A.6.2.2.X Ballistic Coefficient
1
A.6.2.2.X Ballistic Coefficient
A vehicle’s ballistic coefficient1 is a “measure of its ability to overcome air resistance in
flight”2. The coefficient is given as Eq. (A.6.2.2.X.1).
C ballistic 
m
CD S
(A.6.2.2.X.1)
where m is the current mass, C D is the coefficient of drag (calculated by the Aerothermal
group’s solve_cd.m), and S is the reference area (current stage diameter). A larger
Cballistic means that a vehicle is massive enough to overcome air resistance during ascent.
Unlike smaller launch vehicles, the Saturn V, Ariane 4, and Pegasus vehicles are not as
susceptible to the effects of wind and is expected to have much larger Cballistic than our
vehicle.
As another way of validating the trajectory model, we perform an analysis comparing the
ballistic coefficient of our launch vehicle to the much larger vehicles listed above. Our
trajectory code is run with our ground, balloon, or aircraft steering law.
Table
A.6.2.2.X.1 provides the steering angles at the end of each stage for each launch vehicle
ran.
Table A.6.2.2.X.1 Key Performance and Steering Law Characteristics
 3e (ο)
tb1 (s)
 1e (ο)
 2 e (ο)
Saturn V
Ariane4
m 1 (kg/s)
13,360.24
01,112.19
161.0
205.0
87.0
87.0
40.0
40.0
0.0
0.0
Pegasus
SB-HA-DA-DAa
SG-SA-DT-DTb
LG-SA-DT-DTc
00206.14
00006.85
00014.21
0018.391
073.0
196.5
182.4
171.2
87.0
-14.0
0.0
34.0
-25.0
-20.0
-10.0
-26.0
-30.0
-20.0
-10.0
-26.0
Vehicle
a. 200 g balloon: 1st stage hybrid aluminum, 2nd stage solid aluminum, 3rd stage solid aluminum
b. 200 g ground: 1st stage storable aluminum, 2nd stage solid titanium, 3rd stage solid titanium
c. 5 kg ground: 1st stage storable aluminum, 2nd stage solid titanium, 3rd stage solid titanium
Author: Amanda Briden
A.6.2.2.X Ballistic Coefficient
2
For each case the launch vehicles inputs (i.e. tb,m,T etc.) are changed and come from
published data provided by John Tsohas3. The goal of the analysis is to see trends in the
change in ballistic coefficient and is not meant to capture the exact value for every
vehicle.
The ballistic coefficient variation with time for the larger launch vehicles is plotted in
Fig. A.6.2.2.X.1. The coefficient is set to zero after the launch vehicle is out of the
atmosphere.
Trends match expectations as the Saturn V has the largest ballistic
coefficient initially, followed by the Ariane 4 and Pegasus. The Ariane 4 is larger than
the Saturn V after 60s. This is explained by Saturn V’s first stage mass flow rate. It is
much larger than the Ariane 4 and has a burn time that is smaller (refer to Table
A.6.2.2.X.1). The large dips in the plots occur when the vehicle enters the transonic
regime (M = 1) and C D spikes causing a sudden decrease in Cballistic.
Fig. A.6.2.2.X.1: Ballistic coefficient variation with time for larger launch vehicles (i.e. Saturn V, Ariane 4,
Pegasus).
(Amanda Briden)
Author: Amanda Briden
A.6.2.2.X Ballistic Coefficient
3
In comparison to the Saturn V, our large ground launch vehicle’s ballistic coefficient is
approximately 16 times smaller. All of our launch vehicles fit in the lower left hand
corner of Fig. A.6.2.2.X.1.
A blown up view of that region is displayed in Fig.
A.6.2.2.X.2. Within our region, the trends remain consistent as the bigger launch vehicle
still has the largest ballistic coefficient.
Fig. A.6.2.2.X.2: Ballistic coefficient variation with time for our launch vehicles (i.e. SB-HA-DA-DA, SGSA-DT-DT, LG-SA-DT-DT).
(Amanda Briden)
The trends match our expectations and provide a boost of confidence in the modeling of
drag in the trajectory code.
References
1
Longuski, Professor James. “AAE 450 Spacecraft Design Lecture #6 Spring 2008.” Purdue University,
West Lafayette, IN.
2
“Ballistic
coefficient.”
Wikipedia
[online],
January
18,
2008,
http://en.wikipedia.org/wiki/Ballistic_coefficient [cited 27 February 2008].
3
Tshoas, John. Trajectory Input Values for Historical Launch Vehicles interview. February 2008.
Author: Amanda Briden
URL: