Via Satellite, June 1999
Rocket Thrust Equation and Launch Vehicles
The fundamental principles of propulsion and launch vehicle physics
by Robert A. Nelson
A satellite is launched into space on a
rocket, and once there it is inserted into the operational orbit and is
maintained in that orbit by means of thrusters onboard the satellite itself.
This article will summarize the fundamental principles of rocket propulsion and
describe the main features of the propulsion systems used on both launch
vehicles and satellites.
The law of physics on which rocket propulsion is based is
called the principle of momentum. According to this principle, the time rate of
change of the total momentum of a system of particles is equal to the net
external force. The momentum is defined as the product of mass and velocity. If
the net external force is zero, then the principle of momentum becomes the
principle of conservation of momentum and the total momentum of the system is
constant. To balance the momentum conveyed by the exhaust, the rocket must
generate a momentum of equal magnitude but in the opposite direction and thus it
accelerates forward.
The system of particles may be defined as the sum of all the
particles initially within the rocket at a particular instant. As propellant is
consumed, the exhaust products are expelled at a high velocity. The center of
mass of the total system, subsequently consisting of the particles remaining in
the rocket and the particles in the exhaust, follows a trajectory determined by
the external forces, such as gravity, that is the same as if the original
particles remained together as a single entity. In deep space, where gravity may
be neglected, the center of mass remains at rest.
ROCKET THRUST
The configuration of a chemical rocket engine consists of the combustion
chamber, where the chemical reaction takes place, and the nozzle, where the
gases expand to create the exhaust. An important characteristic of the rocket
nozzle is the existence of a throat. The velocity of the gases at the throat is
equal to the local velocity of sound and beyond the throat the gas velocity is
supersonic. Thus the combustion of the gases within the rocket is independent of
the surrounding environment and a change in external atmospheric pressure cannot
propagate upstream.
The thrust of the rocket is given by the theoretical
equation :
F = lm(dot) v_{e} + ( p_{e}  p_{a} )
A_{e}
This equation consists of two terms. The first term, called the
momentum thrust, is equal to the product of the propellant mass flow
rate m(dot)
and the exhaust velocity v_{e} with a correction factor l for nonaxial flow due to nozzle divergence angle. The
second term is called the pressure thrust. It is equal to the difference in
pressures p_{e} and p_{a} of the exhaust velocity
and the ambient atmosphere, respectively, acting over the area
A_{e} of the exit plane of the rocket nozzle. The combined effect
of both terms is incorporated into the effective exhaust velocity c. Thus
the thrust is also written
F = m(dot) c
where an average value of c is used, since it is not
strictly constant.
The exhaust exit pressure is determined by the expansion ratio
given by
e= A_{e} /
A_{t}
which is the ratio of the area of the nozzle exit plane
A_{e} and the area of the throat A_{t} . As the
expansion ratio e increases, the exhaust exit pressure
p_{e} decreases.
The thrust is maximum when the exit pressure of the exhaust is
equal to the ambient pressure of the surrounding environment, that is, when
p_{e} = p_{a}. This condition is known as optimum
expansion and is achieved by proper selection of the expansion ratio. Although
optimum expansion makes the contribution of the pressure thrust zero, it results
in a higher value of exhaust velocity v_{e} such that the
increase in momentum thrust exceeds the reduction in pressure thrust.
A conical nozzle is easy to manufacture and simple to analyze.
If the apex angle is 2a , the correction factor for
nonaxial flow is
The apex angle must be small to keep the loss within
acceptable limits. A typical design would be a =
15° , for which l = 0.9830.
This represents a loss of 1.7 percent. However, conical nozzles are excessively
long for large expansion ratios and suffer additional losses caused by flow
separation. A bellshaped nozzle is therefore superior because it promotes
expansion while reducing length.
ROCKET PROPULSION PARAMETERS
The specific impulse I_{sp} of a rocket is the
parameter that determines the overall effectiveness of the rocket nozzle and
propellant. It is defined as the ratio of the thrust and the propellant weight
flow rate, or
I_{sp} = F / m(dot) g = c / g
where g is a conventional value for the acceleration
of gravity (9.80665 m/s^{2} exactly). Specific
impulse is expressed in seconds.
Although gravity has nothing whatever to do with the rocket
propulsion chemistry, it has entered into the definition of specific impulse
because in past engineering practice mass was expressed in terms of the
corresponding weight on the surface of the earth. By inspection of the equation,
it can be seen that the specific impulse I_{sp} is physically
equivalent to the effective exhaust velocity c, but is rescaled
numerically and has a different unit because of division by g. Some
manufacturers now express specific impulse in newton seconds per kilogram, which
is the same as effective exhaust velocity in meters per second.
Two other important parameters are the thrust coefficient
C_{F} and the characteristic exhaust velocity c*. The
thrust coefficient is defined as
C_{F} = F / A_{t} p_{c}
= m(dot) c /
A_{t} p_{c}
where F is the thrust, A_{t} is the
throat area, and p_{c} is the chamber pressure. This parameter is
the figure of merit of the nozzle design. The characteristic exhaust velocity is
defined as
c* = A_{t} p_{c} / m(dot) = c /
C_{F}
This parameter is the figure of merit of the propellant. Thus
the specific impulse may be written
I_{sp} = C_{F} c* / g
which shows that the specific impulse is the figure of merit of
the nozzle design and propellant as a whole, since it depends on both
C_{F} and c*. However, in practice the specific impulse is
usually regarded as a measure of the efficiency of the propellant alone.
LAUNCH VEHICLE PROPULSION SYSTEMS
In the first stage of a launch vehicle, the exit pressure of
the exhaust is equal to the sea level atmospheric pressure 101.325 kPa (14.7
psia) for optimum expansion. As the altitude of the rocket increases along its
trajectory, the surrounding atmospheric pressure decreases and the thrust
increases because of the increase in pressure thrust. However, at the higher
altitude the thrust is less than it would be for optimum expansion at that
altitude. The exhaust pressure is then greater than the external pressure and
the nozzle is said to be underexpanded. The gas expansion continues downstream
and manifests itself by creating diamondshaped shock waves that can often be
observed in the exhaust plume.
The second stage of the launch vehicle is designed for optimum
expansion at the altitude where it becomes operational. Because the atmospheric
pressure is less than at sea level, the exit pressure of the exhaust must be
less and thus the expansion ratio must be greater. Consequently, the second
stage nozzle exit diameter is larger than the first stage nozzle exit
diameter.
For example, the first stage of a Delta II 7925 launch vehicle
has an expansion ratio of 12. The propellant is liquid oxygen and RP1 (a
kerosenelike hydrocarbon) in a mixture ratio (O/F) of 2.25 at a chamber
pressure of 4800 kPa (700 psia) with a sea level specific impulse of 255
seconds. The second stage has a nozzle expansion ratio of 65 and burns nitrogen
tetroxide and Aerozene 50 (a mixture of hydrazine and unsymmetrical dimethyl
hydrazine) in a mixture ratio of 1.90 at a chamber pressure of 5700 kPa (830 psia), which yields a vacuum specific impulse of 320
seconds.
In space, the surrounding atmospheric pressure is zero. In
principle, the expansion ratio would have to be infinite to reduce the exit
pressure to zero. Thus optimum expansion is impossible, but it can be
approximated by a very large nozzle diameter, such as can be seen on the main
engines of the space shuttle with e = 77.5. There is
ultimately a tradeoff between increasing the size of the nozzle exit for
improved performance and reducing the mass of the rocket engine.
In a chemical rocket, the exhaust velocity, and hence the
specific impulse, increases as the combustion temperature increases and the
molar mass of the exhaust products decreases. Thus liquid oxygen and liquid
hydrogen are nearly ideal chemical rocket propellants because they burn
energetically at high temperature (about 3200 K) and produce nontoxic exhaust
products consisting of gaseous hydrogen and water vapor with a small effective
molar mass (about 11 kg/kmol). The vacuum specific impulse is about 450 seconds.
These propellants are used on the space shuttle, the Atlas Centaur upper stage,
the Ariane4 third stage, the Ariane5 core stage, the H2 first and second
stages, and the Long March CZ3 third stage.
SPACECRAFT PROPULSION SYSTEMS
The spacecraft has its own propulsion system that is used for
orbit insertion, stationkeeping, momentum wheel desaturation, and attitude
control. The propellant required to perform a maneuver with a specified velocity
increment Dv is given by the "rocket
equation"
D m = m_{0} [ 1
 exp( Dv / I_{sp} g) ]
where m_{0} is the initial spacecraft mass. This
equation implies that a reduction in velocity increment or an increase in
specific impulse translates into a reduction in propellant.
In the case of a geostationary satellite, the spacecraft must
perform a critical maneuver at the apogee of the transfer orbit at the
synchronous altitude of 35,786 km to simultaneously remove the inclination and
circularize the orbit. The transfer orbit has a perigee altitude of about 200 km
and an inclination roughly equal to the latitude of the launch site. To minimize
the required velocity increment, it is thus advantageous to have the launch site
as close to the equator as possible.
For example, in a Delta or Atlas launch from Cape Canaveral the
transfer orbit is inclined at 28.5° and the velocity
increment at apogee is 1831 m/s; for an Ariane launch from Kourou the
inclination is 7° and the velocity increment is 1502
m/s; while for a Zenit flight from the Sea Launch platform on the equator the
velocity increment is 1478 m/s. By the rocket equation, assuming a specific
impulse of 300 seconds, the fraction of the separated mass consumed by the
propellant for the apogee maneuver is 46 percent from Cape Canaveral, 40 percent
from Kourou, and 39 percent from the equator. As a rule of thumb, the mass of a
geostationary satellite at beginning of life is on the order of one half its
mass when separated from the launch vehicle.
Before performing the apogee maneuver, the spacecraft must be
reoriented in the transfer orbit to face in the proper direction for the thrust.
This task is sometimes performed by the launch vehicle at spacecraft separation
or else must be carried out in a separate maneuver by the spacecraft itself. In
a launch from Cape Canaveral, the angle through which the satellite must be
reoriented is about 132° .
Once on station, the spacecraft must frequently perform a
variety of stationkeeping maneuvers over its mission life to compensate for
orbital perturbations. The principal perturbation is the combined gravitational
attractions of the sun and moon, which causes the orbital inclination to
increase by nearly one degree per year. This perturbation is compensated by a
northsouth stationkeeping maneuver approximately once every two weeks so as to
keep the satellite within 0.05° of the equatorial
plane. The average annual velocity increment is about 50 m/s, which represents
95 percent of the total stationkeeping fuel budget. Also, the slightly
elliptical shape of the earth's equator causes a longitudinal drift, which is
compensated by eastwest stationkeeping maneuvers about once a week, with an
annual velocity increment of less than 2 m/s, to keep
the satellite within 0.05° of its assigned
longitude.
In addition, solar radiation pressure caused by the transfer of
momentum carried by light and infrared radiation from the sun in the form of
electromagnetic waves both flattens the orbit and disturbs the orientation of
the satellite. The orbit is compensated by an eccentricity control maneuver that
can sometimes be combined with eastwest stationkeeping. The orientation of the
satellite is maintained by momentum wheels supplemented by magnetic torquers and
thrusters. However, the wheels must occasionally be restored to their nominal
rates of rotation by means of a momentum wheel desaturation maneuver in which a
thruster is fired to offset the change in angular momentum.
Geostationary spacecraft typical of those built during the
1980s have solid propellant rocket motors for the apogee maneuver and liquid
hydrazine thrusters for stationkeeping and attitude control. The apogee kick
motor uses a mixture of HTPB fuel and ammonium perchlorate oxidizer with a
specific impulse of about 285 seconds. The hydrazine stationkeeping thrusters
operate by catalytic decomposition and have an initial specific impulse of about
220 seconds. They are fed by the pressure of an inert gas, such as helium, in
the propellant tanks. As propellant is consumed, the gas expands and the
pressure decreases, causing the flow rate and the specific impulse to decrease
over the mission life. The performance of the hydrazine is enhanced in an
electrothermal hydrazine thruster (EHT), which produces a hot gas mixture at
about 1000 ° C with a lower molar mass and higher
enthalpy and results in a higher specific impulse of between 290 and 300
seconds.
For example, the Ford Aerospace (now Space Systems/Loral)
INTELSAT V satellite has a Thiokol AKM that produces an average thrust of 56
kN (12,500 lbf) and burns to depletion in approximately
45 seconds. Onorbit operations are carried out by an array of four 0.44 N (0.1
lbf) thrusters for roll control, ten 2.0 N (0.45 lbf) thrusters for pitch and
yaw control and E/W stationkeeping, and two 22.2 N (5.0 lbf) thrusters for
repositioning and reorientation. Four 0.3 N (0.07 lbf)
EHTs are used for N/S stationkeeping. The nominal mass of the spacecraft at
beginning of life (BOL) is 1005 kg and the dry mass at end of life (EOL) is 830
kg. The difference of 175 kg represents the mass of the propellant for a design
life of 7 years.
Satellites launched in the late 1980s and 1990s typically have
an integrated propulsion system that use a bipropellant combination of
monomethyl hydrazine as fuel and nitrogen tetroxide as oxidizer. The specific
impulse is about 300 seconds and fuel margin not used for the apogee maneuver
can be applied to stationkeeping. Also, since the apogee engine is restartable,
it can be used for perigee velocity augmentation and supersynchronous transfer
orbit scenarios that optimize the combined propulsion capabilities of the launch
vehicle and the spacecraft.
For example, the INTELSAT VII satellite, built by Space
Systems/Loral, has a Marquardt 490 N apogee thruster and an array of twelve 22 N
stationkeeping thrusters manufactured by Atlantic Research Corporation with a
150:1 columbium nozzle expansion ratio and a specific impulse of 235 seconds.
For an Ariane launch the separated mass in GTO is 3610 kg, the mass at BOL is
2100 kg, and the mass at EOL is 1450 kg. The mission life is approximately 17
years.
The Hughes HS601 satellite has a similar thruster
configuration. The mass is approximately 2970 kg at launch, 1680 kg at BOL, and
1300 kg for a nominal 14 year mission.
An interesting problem is the estimation of fuel remaining on
the spacecraft at any given time during the mission life. This information is
used to predict the satellite end of life. There are no "fuel gauges" so the
fuel mass must be determined indirectly. There are three principal methods. The
first is called the "gas law" method, which is based on the equation of state of
an ideal gas. The pressure and temperature of the inert gas in the propellant
tanks is measured by transducers and the volume of the gas is computed knowing
precisely the pressure and temperature at launch. The volume of the remaining
propellant can thus be deduced and the mass determined from the known density as
a function of temperature. Corrections must be applied for the expansion of the
tanks and the propellant vapor pressure. The second method is called the
"bookkeeping" method. In this method the thruster time for each maneuver is
carefully measured and recorded. The propellant consumed is then calculated from
mass flow rate expressed in terms of the pressure using an empirical model. The
third method is much more sophisticated and is based on the measured dynamics of
the spacecraft after a stationkeeping maneuver to determine its total mass. In
general, these three independent methods provide redundant information that can
be applied to check one another.
NEW TECHNOLOGIES
Several innovative technologies have substantially improved the
fuel efficiency of satellite stationkeeping thrusters. The savings in fuel can
be used to increase the available payload mass, prolong the mission life, or
reduce the mass of the spacecraft.
The first of these developments is the electric rocket arcjet
technology. The arcjet system uses an electric arc to superheat hydrazine fuel,
which nearly doubles its efficiency. An arcjet thruster has a specific impulse
of over 500 seconds. Typical thrust levels are from 0.20 to 0.25 N. The arcjet
concept was developed by the NASA Lewis Research Center in Cleveland and
thrusters have been manufactured commercially by Primex Technologies, a
subsidiary of the Olin Corporation.
AT&T’s Telstar 401 satellite, launched in December 1993
(and subsequently lost in 1997 due to an electrical failure generally attributed
to a solar flare) was the first satellite to use arcjets. The stationkeeping
propellant requirement was reduced by about 40 percent, which was critical to
the selection of the Atlas IIAS launch vehicle. Similar arcjet systems are used
on INTELSAT VIII and the Lockheed Martin A2100 series of satellites. INTELSAT
VIII, for example, has a dual mode propulsion system incorporating a
bipropellant liquid apogee engine that burns hydrazine and oxidizer for orbit
insertion and four arcjets that use monopropellant hydrazine in the reaction
control subsystem for stationkeeping.
Electrothermal hydrazine thrusters continue to have
applications on various geostationary satellites and on some small spacecraft
where maneuvering time is critical. For example, EHTs are used on the IRIDIUM
satellites built by Lockheed Martin.
The most exciting development has been in the field of ion
propulsion. The propellant is xenon gas. Although the thrust is small and on the
order of a few millinewtons, the specific impulse is from 2000 to 4000 seconds,
which is about ten to twenty times the efficiency of conventional bipropellant
stationkeeping thrusters. Also, the lower thrust levels have the virtue of
minimizing attitude disturbances during stationkeeping maneuvers.
The xenon ion propulsion system, or XIPS (pronounced "zips"),
is a gridded ion thruster developed by Hughes. This system is available on the
HS601 HP (high power) and HS702 satellite models and allows for a reduction in
propellant mass of up to 90 percent for a 12 to 15 year mission life. A typical
satellite has four XIPS thrusters, including two primary thrusters and two
redundant thrusters.
Xenon atoms, an inert monatomic gas with the highest molar mass
(131 kg/kmol), are introduced into a thruster chamber ringed by magnets.
Electrons emitted by a cathode knock off electrons from the xenon atoms and form
positive xenon ions. The ions are accelerated by a pair of gridded electrodes,
one with a high positive voltage and one with a negative voltage, at the far end
of the thrust chamber and create more than 3000 tiny beams. The beams are
neutralized by a flux of electrons emitted by a device called the neutralizer to
prevent the ions from being electrically attracted back to the thruster and to
prevent a space charge from building up around the satellite.
The increase in kinetic energy of the ions is equal to the work
done by the electric field, so that
½ m v^{2} = q V
where q, m, and v are the charge, mass,
and velocity of the ions and V is the accelerating voltage, equal to the
algebraic difference between the positive voltage on the positive grid and the
negative voltage on the neutralizer. The charge to mass ratio of xenon ions is
7.35 ´ 10^{5} C/kg.
The HS601 HP satellite uses 13centimeter diameter XIPS
engines to perform northsouth stationkeeping and to assist the spacecraft’s
gimballed momentum wheel for roll and yaw control. The accelerating voltage is
about 750 volts and the ions have a velocity of 33,600 m/s. The specific impulse
is 3400 seconds with a mass flow rate of 0.6 mg/s and 18 mN of thrust. Each ion thruster operates for approximately 5 hours
per day and uses 500 W from the available 8 kW total spacecraft power.
The HS702 spacecraft has higher power 25centimeter thrusters to perform all stationkeeping maneuvers and to
complement the four momentum wheels arranged in a tetrahedron configuration for
attitude control. The accelerating voltage is 1200 volts, which produces an ion
beam with a velocity of 42,500 m/s. The specific
impulse is 4300 seconds, the mass flow rate is 4 mg/s, and the thrust is 165 mN.
Each HS702 ion thruster operates for approximately 30 minutes per day and
requires 4.5 kW from the 10 to 15 kW solar array. The stationkeeping strategy
maintains a tolerance of ± 0.005° that allows for the collocation of several satellites at a
single orbital slot.
The HS702 satellite has a launch mass of up to 5200 kg and an
available payload mass of up to 1200 kg. The spacecraft
can carry up to 118 transponders, comprising 94 active amplifiers and 24 spares.
A bipropellant propulsion system is used for orbit acquisition, with a fuel
capacity of 1750 kg. The XIPS thrusters need only 5 kg of xenon propellant per
year, a fraction of the requirement for conventional bipropellant or arcjet
systems. The HS702 also has the option of using XIPS thrusters for orbit
raising in transfer orbit to further reduce the required propellant mass
budget.
The first commercial satellite to use ion propulsion was PAS5,
which was delivered to the PanAmSat Corporation in August 1997. PAS5 was the
first HS601 HP model, whose xenon ion propulsion
system, together with gallium arsenside solar cells and advanced battery
performance, permitted the satellite to accommodate a payload twice as powerful
as earlier HS601 models while maintaining a 15 year orbital life. Four more
Hughes satellites with XIPS technology were in orbit by the end of 1998. In
addition, Hughes also produced a 30centimeter xenon ion engine for NASA’s Deep
Space 1 spacecraft, launched in October 1998.
Another type of ion thruster is the Hall effect ion thruster.
The ions are accelerated along the axis of the thruster by crossed electric and
magnetic fields. A plasma of electrons in the thrust chamber produces the
electric field. A set of coils creates the magnetic field, whose magnitude is
the most difficult aspect of the system to adjust. The ions attain a speed of
between 15,000 and 20,000 m/s and the specific impulse is about 1800 seconds.
This type of thruster has been flown on several Russian spacecraft.
SUMMARY
The demand for ever increasing satellite payloads has motivated
the development of propulsion systems with greater efficiency. Typical
satellites of fifteen to twenty years ago had solid apogee motors and simple
monopropellant hydrazine stationkeeping thrusters. Electrically heated thrusters
were designed to increase the hydrazine performance and the principle was
further advanced by the innovation of the arcjet thruster. Bipropellant systems
are now commonly used for increased performance and versatility.
The future will see a steady transition to ion propulsion. The
improvements in fuel efficiency permit the savings in mass to be used for
increasing the revenuegenerating payloads (with attendant increase in solar
arrays, batteries, and thermal control systems to power them), extending the
lifetimes in orbit, or reducing the spacecraft mass to permit a more economical
launch vehicle.
____________________________________________
Dr. Robert A. Nelson, P.E. is president of Satellite
Engineering Research Corporation, a satellite engineering consulting firm in
Bethesda, Maryland, a Lecturer in the Department of Aerospace Engineering at the University of
Maryland and Technical Editor of Via Satellite magazine. Dr. Nelson is the instructor for the ATI course Satellite
Communications Systems Engineering. Please see our Schedule for dates and
locations.
