Homework Assignment III
Instructions
Write your name at the top of each sheet. Staple the sheets together. Show your work and
circle your answers. Here is a recommendation for how to work the problems, but is not
absolutely necessary:
1. Make appropriate sketches of the problem
2. Make a table of data
3. Write down what is given, and assumptions
4. Write down what you need to find.
5. Write down and apply other appropriate equations relevant to particular parts of problem
as needed, and have fun
Undergraduates (1-3), Graduates (1-4)
1. Elliptical orbits.
a) Sketch an elliptical orbit and label the semi-major axis, semi-minor axis, eccentricity,
true anomaly, and semi-latus rectum.
b) For a perigee of 500 km, eccentricity of 0.7, calculate the velocity for a mean
anomaly of 0, 90, 180, and 270 degrees.
c) If the satellite passes the perigee at noon on a Monday, at what time will it return to
this position?
d) What is the velocity for circular orbits of radius equal to the perigee and apogee?
2. Parabolic orbits
a) For a perifocal length of 500 km, how long does it take for the satellite to leave the
sphere of influence of the earth?
b) If the satellite initially starts in a circular orbit, what is the ΔV required to be put on
this departure orbit?
3. Hyperbolic orbits
a) The satellite is in a circular orbit around earth at 500 km again. This time it burns to
put it on a heliocentric trajectory with e = 2.05 and a = -1.4 ⨉ 10 8 km. Neglecting the
gravitational pull from Earth or Mars, how long does it take to traverse the distance
from Earth to Mars orbit?
b) What is the ΔV required from its initially circular orbit?
4. Grad students only. Brock Blastoff has decided to take a vacation on Makemake as he
has heard that the beaches on the Kuiper Belt object are out of this world. First, he has to
figure out the orbit and location. Look up the orbital elements for MakeMake. For
example, try here: http://www.princeton.edu/~willman/planetary_systems/Sol/. Plot
Makemake's trajectory and current position along with that of Earth's so Brock knows
which direction he needs to set a course for his spacecraft, the Century Buzzard.
Due date: 2/19/2015
Table of Contents
Problem
Problem
Problem
Problem
#1
#2
#3
#4
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Problem #1
Elliptic orbit
%periapsis
rp = 500e3 + getradius('earth');
%eccentricity
ec = .7;
%semimajor axis
a = getsemimajoraxis(rp,ec);
%apoapsis
ra = 2*a-rp;
%true anomaly
theta = [0:.1:360];
orbit = ellipse(theta,a,ec,'earth');
figure(1), plot(orbit.x,orbit.y); axis equal
Re = getradius('earth');
x = Re*cosd(theta);
y = Re*sind(theta);
hold on
plot(x,y,'color',[0 1 1])
hold off
figure(2), plot(theta,orbit.V/1000)
xlabel('\theta'), ylabel('V (km/s)')
%current time
c = clock;
%reset
c(4) =
c(5) =
c(6) =
hour and second
12; %hour
0; %minute
orbit.tau; %seconds
%ellapsed time
datetime(c)
%returns 14-Feb-2015 21:34:54, so it's about 9.5 hours later still on
%Monday
1
1
3
5
8
ans =
22-Feb-2015 21:34:54
2
The circular orbits at perigee and apogee are
7.6167 and 3.1997 km/s, respectively.
Problem #2
Parabolic orbit
Sphere of influence is actually a chapter 5 concept, but is the distance at which the gravitational forces
between two objects is equal. Thus,
(2.1)
While an approximation, if we just estimate this as the point between the sun and the earth where
these forces balance, then if the orbital radius of Earth is
, then
, and thus solution for r_E in Eq. (2.1) gives
r_soi = getsphereofinfluence('earth','sun');
display(['Radius from center of earth to sphere of influence is approximately ' num
%perifocal distance
rp = 500e3 + getradius('earth');
%eccentricty
ec = 1; %always so for a a parabola
theta = -180:1:180;
orbit = parabola(theta,rp,'earth');
figure(3), plot(orbit.x,orbit.y); axis equal
Re = getradius('earth');
x = Re*cosd(theta);
y = Re*sind(theta);
hold on
plot(x,y,'color',[0 1 1])
hold off
figure(4), plot(theta,orbit.V/1000)
xlabel('\theta'), ylabel('V (km/s)')
Radius from center of earth to sphere of influence is approximately 924722538.0586
3
4
The time required to leave the sphere of influence is
D = calendarDuration(0,0,0,0,0,orbit.t(end))
% delta v required
deltav = [num2str((orbit.Vmax - orbit.Vcirc_rp)/1000) ' km/s'];
D =
186h 28m 23.9429146040929s
The
required to leave the sphere of influence is
display(deltav)
deltav =
3.155 km/s
Problem #3
Hyperbolic orbit
%eccentricity
ec = 2.05;
%semimajor axis
a = -1.4e11;
%radial points of interest in orbit
r1 = getorbitradiusofcelestialbody('earth');
r2 = getorbitradiusofcelestialbody('mars');
%get orbit data
dr = (r2-r1)/100;
orbit = hyperbola([],a,ec,'sun',(r1:dr:r2));
figure(5), plot(orbit.x,orbit.y); axis equal
Rs = getradius('sun');
x = Rs*cosd(0:1:360);
y = Rs*sind(0:1:360);
hold on
plot(x,y,'color',[.5 .5 .1])
Re = getorbitradiusofcelestialbody('earth');
x = Re*cosd(0:1:360);
y = Re*sind(0:1:360);
plot(x,y,'color',[0 1 1])
5
Rm = getorbitradiusofcelestialbody('mars');
x = Rm*cosd(0:1:360);
y = Rm*sind(0:1:360);
plot(x,y,'color',[1 .2 .2])
hold off
figure(6), plot(orbit.theta,orbit.V/1000)
xlabel('\theta'), ylabel('V (km/s)')
6
The time required to go from Earth to Mars is
t = seconds2human(orbit.t(end)-orbit.t(1))
% delta v required
deltav = [num2str((orbit.V(1) - getcircularvelocity(5e5,'earth') - getcircularveloc
t =
years:
months:
days:
hours:
minutes:
seconds:
The
0
1
10
0
4
1.611795342527330
required to leave the sphere of influence is
display(deltav)
deltav =
14.841 km/s
7
Problem #4
Set a Course for Makemake
bodies = [...
{'Mercury'};
{'Venus'};
{'Earth'};
{'Mars'};
{'Ceres'};
{'Jupiter'};
{'Saturn'};
{'Uranus'};
{'Neptune'};
{'Pluto'};
{'Makemake'};
];
color = [ ...
.9 .6 .1;
1 .5 0;
0 1 1;
1 .4 .1;
.8 .6 .3;
1 .5 .35;
1 .9 0;
0 .8 1;
.2 0 1;
.5 .5 .5;
.6 0 .6];
figure(7)
%plot the orbit
for ii = 1:length(bodies)
orbit = Kepler2cart(char(bodies(ii)));
ph(ii) = plot3(orbit.x,orbit.y,orbit.z,'-'); axis equal
set(ph(ii),'color',color(ii,:),'linewidth',1)
hold on
end
%now plot the position
for ii = 1:length(bodies)
orbit = Kepler2cart(char(bodies(ii)),juliandate(now));
ph(ii) = plot3(orbit.x,orbit.y,orbit.z,'o'); axis equal
set(ph(ii),'color',color(ii,:),'markerfacecolor',color(ii,:))
hold on
end
hold off
xlabel('x')
ylabel('y')
8
lh = legend(bodies,'location','northeastoutside'); legend boxoff
Published with MATLAB® R2014b
9