Launch in to Low Earth Orbit
Objective
Perform a preliminary design study for a vehicle to carry humans from the surface of Earth to low earth
orbit. The vehicle must be subject to an acceleration not to exceed 5 times earth gravity and must be
moving at orbital speed, 7.8 km/s, between 80 and 150 km in altitude. Assume a rail launch using a
ramjet brings the vehicle to a speed between Mach 1 to 5 before the rocket is ignited.
Description
For years NASA has been investigating methods for low cost launch of piloted vehicles into low earth
orbit (LEO). The design challenge has been the insertion of a large mass (80,000 kg) into LEO with a
minimum launch mass. The volume required for the payload is at least 450 m3 . A design study to size
the 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 subject to thrust ( T ) , drag ( D ), weight ( W ), and

centrifugal ( C ) forces.
Fig. 1. Forces for launch into LEO



The forces acting on the projectile will govern the trajectory, r t   xt i  z t k , of the projectile,
 
where xt  is the horizontal location and z t  is the altitude. The unit vectors i , k are along the
horizontal and altitude coordinates respectively.


The drag, D , is opposed to the direction of motion and the thrust, T , along the direction of motion.


The weight, W , is down and the centrifugal force, C , is up.
The drag can be evaluated using


1
D   C D v 2 S 
2

where


v

v




is the density of the atmosphere of Earth at the vehicle,
(1)

v is the
vehicle velocity,
S   (D0 / 2) 2 is the base area of the vehicle and C D is the drag coefficient. The drag coefficient is
generally a function of the vehicle’s Mach number
M  , Reynolds number Rn  and geometry. The
diameter D0 is a parameter to be chosen. The length is then found from the total volume (450 m3).

The weight, W , acting along the negative z direction. Then
 R02


k
W  mg 0 
 R  z 2 
 0

where
m is the mass of the vehicle (80,000 kg), g 0
(2)
is the acceleration of gravity at the surface of Earth
(9.81 m/s2) and R0 is radius to the surface of Earth (6371 km). The centrifugal force, for an eastward
launch along the equator ( v  0 ), is
2


 v

C  m
   R0  z k
 R0  z

(3)
where  is angular velocity of the earth ( 2 /24  60  60/ sec  7.2722e-5 / sec ).
The equation of motion for the projectile in the rotating coordinate system is
m

d 2r
dt 2
   
 T W  D  C .
(4)
It is most efficient to have the thrust of the rocket aligned with the velocity. Then


v
T  m ve 
v
(5a)
 is the mass flow rate out of the
where ve is the exhaust velocity of the rocket (4.45 km/s) and m
rocket, that is,
dm
 m .
dt
(5b)
Assume the thrust cannot exceed 22.25 MN and is adjusted to keep the acceleration at or below five
times standard gravity,




d 2r d 2 x  d 2 z 
dr  dx  dz 

Recall that r t   xt i  z t k , then
v 
i  k , and 2  2 i  2 k .
dt
dt
dt
dt
dt
dt
Drag Coefficient
The drag coefficient is a function of the Mach number, M  v / c , d/D0 (d is the diameter of the
spherical cap), the conical half angle,,Reynolds number based on the base diameter, Rn  vD0 /  ,
and the local speed of sound, c  RT , where  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). If you like, you can use Sutherland’s
formula for the viscosity
 T  C  T 
  0  0
 
 T  C  T0 
3/ 2
(6)
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
D    C D f  2 B  v 2 S 
2
v / 2 



v

v

.


(7)
Earth Atmosphere
The density and pressure variation of the atmosphere of Earth with altitude can be found using
measured data. Not only is the density a function of the altitude, but c,  , M , T , and  are also.
Since the atmosphere of Earth ( 78% N2, 21% O2, and 1% Ar, at the surface),
  1.4 , M  28.97
g/mol. You can use the perfect gas law to get accurate values relating pressure temperature and
density.
Initial Conditions
The initial conditions can be found using Figure 1. From the figure the initial position is

r 0
(8)



vi  vi sin  i  vi cos  k ,
(9)
and the initial velocity is
where vi is the Mach number times the speed of sound as the vehicle leaves the launch rail and
 is
the angle from the horizontal with which it leaves the launch rail (i.e.,    at t  0 ).
Summary
The specific objective is to find values for: 1) the launch angle,  , and the launch speed, vi , 2) the
initial mass, mi , and, 3) the spherical diameter of the tip and the conical base diameter.
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 Earth. This will require a one-dimensional interpolation.
Part 2
Build a function to find the drag coefficient and the back pressure as a function of altitude and speed
and the dimensions of the vehicle. This will require up to a three dimensional interpolation, since fore
body drag coefficient is a function of Mach number v/c, cone half angle  and diameter ratio d/D0.
Part 3
Build a code to integrate the equations of motion for a given launch velocity, launch angle, initial mass
and vehicle geometry.
Final Report
Find values for the launch speed vi, , , d, and D0. Complete the report. The project score sheet can
be found at: ProjectScoreDescription.doc.