AAE251 HW 3
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Purdue University *
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Course
251
Subject
Aerospace Engineering
Date
Dec 6, 2023
Type
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12
Uploaded by JusticeAtomBear31
1
Name Team Number AAE 251: Introduction to Aerospace Design
Homework 3—Orbits and Launch to LEO
Due Tuesday February 8
th
, 11:59 PM ET on Gradescope
Instructions
Show all the work and clearly box your final answer. You will not receive full credit for a correct final answer if you don’t show your work. You must use the template provided. Anything that you want the graders to grade must be written legibly within the boxes provided to you. There are multiple steps involved to complete your submission on Gradescope. You can follow the step-by-step guide posted on Brightspace. Select the relevant pages of your final answer for each question in your Gradescope submission.
List of Constants:
Mar’s radius
(
!
!
)
is 3389.5 km
Earth’s gravitational constant or parameter (μ
E
or GM) is
3.986 × 10
"
km
#
/s
$
Mars’ gravitational constant (μ
Mars
01 23
!
) is
4.283 × 10
%
km
#
/s
$
Earth’s rotational speed (angular velocity) is
7.292 × 10
&"
rad/s
Mars’ rotational speed (angular velocity) is
7.095 × 10
&"
rad/s
2
Question 1 [15 points]
In class, we saw that the equation of a conic section is given by: ! =
#
1 + & cos(+
∗
)
(1)
where 8 = :(1 − =
$
)
is a geometrical constant of the conic called the “parameter” or “semi-latus rectum”, =
is the eccentricity, and >
∗
is the true anomaly. NASA, in further attempts to help foster the commercial space industry, has awarded two companies contracts to launch one astronaut each to Mars. Both launches depart on the same day, to take advantage of the launch window. Once the missions arrive at Mars, the plan is to go into elliptical parking orbits and then have each one executes several reverse thrust burns to slow down and subsequently arrive on the Martian surface. Your team at RocketsAreUS has launched Jebediah, while the other company, GoThrusters, has launched Valentina. Nine months or so later, each spacecraft is now in a parking orbit. The orbits differ slightly, so let’s develop some expressions that we can use to calculate values for both orbits. Assume that the apogee for each orbit is at some height ℎ
(
above the surface of Mars, and an eccentricity of =
. PART 1:
Derive expressions for the following quantities in terms of the apogee height (
ℎ
(
), eccentricity (
=
), Mars’ radius (
!
!
)
, and Mars’ gravitational constant (
@
!()*
). Make sure to simplify or formulate your expressions such that they contain no quantities other than the four mentioned here. Show your work. (a)
The perigee radius
(b)
The semimajor axis
(c)
The perigee velocity
(d)
The apogee velocity
3
Answer 1a: Answer 1b:
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4
Answer 1c: Answer 1d:
5
PART 2: Calculate the quantities (a) to (d), if the apogee is at a height of 450 km above the Martian surface and the eccentricity is 0.001. Fill in the calculated values in the table below: (a)
The semimajor axis
(b)
The perigee radius
(c)
The velocity of satellite at perigee
(d)
The velocity of the satellite at apogee
6
Question 2 [25 points]
The team from GoThrusters, who did not attend Purdue, but rather a school in the
South, underestimated the fuel required for a mission to Mars, and now poor
Valentina is stuck orbiting Mars, without even enough fuel to get out of orbit. Jebediah
is safely in a circular orbit with an altitude of 340 km above the Martian surface. It’s
up to your team at RocketsAre US to rescue Valentina. You come up with a daring plan
to rendezvous with her spacecraft in orbit, and subsequently deorbit the two
spacecraft together. Thanks to your superior math skills, GRIT
TM
, and cautious risk
management, you have plenty of fuel.
(a)
Valentina is in an elliptical orbit with an apogee of 450 km above the Martian
surface, and an eccentricity of 0.055. Fill in the calculated values in the table
below:
(e)
The semimajor axis
(f)
The perigee radius
(g)
The velocity of satellite at perigee
(h)
The velocity of the satellite at apogee
(b)
To design the rescue mission, you decide to plot the orbits of Jebediah and
Valentina in MATLAB. Your colleague says you need to use the conic equation
as a function of the true anomaly,
Plot the Martian surface, with
its center at the origin of your plot and the orbit perigees in the
+B
direction.
Do NOT forget to attach your MATLAB code at the end. Use ‘axis equal’ when
plotting your orbits.
For better bookkeeping, include the following details at the start of all your code submissions for AAE251: %AAE 251 Spring 2022
%Homework <Enter Homework #>
%Title of the code
%Authors: <Enter your and your collaborators’ names>
*
0
360 .
q
°
°
£
£
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7 Plot:
8 Matlab Code: %AAE 251 Spring 2022 %Homework <3> %Orbits of Jebediah and Valentina %Authors: <Andrea Goh> %Constants mars_radius = 3389.5; %km earth_grav = 3.986 * power(10,5); %km^3/s^2 mars_grav = 4.283 * power(10,4); %km^3/s^2 earth_rot = 7.292 * power(10,-5); %rad/s mars_rot = 7.095 * power(10,-5); %rad/s %Given jabediah_radius = 340 + mars_radius; valentina_radius_apogee = 450 + mars_radius; valentina_eccentricity = 0.055; %Martian surface x_surface = -mars_radius:1:mars_radius; %range of x values y_surface = sqrt(power(mars_radius,2)-power(x_surface,2)); %y values %Jebediah's circular orbit x_jebediah = -jabediah_radius:1:jabediah_radius; y_jebediah = sqrt(power(jabediah_radius,2)-power(x_jebediah,2)); %Valentina's orbit anomaly = 0:1:360; valentina_radius = ((valentina_radius_apogee/(1+valentina_eccentricity)) * (1 - power(valentina_eccentricity,2))) ./ (1 + valentina_eccentricity .* cos(anomaly .* pi ./ 180)); size = length(valentina_radius); max_valentina_radius = max(valentina_radius);
9 y_valentina = -max_valentina_radius:2 * max_valentina_radius / (size - 1):max_valentina_radius; x_valentina = sqrt(power(valentina_radius,2)-power(y_valentina,2)); y_valentina_plot = y_valentina(19:size-18); x_valentina_plot = x_valentina(19:size-18); %Plot Martian surface plot(x_surface,y_surface) title(
"Valentina's and Jebediah's orbit around Mars"
,
"FontSize"
,18) hold on grid on plot(x_surface,-y_surface) %Reflect graph across x axis hold on %Plot Jebediah's orbit plot(x_jebediah,y_jebediah) hold on plot(x_jebediah,-y_jebediah) %Reflect graph across x axis hold on %Plot Valentina's orbit plot(x_valentina_plot,y_valentina_plot) hold on plot(-x_valentina_plot,y_valentina_plot) %Reflect graph across y axis legend(
"Martian surface"
, "Martian surface"
, "Jebediah's orbit"
, "Jebediah's orbit"
, "Valentina's orbit"
, "Valentina's orbit"
,
"Fontsize"
,18)
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9
(c)
While waiting for you to rescue them, since Valentina and Jebediah still have
radio communication between them, they decide to race around Mars.
i.
At their fastest, who is faster?
Answer 2c i): ii.
Who takes the shortest amount of time to go around Mars?
Answer 2c ii):
10
iii.
At what distance from the Martian surface would Valentina orbit at the same
speed as Jebediah? What value would true anomaly take at this location?
Answer 2c iii):
11
Question 3 [8 points] Among the three following launch sites, which one would give the maximum boost from Earth’s rotation for a launch due east? Why is it advantageous to have maximum boost from Earth’s rotation for a launch due east? Show your calculations for each launch site. a.
SpaceX Starbase Spaceport, Texas (25.996°N 97.154°W)
b.
Vikram Sarabhai Space Centre (VSSC), India (8.5314°N 76.8690°E)
c.
Jiuquan Satellite Launch Center (JSLC), China (40.96056°N 100.29833°E)
Answer 3):
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