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论文研究 - 火星最短霍曼转移轨道的仿真和计算.pdf
The Simulation and the Calculation of the Shortest Hohmann Transfer Orbit to Mars
Abstract
Keywords
1. Introduction
2. Organization
3. Materials and Methods
3.1. Experimental Tools
3.2. The Initial Data
3.3. Depending Theory
4. Simulation and Analysis
4.1. Simulating Gravitational Environment
4.1.1. Modeling Gravitational Effect
4.1.2. Modeling the Revolution Orbit of Earth and Mars
4.2. Simulating a Trajectory
4.2.1. Determining the Trajectory
4.2.2. The Velocity of Rocket to Transfer into the Hohmann Orbit
4.2.3. The Principle to Draw a Trajectory
4.2.4. Orbit from Low Earth Orbit
4.2.5. Simulating the Transfer
4.3. Orbiting from Hohmann Transfer Orbit to Mars Reconnaissance Orbit
4.3.1. Simplifying the Original Problem
4.3.2. Perturbation and Correction
4.4. Fuel Consumption
4.5. The Date the Satellite Escape the Orbit of the Earth
5. Conclusions
Acknowledgements
Conflicts of Interest
References
Journal of Applied Mathematics and Physics, 2019, 7, 2384-2400
https://www.scirp.org/journal/jamp
ISSN Online: 2327-4379
ISSN Print: 2327-4352
The Simulation and the Calculation of the
Shortest Hohmann Transfer Orbit to Mars
Yifan Wei1, Yuke Zhang2
1San New School, Wuhan, China
2Purdue University, West Lafayette, USA
How to cite this paper: Wei, Y.F. and
Zhang, Y.K. (2019) The Simulation and
the Calculation of the Shortest Hohmann
Transfer Orbit to Mars. Journal of Applied
Mathematics and Physics, 7, 2384-2400.
https://doi.org/10.4236/jamp.2019.710162
Received: August 19, 2019
Accepted: October 19, 2019
Published: October 22, 2019
Copyright © 2019 by author(s) and
Scientific Research Publishing Inc.
This work is licensed under the Creative
Commons Attribution International
License (CC BY 4.0).
http://creativecommons.org/licenses/by/4.0/
Open Access
Abstract
This paper will present the results and analyses of a simulation to send a sa-
tellite from the Earth to Mars. We use Python to simulate the orbit of the
rocket. Our goal is to find the least energy-cost trajectory, with the least initial
velocity. We find the date which allows the satellite to go from the Earth to
Mars in the shortest distance based on a Hohmann transfer orbit considering
the gravity of the Sun, Earth, and Mars.
Keywords
Trajectory, Earth, Mars, Hohmann Transfer, Energy Efficient
1. Introduction
Mars, the closest planet to earth, has a lot in common with and has long been
exploring. Since ancient times, “Mars has fascinated mankind” [1]. Finding
retrograde motion of Mars not only improved Ptolemy’s geocentric theory, but
also supported the heliocentric theory of Copernicus. “From then on, natural
science liberated from theology” (Copernicus, 1543). Moreover, almost all of
Kepler’s theories were based on observations of Mars for eighteen years. With the
idea of “artificial canals” on the surface of Mars in 1877 [2] and the same year, A.
Hall discovered two small satellites of Mars—Phobos and Phobos. In recent years,
people increasingly considered Mars as another habitable planet, because of the
environmental degradation on the earth and the evidence found on Mars. Radar
evidence of subglacial liquid water on Mars found by the MARSIS instrument
provided strong support to the importance of exploring Mars [3].
With the attraction of this precious red planet, we do want to learn more
about it. However, the high risk of sending the probes from the Earth to Mars is
DOI: 10.4236/jamp.2019.710162 Oct. 22, 2019
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Journal of Applied Mathematics and Physics
Y. F. Wei, Y. K. Zhang
a big obstacle. Since the Soviet Union launched its first Mars probe in 1960, 46
missions have been launched around the world, with less than a 45 percent suc-
cess rate. In the last century, a total of thirty-three detectors were launched
worldwide, but only 9 of them were completely successful. The key concerns of
launching process are the calculation of the track and the unknown accidents.
When the detector approaches Mars, a slight deviation in the orbital calculation
will cause amplified error enough to pass the planet. Also to be practical, fuel is
another serious problem, which means the limited size of the aircraft restricted
the amount of fuel carried. Then in order to solve these two problems, we try to
figure out a better trajectory that can minimize the use of fuel with our careful
calculation. We chose the Hohmann transfer as the basic model of our trajectory,
because it’s undoubtedly the most fuel-efficient one of all the orbital paths,
which has been proved hundreds of times, such as the article Journey to Mars:
The physics of travelling to the red written by Stinner and Begoray (2005) [4]. In
this paper, we simulate the launch of a detector from the Earth’s low Earth orbit,
continuous moving by passing the sun and entering the orbit of Mars under so-
lar gravitation with the help of PYTHON. In addition, on the basis of Hoh-
mann’s orbit, we compared the 365 launch dates, the path generated, and the
shortest distance path on a daily basis, which allows a relatively small amount of
required fuel and short time. And then we calculated the time required, and the
initial position of the Martian Earth, and chose the ideal launch date.
2. Organization
During our brainstorm, there were two routines of the probe from Earth to Mars
that can travel for solving this problem: directly from the Perihelion of the Earth
move to the Perihelion of Mars.
Because our simulation depends on the Hohmann transfer, we first simulated
an elliptic trajectory from lower orbit (the earth orbit around the sun) to the
higher orbit (the mars orbit around the sun). In order to be simple, we chose to
simplified the whole solar system with seven planets to be only Sun, Earth, and
Mars, and also treat Earth and Mars orbits as two circular orbits with the same
center. Because we want the trajectory from Earth to Mars to be the shortest.
Then we treat the Perihelion coordinate to be the starting point from Earth, and
the aphelion coordinate to be the ending point approaching Mars. We used the
data (speed and distance from sun of the Perihelion coordinate and the aphelion
coordinate of two planets) to dot a point on every final positions of mars and
earth in a linear motion with the gravitational force, and gain two oval orbits of
them. However, in fact, those two points, aphelion and Perihelion, are not in the
same line, because there is an angle between the long-axis of two ellipses. We use
the knowledge in geometry to figure out the coordinate of one point in mars or-
bit and successfully “turn the orbit of Mars in a certain angle”. The orbits of two
planets were fixed.
The next step of the research was to add the probe into the simulation. We
2385
Journal of Applied Mathematics and Physics
DOI: 10.4236/jamp.2019.710162
Y. F. Wei, Y. K. Zhang
assume an initial altitude of probe and calculate the initial angle and the speed
which allow the probe to reach Mars. We simulate the first situation (move di-
rectly toward mars). The result was it requires a numerous amounts of energy
allowing the probe move toward mars, resisting the gravitational force exerted
by Sun. Then, we considered Sun’s gravity at first, and gain an approximate ini-
tial speed. Then, we tended to figure out the shortest distance for probe to reach
Mars, and enter the orbit to rotate around mars. We used recursive algorithm
again, to simulate every position of Earth on daily basis, found the shortest dis-
tance it need to take, and return the position of the earth. After we finished the
final orbit of probe with the gravity of earth, we located the shortest position
between the orbit of Mars and probe, and calculate the time of probe takes to
reach this point. Finally, we calculated the initial position of mars. We used the
possible mass of probe provided by professor to evaluate the altitude of the low
mars orbit, and speed. We tried to figure out the shortest time period needed
from launching from Earth and arriving on mars, but our computer was unable
to finish such huge amount of computation. Then, we found the target launch
date in the lists of the coordinates of earth and mars for our shortest routine
plan and end up our research.
3. Materials and Methods
3.1. Experimental Tools
We used Python and Matlab to do the calculation and draw graph to simulate
the orbits.
3.2. The Initial Data
In order to simulating the revolution orbit of Earth and Mars, we need initial
data, the data is listed:
1) The mass, radius of Mars, Earth, and Sun respectively, and Distance to Sun,
Perihelion Speed, Perihelion coordinates, and Period of Mars and Earth are dis-
played in Table 1 and Table 2.
2)
G
=
Gravity
constant
=
6.673
3.3. Depending Theory
11
−×
10 N m kg
⋅
2
⋅
2
Hohmann transfer
a certain trajectory from the Earth to Mars.
law of universal gravitation
Newton’s second law
F ma=
Law of conservation of energy
the total energy of an isolated system remains constant.
Einitial
=
E
1
=
E
2
=
E
3
=
=
E
n
2386
Journal of Applied Mathematics and Physics
DOI: 10.4236/jamp.2019.710162
Y. F. Wei, Y. K. Zhang
Table 1. Mass and radius.
Mass
Radius
Mars
Earth
Sun
6.4219 × 1023 kg
5.965 × 1024 kg
1.9891 × 1030 kg
3397 km
6378 km
6.6 × 105 km
Table 2. Distance and Perihelion coordinates and speed.
Distance to Sun
Perihelion
coordinates
Perihelion speed
Mars
2.0662 × 108 km
Earth
1.471 × 108 km
(−126, 169, 776, 453 m, 163, 625, 969, 530 m) (0 m, −147, 098, 074, 000 m)
26,499 m/s
30,286 m/s
Law of conservation of angular momentum
The total angular momentum of a system remains constant unless acted on by
an external torque.
x,y components
We separated the x component and y component of Force, speed, accelera-
tion.
4. Simulation and Analysis
4.1. Simulating Gravitational Environment
4.1.1. Modeling Gravitational Effect
G∗
=
6.67 10
×
11
−
We calculate the gravitational forces between the Sun, Earth and Mars. The
gravitational force between two masses is given by:
F G
=
ˆMm
r
r
2
(1)
According to the Newton’s second law F ma=
, we obtain the acceleration:
=
a G
2 ˆM
r
r
(2)
4.1.2. Modeling the Revolution Orbit of Earth and Mars
In order to simulate the revolution orbit of Earth and Mars, we need Perihelion
coordinates and Perihelion speed of Mars and Earth in the initial data.
Choose the time step (t) as one minute. Combined with gravitational effect,
we can get the acceleration:
a
n
1
−
=
G
M
2
R
n
1
−
ˆ
r
(3)
Therefore, the velocity varies by acceleration:
v
n
1
−
Also, the position varies by velocity:
v
n
=
+
a t
n
1
−
(4)
DOI: 10.4236/jamp.2019.710162
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Journal of Applied Mathematics and Physics
Y. F. Wei, Y. K. Zhang
x
n
y
n
(5a)
(5b)
v t θ
n
v t θ
n
sin
cos
+
+
x
−=
n
1
y
=
1
−
n
Finally, the distance between the two objects is:
(5c)
Because we have known the initial theta, we can draw the elliptical revolution
2
x
n
1
−
R
n
=
+
y
1
−
1
−
n
orbit of Earth and Mars by using these equations (see from Figure 1).
4.2. Simulating a Trajectory
4.2.1. Determining the Trajectory
In order to find the least amount of fuel, we need to find the shortest distance
from the initial position to the final position of the rocket.
Because of the Kepler’s first law: all planets move around the Sun in elliptical
orbits, having the Sun as one of the foci, the rocket’s trajectory is elliptical and
focuses on the Sun. As a result, the line connecting the initial position to the fi-
nal position passes through the center of the Sun.
We have known that the initial position of the rocket is on the line between
the Sun and Earth and 6800 km from the center of Earth and also on the line
between the Sun and Mars and 3800 km from the center of Mars.
To find the shortest distance from the initial position to the final position, we
use python to select it and print the coordinates of the initial and final position
of the rocket. The results are below:
)
x
1
Rocket initial coordinate (
distance
=
; rocket final coordinate (
−
(6a)
,x y
1
1
(
x y
,n
x
n
y
1
+
−
y
)
2
)
(
)
n
2
n
a
=
semi-longaxis
=
Distance
2
=
(
x
1
−
x
n
2
)
+
2
(
y
1
−
y
n
2
)
(6b)
DOI: 10.4236/jamp.2019.710162
Figure 1. The orbit of Earth and Mars.
2388
Journal of Applied Mathematics and Physics
Y. F. Wei, Y. K. Zhang
c
=
focallength
= −
a
2
x
1
+
2
y
1
=
(
x
1
−
x
n
2
)
+
2
(
y
1
−
y
n
2
)
−
2
x
1
+
2
y
1
(6c)
b
=
2
a
−
2
c
(6d)
4.2.2. The Velocity of Rocket to Transfer into the Hohmann Orbit
A) Simplifying the original problem
The problem in reality is quite difficult to solve by our hand calculation, we
started from a simplified form that we ignore the competing gravitational effect
from the Earth and Mars. In this situation, the rocket moves noticeably in a sim-
ple Keplerian elliptical orbit under the gravitation of the Sun, which we can use
the Kepler’s law to solve the problem.
According to the Kepler’s second law, a line segment joining a planet and the
Sun sweeps out equal areas during equal intervals of time, which represents the
ratio of area to time as a constant.
1d = distance from the center of the earth to the rocket = 6878 km.
2d = distance from the center of mars to the satellite’s orbit which orbit
around the mars = 4185 km.
id = distance from the initial position to the center of sun= 1.473459836 ×
1011 m.
fd = distance from the final position to the center of sun= 2.034171152 ×
1011 m.
iv = velocity we need at the initial position.
lv = the lower velocity at the initial position.
ilv = the ideal initial velocity.
fv = the final velocity at the final position.
flv = the ideal final velocity.
a = semi-long axis.
b = semi-short axis.
m = rocket (satellite).
av = velocity at the point a.
bv = velocity at the point b.
Area swept in a very small time interval at the initial position:
v
∆ =
v
a
−
v
b
(7)
d
δ
i
v t
δ=
i
1
2
(8)
Area swept in a period:
(9)
According to the Kepler’s third law, Square of the orbital period of a planet is
proportional to the cube of the semi-major axis of its orbit. Combining with the
Newton’s gravitation law, we can get the equation:
= π
ab
ta
p
t
=
2
3
a
π
4
GM un
s
(10)
DOI: 10.4236/jamp.2019.710162
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Journal of Applied Mathematics and Physics
Y. F. Wei, Y. K. Zhang
Finally we get the equation:
a
t
p
t
a
δ
i
t
δ
=
The result is below:
ilv =
32786 m s
=
=
ab
π
2
3
a
4
π
GM un
s
ab GM un
s
2
3
a
(11a)
v d
il
i
=
v d
fl
f
(11b)
23734 m s
flv =
B) Perturbation and Correction
Considering the competing gravitation of Sun, Earth and Mars, we can first
get a range of the velocity.
-To determine the initial velocity
We start with velocity
lv , which is equal to the orbital speed of the earth
around the sun plus the escape velocity from the Earth:
lv = the lower speed
a) The Lower Speed
Because we ignore the Earth gravitational effect on the rocket, we can add the
energy needed to escape from the earth to find a velocity which has lower mag-
nitude than the actual velocity.
= escape velocity from earth
v
Escape
v
Escape
=
11169 m s
The kinetic energy:
1
2
mv
2
Escape
−
G
M m
earth
R
earth
= (12)
0
1
2
2
mv
l
=
1
2
2
mv
il
+
1
2
mv
2
Escape Velocity
(13)
34619 m s
lv =
b) The Higher Speed
Because planets conserve energy at any point as it moves around the same star
in an elliptical orbit, we can use the conservation law of energy to calculate the
speed difference between the initial position and final position.
E
a
=
1
2
(
m v
a
−
v
b
)2
−
G
M m
sun
d
E
b
= −
G
M
d
sun
−
G
f
−
G
i
M m
mars
d
2
M m
earth
d
l
(14a)
(14b)
According to the conservation law of energy:
The figure of elliptical revolution orbits of Earth and Mars with pointing a and
E
a
E=
b
(14c)
b is depicted in Figure 2.
We obtain the value of
v∆ =
18117 m s
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Journal of Applied Mathematics and Physics
DOI: 10.4236/jamp.2019.710162
Y. F. Wei, Y. K. Zhang
Figure 2. Orbits with perihelion and aphelion.
The sum of
fv and
v∆ is greater than the true speed, because the ignor-
ance of Earth and Mars gravitation.
As a result, we obtain a range of speed at initial position
Using the Newton’s law of motion (Rocket Equation), we can get these func-
v v
,l
∈
v
fl
(
)
+ ∆ .
v
tions:
x
(
t
+∆ =
t
)
x
0
+
t
∆ (15a)
x
t
(
,x y
0
0
−
Input the initial position (
x
mars
)
, the range of initial velocity (get results
every 50), and the duration of time steps, these functions help us to modeling the
trajectory. We can use python to add up these steps and using numerical method
for integration. The result is below:
mars
−
y
)
2
41600 m s
iv =
We then increase the velocity
lv from velocity in small steps to find the tra-
jectory that approximately intersects the designated point. From this process, we
find the required change in velocity to be 648 m/s and the total velocity to be
2391
Journal of Applied Mathematics and Physics
DOI: 10.4236/jamp.2019.710162
y
(
t
+∆ =
t
)
y
0
+
t
∆ (15b)
v
x
(
t
t
+∆
)
=
v
x
0
+
+
v
y
(
t
t
+∆
)
=
v
y
0
M
sun
2
2
x
y
+
t
t
+
G
a
(
t
t
δ+
)
=
G
+
G
t
∆ (15c)
t
∆ (15d)
M
earth
)
2
y
+
t
(
−
y
e
2
)
(15e)
x
δ
t
δ
y
δ
t
δ
v
δ
x
t
δ
v
δ
y
t
δ
x
e
−
(
x
t
M
mars
)
2
y
+
t
(
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