Piloted Mars Entry Vehicle
Objective
Perform a preliminary design study for a vehicle to carry humans from low Mars orbit to the surface of
Mars. The vehicle must be subject to a deceleration not to exceed 8 times earth gravity and must be
moving at less than the local speed of sound at two kilometers altitude. Assume parachutes and rockets
will be used once the vehicle is below the speed of sound.
Description
NASA has determined that large spacecraft landing on the surface of Mars will require aerodynamic
decelerators that are massive for bringing a piloted craft to the surface of Mars. The design challenge
has already been encountered with the Mars Curiosity lander for which a complex landing system had to
be developed. Yet the mass of the rover, at 900 kg, is well below the mass that must be used for a brief
piloted landing (62,000 kg and a
volume of approximately 400
m3).
A design study to size the entry
vehicle will require a prediction
of the speed and altitude of the
vehicle. Assume that the vehicle
is spherically blunt cone as
shown in Figure 1.
The forces acting on the
projectile will govern the



trajectory, r t   xt i  yt  j ,
of the projectile. There are two
forces acting on the vehicle: the
gravitational force of Mars and
the drag due to its atmosphere.
Figure 1 also illustrates the
geometry for the motion.
Figure 1. Geometry and coordinate system.
Let the altitude of the entry vehicle be z, the radius of Mars be R0, and the entry angle be . The forces
acting on the entry vehicle are illustrated in Fig. 2. The drag,

D0 , is opposed to the direction of motion
and is given by


1
v
2 
D0   C D v S  
2
v




(1)
where

is the density of the
atmosphere of Titan at the entry

vehicle, v is the entry vehicle
velocity,
S   (D / 2) 2 is the
frontal area of the sphere (with
radius
a ) and C D
is the drag
coefficient. The drag coefficient
is generally a function of the
entry vehicle’s Mach number
M  , Reynolds number Rn 
and geometry. The diameter
2a
is a parameter to be
chosen.
Fig. 2. Forces on Entry Vehicle

The second force is the weight, W , acting along the negative z direction. Then


R2



0
W  mg 
j
0
 R0  z 2 


where
m is the mass of the entry vehicle (62000 kg), g 0
(2)
is the acceleration of gravity at the surface of
Mars (3.711 m/s2) and R0 is radius to the surface of Mars (3390 km).
The equation of motion for the projectile in the rotating coordinate system is
m

d 2r
dt 2
 
 W  D0 .
(3)




dr  dx  dy 
d 2r d 2 x  d 2 y 


i

j
Recall that r t   xt i  yt  j . Hence
, and
v  i 
j.
2
2
2
dt
dt
dt
dt
dt
dt
Drag Coefficient
The drag coefficient is function of the Mach number, M  v / c , d/D, and Reynolds number based on
the base diameter, Rn  vD /  and D is the base diameter, c  RT is the local speed of sound,
 is the viscosity coefficient, T is the temperature,  is the ratio of the specific heats, R  Ru / M is
the gas constant, M is the average molecular mass of the atmosphere, and Ru is the universal gas
constant (8.3145 J/K). Not only is the density a function of the altitude, but c,  , M , T , and  are also.
Use Sutherland’s formula for the viscosity
 T  C  T 
  0  0
 
 T  C  T0 
3/ 2
(4)
where C  120 K, T0  291.15 K and  0  18.27 Pa-s.
See NASA TN D-3088 for the drag coefficient of the fore body ,
CD f
, and Ballistic Research Laboratory
(BRL) Memorandum Report No. 1709 for an estimate for the base pressure, p
dynamic pressure (i.e.,
v 2 / 2 ).
B
, relative to the
The drag can now be written as

p 


1
B
D0    C D 
v 2 S 

f
2

v 2 / 2 


v

v

.


(5)
Mars Atmosphere
The density and pressure variation of the atmosphere of Mars with altitude can be found using Viking 1
data. Note that Viking 2 data is not as accurate. Not only is the density a function of the altitude, but
c,  , M , T , and  are also. Since the atmosphere of Mars (95.3% CO2, 2.7% N2, 1.6% Ar, .2% O2 and
0.1% CO),
  1.4 , M  43.42 g/mol.
You can use the perfect gas law to get more accurate values.
Initial Conditions
The initial conditions can be found using Figure 1. From the figure the initial position is


r  H  R0  j
(5)
where H is the entry altitude (which must be chosen sufficiently large, assume 600 km) and the initial
velocity is


vi  v0 i .
(6)
where v0 is the entry speed (which is less than the orbital speed, 3.269 km/s at 600 km altitude ). For a
low point in the orbit at the surface of Mars the speed at 600 km is 3.113 km/s. Hence a speed change
of about 0.156 km/s is required to arrive at the surface. Note that the speed will be 3.617 km/s at the
surface if there was no drag from the atmosphere.
Summary
The specific objective is to find values for: 1) the entry angle and speed and, 2) the spherical diameter of
the tip, the conical base diameter and the conical half-angle for a projectile with a maximum
acceleration of 8 times the acceleration of gravity at the surface of the earth and a final speed less than
the local speed of sound.
Project Components
There are three parts to the project plus the final report. Each part is due according to the schedule for
the course. The due dates for the parts and the report will not change. Each part will be worth 4 points
and the final report will be 18 points for a total of 30 points. Each of the three parts consists of the
code, a discussion of the algorithms used and test cases validating the algorithms and code.
Part 1
Build a function to find the density, temperature, speed of sound and the viscosity as a function of
altitude above the surface of Mars. This will require a one-dimensional interpolation.
Part 2
Build a function to find the drag coefficientand the back pressure as a function of altitude and speed and
the dimensions of the entry vehicle. This will require a three dimensional interpolation, since fore body
drag coefficient is a function of Mach number v/c, cone half angle  and diameter ratio d/D.
Part 3
Build a code to integrate the equations of motion for a given entry velocity, angle and entry vehicle
geometry.
Final Report
Find values for the entry speed v0, , , d, and D. Complete the report. The project score sheet can be
found at: ProjectScoreDescription.doc.