variables
  1. Highlight the nested Maneuver and make certain that the Maneuver Type is set to Impulsive.
  2. Select Thrust Vector for Attitude Control.
  3. Select Cartesian as the vector type.
  4. Select VNC(Earth) Thrust Axes.
  5. Select the X (Velocity) and Z (Co-Normal) components as the independent variables.
  6. On the Engine tab, select the Update Mass Based on Fuel Usage checkbox.
  7. Click Results... and select Eccentricity (Keplerian Elements folder) and Flight Path Angle (Spherical Elements folder) as the dependent variables.
Set up the Targeter
  1. Select the Target Sequence, highlight the default Profile (Differential Corrector), and open its Variables page by clicking Properties....
  2. Select the Use options for both independent variables under Control Parameters and for both dependent variables under Equality Constraints.
  3. Leave the Desired Values for Eccentricity and Flight Path Angle at their default values of zero.
  4. Set Max. Step (under Control Parameters) for each independent variable to 0.3 km/sec.
  5. Display the Convergence page, increase the Maximum Iterations amount to 100, and select the Display Status option. Click OK to close the Properties window for the Profile.
  6. Set the Mode for the Profile to Iterate.
  7. Make sure the targeter is turned on (select Run active profiles in the Action field).

Propagate the Outer Orbit

  1. Insert a Propagate segment after the second Target Sequence.
  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 (Trip value) 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 and observe the targeting process as displayed in the Status window. When the process is finished, the 3D Graphics window should show a sharp turn from the transfer trajectory into the final orbit, similar to that shown in the illustration. Because of the change in direction, it is necessary to select two components of the second ΔV as independent variables.

In the second Target Sequence segment (Finish Transfer), click Apply Changes. Then, in the nested Impulsive Maneuver (DV2), note the current values for Cartesian X and Z. Using these values to compute the value of ΔV2,

yields a result that is very close to that reached via the Law of Cosines.

Note: The current values you observe for Cartesian X and Z may differ slightly from those shown here, depending, e.g., on the Tolerance you use for Eccentricity in the second Target Sequence.

Incidentally, why are the values for X and Z negative in DV2? Because you are transferring into a lower energy orbit and slowing down.

As the technical notes for this exercise show, a fast transfer is more expensive in terms of ΔV, but takes considerably less time, than a Hohmann transfer. You can also confirm this by running a Summary for the final Propagate segment and comparing the final Fuel Mass value with that for the Hohmann Transfer exercise.