ASEN 6008: Interplanetary Mission Design Lab

Spring, 2015

Lab 3: Introduction to Pork Chop Plots

Assigned: Feb 5

Due: Feb. 12

Name:______

Introduction

In the last lab you began the mission design process for the Mars Reconnaissance Orbiter, taking advantage of the Earth-Mars 2005 transfer opportunity. You used STK to target many trajectories from Earth-Mars, departing Earth on August 10th, 2005. You then used your coded solution to Lambert’s Problem to do the same thing. Hopefully you found similar results between the two techniques. Hopefully you also noticed how quickly the Lambert’s problem code converged on solutions compared with STK’s software. STK has many capabilities to simulate very realistic trajectories, which is very useful once a specific mission design has been chosen, but it is slow to explore trajectory options. The coded solution to Lambert’s problem can be easily implemented to do the rough design studies that STK cannot do well. In this lab you will use your code to Lambert’s Problem to produce Pork Chop Plots, a tool that easily analyzes large windows of trajectory options, which can then be used as a starting point for further analysis with software like STK.

Pork Chop Plots

An example pork chop plot is shown in Figure 1, below. It is a contour map displaying various mission design parameters that can be visually inspected to determine candidate launch and arrival windows. The pork chop plot shown is for the Mars 2005 transfer opportunity. The mission design parameters plotted include C3 @ Earth, V∞ @ Mars and Time of Flight.

Figure 1. An example pork chop plot of the Earth-Mars 2005 opportunity.

Your assignment, given below, is to build several pork chop plots. Here is some pseudo-code to build them, followed by some notes on the process:

Departure_Dates = vector of departure dates

Arrival_Dates = vector of arrival dates

Loop Departure_Dates

Loop Arrival_Dates

[R_Earth, V_Earth] = Earth_RV(JD_Depart)

[R_Mars, V_Mars] = Mars_RV(JD_Arrive)

[V_initial, V_final] = Lambert(R_Earth, R_Mars, dt)

Find mission design parameters and stick them into their

appropriate arrays

End Loop

Plot contours in a figure

Notes:

If you don’t know where good opportunities exist, begin with large departure and arrival windows and keep the resolution low. Then work to zoom in on the interesting opportunity windows.
If for some reason the Lambert solver can’t converge on a solution for transfer times greater than something like 500 days, put a catch in there to ignore transfer times greater than that length. A similar catch can be used for very short transfer times.
Matlab’s “contour” function produces good contour plots (type “help contour” in the Matlab prompt to get more information). That is what was used to produce the plot shown in Figure 1.
Plot different variables in different colors and/or different line styles
Use Matlab’s capabilities to put labels on the contours.

Example contour-plotting code for Matlab:

figure(1)

hold on

[cs1,h1] = contour(x_vals,y_vals,C3,C3_contours,'r');

clabel(cs1,h1);

[cs2,h2] = contour(x_vals,y_vals,V_inf,V_inf_contours,'b');

clabel(cs2,h2);

[cs3,h3] = contour(x_vals,y_vals,dt,dt_contours,'k');

clabel(cs3,h3);

Definitions:

“x_vals” is a vector of the values that show up on the x-axis (Julian Dates, real dates, or “days past Jan 1st”-type values),
“y_vals” is a similar vector for the y-axis values,
“C3,” “V_inf,” and “dt” are 2D arrays holding the mission design parameter values for the departure and arrival windows (you may need to transpose them)
“C3_contours” and the other “x_contours” are vectors that hold the values for the contours that you want displayed (if you don’t include these, Matlab will default to displaying some 50 contours at even increments)
“clabel” does the labeling

Assignment

The first problem concerns reading Pork Chop Plots. Please answer the following questions about Figure 1, the 2005 Earth Mars opportunity. Note: For all answers, please give a calendar date or Julian date. Do NOT use the form “30 days past June 4th)

a)What do you think is the optimal departure/arrival combination for a robotic mission that wishes to minimize total ΔV without using any aerobraking? (Just eyeball it – a range of answers is certainly acceptable. Write your answer on the attached answer sheet). Is this a Type I or a Type II trajectory?

b)What is the optimal departure/arrival combination for a robotic mission that wishes to minimize total ΔV and can use aerobraking to slow down at Mars? Why? Is this a Type I or a Type II trajectory?

c)What is the optimal departure/arrival combination for a human mission to Mars, given a maximum flight duration of 150 days? Assume that the mission will implement a direct transfer descent to the Martian surface, i.e., it will not enter into a Mars orbit. Also assume that the entry vehicle can only withstand the heat generated from a trajectory with a V∞ < 4.5 km/s.

Generate a pork chop plot for the 2018 Earth to Mars opportunity. Use Earth departure dates from JD = 2458200.0 (March 22, 2018) to JD = 2458320.0(July 20, 2018) and Mars arrival dates from JD = 2458350.0(Aug. 19, 2018) to JD = 2458600.0(Apr 26, 2019). Plot C3 at Earth (km2/s2), V∞ at Mars (km/s), and time of transfer (days). The plot will look similar to that in Figure 1. Use the plot you generated to answer the following questions. Please turn in the plot with the homework.

a)Find the optimal departure/arrival combination for both a Type I and Type II transfer to Mars for a robotic mission that does use aerobraking.

b)Concerning your answer to part a: Which trajectory, the Type I or the Type II, do you think is more optimal? Why?

c)What is the optimal departure/arrival combination for a mission to Mars, given a maximum flight duration of 200 days? Assume that the mission will implement a direct transfer descent to the Martian surface, i.e., it will not enter into a Mars orbit. Also assume that the entry vehicle can only withstand the heat generated from a trajectory with a V∞ < 4.5 km/s.

Generate a pork chop plot for the 2016 Earth to Mars opportunity. Use Earth departure dates from JD = 2457389.0 (Jan 1, 2016) to JD = 2457509.0(Apr 30, 2016) and Mars arrival dates from JD = 2457570.0 (Jun. 30, 2016) to JD = 2457790.0(Feb 5, 2017). Plot C3 at Earth (km2/s2), V∞ at Mars (km/s), and time of transfer (days). Use the plot you generated to answer the following questions. Please turn in the plot with the homework.

a)Find the optimal departure/arrival combination for both a Type I and Type II transfer to Mars for a robotic mission that does use aerobraking.

b)Concerning your answer to part a: Which trajectory, the Type I or the Type II, do you think is more optimal? Why?

Lab 3: Introduction to Pork Chop Plots

Assigned: Feb. 5

Due: Feb. 12

Answer Sheet

Name:______/100

Analyzing the 2005 Earth to Mars opportunity.

a)Departure: ______(2 pts)
Arrival: ______(2 pts)
Is this a Type I or a Type II trajectory? ______(2 pts)

b)Departure:______(2 pts)
Arrival:______(2 pts)
Why? (2 pts)
Is this a Type I or a Type II trajectory? ______(2 pts)

c)Departure:______(2 pts)
Arrival:______(2 pts)
Is this a Type I or a Type II trajectory? ______(2 pts)

Generate a pork chop plot for the 2018 Earth to Mars opportunity. Please turn in the plot with the homework. (30 pts)

a)Departure Type I: ______(2 pts)
Arrival Type I: ______(2 pts)
Departure Type II: ______(2 pts)
Arrival Type II: ______(2 pts)

b)Which is better?(2 pts)

c)Departure:______(2 pts)
Arrival:______(2 pts)
Is this a Type I or a Type II trajectory? ______(2 pts)
Is this a reasonable trajectory? Why/Why not? (2 pts)

Generate a pork chop plot for the 2016 Earth to Mars opportunity. Please turn in the plot with the homework. (30 pts)

a)Departure Type I: ______(2 pts)
Arrival Type I: ______(2 pts)
Departure Type II: ______(2 pts)
Arrival Type II: ______(2 pts)

b)Which is better? (2 pts)

c)Departure:______(2 pts)
Arrival:______(2 pts)
Is this a Type I or a Type II trajectory? ______(2 pts)
Is this a reasonable trajectory? Why/Why not? (2 pts)