ASEN5519: Interplanetary Mission Design

ASEN 6008: Interplanetary Mission Design

Spring, 2015

Galileo’s Resonant Earth Orbits

Assigned Feb. 24

Due: Mar. 3

Name:______

What to Expect

This homework will reconstruct the Earth resonant orbits of the Galileo mission to Jupiter. Table 1 shows the baseline dates for the events en-route to Jupiter. You will explore the resonant Earth orbit between EGA1 and EGA2.

Table 1. The baseline design of Galileo’s VEEGA transfer to Jupiter.
Event / Calendar Date / Julian Date
Launch / October 8, 1989 00:00:00 / 2447807.5
Venus Flyby (VGA) / February 10, 1990 00:00:00 / 2447932.5
Earth Flyby (EGA1) / December 10, 1990 00:00:00 / 2448235.5
Earth Flyby (EGA2) / December 9, 1992 12:00:00 / 2448966.0
Jupiter Arrival (JOI) / March 21, 1996 12:00:00 / 2450164.0

Resonant Orbits

Many interplanetary missions implement resonant orbits such that the same planet is visited several times in a row. In the final project it may be desirable to visit Earth twice (or more) in a row. The following are instances when such resonant transfer orbits are desirable:

•A single gravity swingby would impact the surface of the planet.

•A deep space maneuver needs to be added to reduce the overall cost of the mission.

•The relative positions of the planets are not ideal for a single encounter, but may be more ideal after another encounter with the same planet.

All of these instances are common when attempting to reach the outer planets in the solar system via swingbys with the inner planets. Cassini needed to implement a deep space maneuver, Galileo would have impacted Earth with a single flyby, a future Saturn mission that flies by Jupiter may require delays in the inner solar system to make the planets line up better, etc.

There are many ways to construct resonant orbits. Refer to the handout “Resonant Orbits” for the method that will be implemented in this homework.

Galileo’s VEEGA

In order to construct Galileo’s 2-year Earth resonant orbit, we will need several things.

Problem 1a: Use your code and the dates shown in Table 1 to calculate the hyperbolic excess velocity (V∞) that Galileo had upon arrival at the first Earth gravity assist (EGA1), i.e., , and record it on the Answer Sheet. In the Resonant Orbit handout, this would be .

Problem 1b: Use your code and the dates shown in Table 1 to calculate the hyperbolic excess velocity that Galileo had upon departure of the second Earth gravity assist (EGA2), i.e., , and record it on the Answer Sheet. In the Resonant Orbit handout, this would be .

Problem 1c: What do you notice about the magnitudes and components of and ? Did Galileo need to perform any maneuvers during the resonant orbit?

Assume that the closest Galileo is allowed to get to Earth is 300 km. Use the handout to construct the 2-year resonant orbit for the dates shown in Table 1.

Problem 2: Construct and turn in a plot showing the periapse radius of each gravity assist as functions of φ. Please include a line showing the minimum acceptable altitude that a gravity assist may have. Please indicate in the plot the region of acceptable gravity assists. See the handout for an example plot.

Please answer the following questions on the Answer Sheet.

Problem 2a: What is the smallest acceptable value of φ? Is this dictated by the 1st or 2nd gravity assist?

Problem 2b: What is the largest acceptable value of φ? Is this dictated by the 1st or 2nd gravity assist?

Problem 2c: What do you think the optimal value of φ is to produce two acceptable gravity assists? Why? What are the altitudes of closest approach for the two gravity assists for this value of φ?

Problem 3: Thought exercise. Let’s say for a future VEEGA mission to Jupiter, the spacecraft requires a value for of at least 9 km/s in order to reach Jupiter with the correct parameters, but it can only obtain a maximum value for of 6 km/s from a flyby of Venus using the best launch engines available. So let’s say there’s no way to increase the value of beyond 6 km/s and no way to reduce the value of below 9 km/s. We know that a ballistic resonant orbit does not change the magnitude of the V∞ vector between Earth encounters more than a little. How can we increase the magnitude of the V∞ vector between successive flybys of Earth? (Hint: Cassini had to do this to reach Saturn).

Galileo’s Resonant Earth Orbits

Answer Sheet

Name:______/100

Problem 1

V∞in,EGA1: x:______(4 pts)

y: ______(4 pts)

z: ______(4 pts)

|V∞in,EGA1|: ______(4 pts)

V∞out,EGA2:______x:_____ (4 pts)

y: ______(4 pts)

z: ______(4 pts)

|V∞out,EGA2|: ______(4 pts)

What do you notice about the magnitudes and components of (V∞in,EGA1) and (V∞out,EGA2)? Did Galileo need to perform any maneuvers during the resonant orbit?

______(8 pts)

(Turn Over for Answers to Problems 2 and 3)

Problem 2:

Gravity Assist Trade-Study Plot______(20 pts)

Smallest φ:______(4 pts)

1stor 2nd GA:______(4 pts)

Largest φ:______(4 pts)

1st or 2nd GA:______(4 pts)

Optimal φ:______(4 pts)

Why?______(6 pts)

hp for EGA1:______(4 pts)

Problem 3:______(6 pts)

How can we raise the magnitude of the V∞ vector between successive Earth flybys?