Orbital Mission Design Using Low-Thrust Devices
By
ASEN 5050 Research Paper
December
15, 2005
1. Abstract
Low thrust, high specific impulse propulsion devices have the advantage of delivering a payload to its operational orbit using a fraction of the propellant that would be used for conventional systems. This translates into the ability to the delivery of more payload to orbit, more propellant to orbit (thus increasing lifetime), or to scale down the launch vehicle (reducing the launch costs). Since these devices typically provide a very low thrust for an extremely long duration, they are most suited for station keeping, orbit repositioning, and orbit transfer applications. However, low-thrust devices present a number of challenging obstacles from the mission design standpoint. The trajectories are particularly susceptible to, and are often dominated by, the effects of gravitational non-uniformity and third-body perturbations. This paper surveys the current state of the art in low-thrust technologies. A detailed mission design is also included to highlight the concepts of low-thrust trajectories, and compared to a similar mission using conventional propulsion technologies.
2. Introduction
The origins of spacecraft propulsion can be traced back to the use of black powder rockets produced by the Chinese around the 12th century. The earliest designs were used primarily as fireworks for religious festivals. However, the military application became immediately apparent, and rockets were used in war by the Chinese, Mongolians, and Arabs. Designs of solid rockets progressed through the centuries, but improvements in cannon technologies made rockets less desirable as a weapon.
Early rockets were crude devices
that were unpredictable, dangerous, and poorly understood. A major scientific advance was made when Isaac
Newton formulated his Laws of Motion in the 17th century. For the first time, flight dynamics were
understood and the notion of space travel became feasible.
Today, liquid and solid propulsion
systems are overwhelmingly used in space applications. However, other maturing technologies have
presented engineers with more possibilities.
What was once only a concept of science fiction novels (electric
propulsion and solar sails, specifically) are now becoming viable designs. Flight tests of SERT I and II in the 1970’s
validated ion thruster operation in space, and opened the door for more
ambitious missions. By the 1990’s, ion
propulsion systems became more common while other advanced propulsion systems
were manufactured and flight tested. The
collapse of the
The discussion of advanced propulsion concepts and their associated mission designs mentioned in this paper is limited to the low-thrust variety. As a general rule, a propulsion system can be considered low thrust if thrust values are on the order of milli-Newtons. Low-thrust devices are an attractive option for stationkeeping, orbit raising/lowering, and interplanetary travel. These systems are characterized by extremely high values of specific impulse, the rocket equivalent to an automobile’s “miles per gallon”. The efficiency in propellant usage translates into reduced launch costs and/or increased payload.
The choice of a low-thrust propulsion system is not without drawbacks, however. Most obvious is the time required to perform a specific mission. There are also limits placed on the missions attainable via low-thrust trajectories. For orbital missions, a spiral trajectory is usually the method employed to raise/lower the orbit. However, other trajectories are being considered, such as one that uses a chemical rocket to place the spacecraft in a highly eccentric, super-synchronous orbit then an electric propulsion system that fires continuously to attain a geosynchronous orbit. Lastly, it is important to point out that low-thrust orbital missions are often susceptible to gravitational perturbation effects.
3. Review of Literature
A simple internet search for
advanced low thrust propulsive devices yields the myriad websites devoted
mostly to the operational aspects of each device. Websites such as those maintained by NASA and
ESA provide excellent descriptions of the various concepts as well as basic
physics. There are also a vast number of
hobby/general interest websites available, but the content of these sources
must be approached with some degree of caution.
Ample coverage of low-thrust devices is also available in print. However, since many of these devices are
still only conceptual (or in early development) the researcher must seek out
the most recent texts available. Of
particular importance to this paper were Rocket
Propulsion Elements by George Sutton and Oscar Bliblarz
and Space Propulsion Analysis and Design
by Ronald Humble et. al. Both of these texts are considered essential
in the field of rocket propulsion and cover the design and analysis of all
types of propulsion systems. Additional
information concerning the propulsion devices came from the course materials
for ASEN 5053, Rocket Propulsion through the
While the types and descriptions of state-of-the-art propulsion devices are readily described on the internet, a solid treatment of their astrodynamics is not. The mission analysis, though conceived decades ago, proved to be somewhat elusive through both internet searches and in print. In most cases, a brief mention of spiral trajectories was found in a section of a chapter of a book dealing with a particular type of propulsion device. Journal papers published within the last ten years tend to focus on the optimization or computational analysis of specific missions, but fail to provide a simple derivation of the equations of motion. Most of the derivations found in this paper follow the work of J. W. Cornelisse in Rocket Propulsion and Spaceflight Dynamics. Other helpful sources include Orbital Dynamics of Space Vehicles by Ralph Deutsch and Ion Propulsion for Space Flight by Ernst Stuhlinger.
4. Types of Low Thrust Propulsion Devices
4.1. Electric Propulsion
In conventional chemical propulsion systems, the energy source used to heat the working propellant comes from the chemical bonds within the propellant itself. The ideal propellant is one that produces low molecular mass exhaust products and high operating temperature. These are often two competing constraints on the choice of propellant, and therefore can limit the performance of the propulsion system. The dilemma is avoided when a separate external energy source (nuclear fission, solar array, electric arcjet, etc.) is available to heat a low molecular mass propellant. This is the basis for eclectic propulsion devices.
4.1.1. Electrostatic Devices
Electrostatic devices derive their thrust from accelerating positively charged propellant particles (ions) through an electrostatic field. These devices can be further classified by the ionization process as electron bombardment thrusters, contact ion thrusters, and field emission/colloid thrusters. The first two are termed ion propulsion devices since they involve the production and acceleration of charged ions. Field emission/colloid thrusters involve the production and acceleration of charged liquid droplets. Of these ionization processes, only electron bombardment has been successfully implemented on a spacecraft.
Figure 1. Electron Bombardment Device [Ref 11]
Figure 1 shows a schematic of the essential elements of an ion thruster. The propellant atoms (or particles) are ionized, and then accelerated out of the thruster by an electrostatic field. The exhausted ions are neutralized by electrons that are emitted from an external cathode.
4.1.2. Electrothermal Devices
Electrothermal devices rely on the thermodynamics expansion of the propellant gas through an exhaust nozzle. The propellant is heated using electrical energy produced by various methods. For resistojets, the propellant is fed into a thruster and heated while flowing over a resistance heater or similar structure. Arcjets use an electrical discharge, or “arc” that is generated between a cathode and an anode. In the most common design, the cathode is oriented coaxially with the anode and the anode acts as the thruster’s nozzle.
Figure
2. Resistojet engine [Ref 13]
Figure 3 Arcjet
Engine [Ref 12]
4.1.3. Electromagnetic Devices
Electromagnetic devices operate by accelerating a plasma (electrically neutral collection of ions, electrons, and neutral atoms) by electric and magnetic field forces. Unlike electrostatic and electrothermal devices, electromagnetic thrusters come in a wide assortment of design configurations. Applied electric/magnetic fields and internal currents can be steady or pulsed, and the choice of working propellant can include solid or liquid types. Although there are many unique configurations, electromagnet devices can be concisely classified as Hall-effect, pulsed plasma, and magnetoplasmadynamic thrusters.
In general, Hall effect thrusters employ a radial magnetic field that is set up between two concentric annular magnet pole structures. The interior volume contains a propellant gas through which a continuous electric discharge passes between the two electrodes. The axial electric field developed between the electrodes interacts with the radial magnetic field to produce a body force (Hall effect) that is transmitted to the ions and neutral atoms by collisions. A simplified schematic showing the Hall phenomenon is shown below. Figure 5 simplified Hall effect thruster.
Figure 4. Hall effect [Ref 2]
Figure 5
Hall effect Thruster [Ref 12]
A pulsed-plasma thruster works by eroding (ablating) and then ionizing material from a fuel mass by using the discharge of a storage capacitor across a pair of electrodes. As shown in Figure 6, the fuel is forced forward by a spring as it is ignited on one end. The propellant plasma is transferred though the thruster via the Lorentz force. This force arises when a magnetic field interacts with an electric field.
Figure 6 Pulsed-plasma thruster [Ref 12]
The magnetoplasmadynamic (MPD) thruster currently produces the most thrust out of all electromagnetic devices. The thruster has two metallic electrodes: a rod-shaped cathode and a concentric anode. A high-current electric arc is produced between the electrodes, heating up the anode. The hot anode emits electrons which collide with propellant gas atoms. As the atoms collide, ionization occurs and plasma is created. A self-induced magnetic field is created when the electric current returns to the power supply through the cathode. The magnetic field interacts with the electric current flowing between the electrodes to produce a Lorentz force that pushes the plasma out of the engine, creating thrust.
Figure 7 Magnetoplasmadynamic thruster [Ref
9]
4.2. Space Sails
The category of space sails includes devices that use solar radiation pressure or laser light generated by sunlight to propel the spacecraft. The most well-known concept is the solar sail, which consists of a large, flexible, yet lightweight structure that is coated on one side with a highly reflective surface. As photons bounce off the reflective surface, thy transfer momentum to the sail. This results in a constant pressure on the sail and therefore a constant acceleration. Other concepts that should be mentioned here are magnetic sails, diode sails, differential sails, and induction sails.
Figure 8. Solar Sail [Ref 9]
4.3. Beamed Energy devices
Beamed energy propulsion devices use a remote energy source, such as a ground- or space-based laser or microwave transmitter, to send power to a space vehicle via a beam of electromagnetic radiation. As was pointed out about solar sails, this concept has become very popular recently because the energy source does not have to be carried on board the vehicle. However, this technology is in its infancy and requires significant improvements in laser capability.
5. Low
Thrust Mission Design and Analysis
The nature of continuous, low
thrust devices facilitates the need for a modified approach when describing
their motion. Conventional high-thrust propulsion
systems are often analyzed (for preliminary mission design) using the
assumption of impulsive maneuvering. In
that case, the DV
produced by the rocket occurs instantaneously.
Obviously, when analyzing low-thrust devices, we must take into account
the slow change in velocity and mass expenditure. The analytical approach that follows is by no
means a rigorous treatment of the subject of low thrust propulsion. It can, however, be used an initial step
before more sophisticated methods are implemented.
5.1. Equations of Motion
.Where 
radial distance from
the center of attracting body to the satellite
unit
vector directed from the center of attracting body to the satellite
Gravitational
parameter of attracting body
However, when there exists an acceleration due to thrust, the right hand side is no longer zero:
Equation 2
Where 
thrust
M = instantaneous mass of the spacecraft
acceleration
due to thrust
For propulsion devices that carry and expend their own propellant, the thrust can be constant but the acceleration is generally not constant because the spacecraft is continuously ejecting mass. It should be pointed out here that devices such as solar sails do not carry their own “propellant” so there is not an acceleration change due to propellant consumption. In that case, an expression could be developed for the thrust acceleration that includes parameters pertinent to the operation of solar sails. This is not within the scope of the current paper, so all of the following derivations will assume mass loss.
The thrust acceleration is time dependent, and can be related to a value at t = 0 by the relations
In these equations, the “dotted” values for mass flow rate are used to be consistent with the literature. Also, those values with a zero subscript represent the initial value. The three relations above can be manipulated to give a final expression for acceleration as a function of time:
Equation 3
If we cross multiply Equation 2 by 
, we obtain an expression for the rate of change in orbital
angular momentum per unit mass:
Equation 4
Where 
orbital angular
momentum per unit mass
thrust
acceleration
Similarly, if we scalar multiply Equation 2 by the velocity 
Equation 5
Where 
orbital angular
velocity
ε = total energy per unit mass.
The energy per unit mass can also be expressed as the sum of potential and kinetic energy:
Equation 6
For two-dimensional motion, i.e. the thrust is always directed in the original plane of motion, the expressions for angular momentum and energy can be expressed in terms of flight path angle and thrust angle.
Equation
7
Equation 8
The angles in Equation 5 and Equation 6 are defined in below. The true anomaly q describes the angular position of the spacecraft.
Figure 9
If the spacecraft is initially in a circular parking orbit, the following initial conditions are established:
t = 0; r = r0; q = 0; and 
.
At this point, the equation of motion shown in Equation 2 can be integrated to give the trajectory of the spacecraft (given acceleration and thrust angle as a function of time).
Two simplified cases are of particular interest: constant radial thrust and constant tangential thrust. The constant radial thrust situation gives the angular momentum and energy as:
Equation
9
Equation
10
Thus the momentum per unit mass is constant (just as in Keplerian orbits). For the case of tangential thrust, the angular momentum and energy equations simplify to:
Equation
11
Equation 12
Hence, the tangential thrust case represents the maximum rate of change of energy. Furthermore, if the orbit is circular or nearly circular, a tangential thrust would maximize the rate of change of angular momentum since the thrust is applied perpendicular to the position vector.
Compared to tangential thrust, the radial thrust case is not as effective in raising/lowering circular orbits, nor is it relevant for interplanetary missions. For this reason, constant radial thrust will not be mentioned further in this paper. Constant radial thrust missions do, however, have some application for certain other orbital maneuvers.
5.2. Spiral
Orbit Mission
For a spacecraft initially in a circular parking orbit, application of a low-magnitude, constant tangential thrust will slowly change the orbital radius (semi-major axis). Since each pass closely approximates a circular orbit, the velocity can be calculated as
Equation 13
This approximation allows us to find expressions for the
radial distance as a function of time, and for the amount of propellant
consumed for the maneuver.
By substituting Equation 13 into Equation 6 and Equation 12, we can obtain following expressions:
Equation
14
and
Equation
15
Now energy can be eliminated from the preceding two equations to give:
Equation 16
and the instantaneous acceleration can be inserted via Equation 3. Using the initial conditions described earlier, a final expression for radial distance is found by integrating Equation 16:
Equation 17
The propellant consumption can be determined by assuming constant thrust and exhaust velocity (two assumptions necessary for the derivation of the radial distance relationship). For the simple case of raising (or lowering) an orbit, the propellant mass is calculated by
Equation 18
It is imperative to understand the
underlying assumptions that were built into Equation
17 and Equation
18. The equations
are valid only for constant low-thrust, circular or nearly circular orbits. The thrust vector is assumed to be oriented
along the tangential direction, and there are no out-of-plane thrust
components. Although it is easy to
extend the formulations to consider escape conditions, the methodology changes
when the satellite begins traveling on a non-circular orbit. The formulation neglects the effects of
atmospheric drag and assumes the central attracting body is a point mass.
6. Extension:
Interplanetary Mission
The simplified analysis that led to the expression for spiral radius as a function of time can be used to analyze an interplanetary mission to Mars. Before proceeding, some assumptions must be made about the technique. First, the orbits of Earth and Mars around the sun are assumed to be circular and coplanar. The two planets are assumed to be point masses with no gravitational perturbations, and perturbations from solar radiation pressure and atmospheric drag are ignored as well. Next, the escape velocity and capture events are assumed to occur just when the spacecraft crosses the sphere of influence of the respective planet. We also assume that the Earth-centered spiral trajectory becomes hyperbolic (elliptical with respect to the sun) exactly at the periapse of the transfer orbit, as shown in Figure 10. Likewise, it is assumed that Mars capture occurs exactly at apoapse. This means that we have pre-determined the future positions of the planets and timed the start of the escape spiral accordingly. Furthermore, the propulsion system is assumed to provide constant thrust along the tangent of the circular orbit. Finally, the transfer orbit is assumed to require some additional DV for a Hohmann-type trajectory and all other corrective maneuvers are ignored.
The mission starts at a 400 km parking orbit around earth. The spacecraft is assumed to use an ion thruster with an initial acceleration of 1 x 10-6 km/sec and an exhaust velocity of 150 km/sec. At each timestep, the new orbital radius (about Earth) is calculated using Equation 17. The new velocity is calculated using the equation for circular velocity. This procedure is repeated until the radius equals the radius of the sphere of influence. The velocity at the end of the time interval is calculated and the initial velocity is subtracted off, giving an escape DV = 7.0124 km/sec. Next, the procedure is repeated for the Mars spiral. In this case, the initial acceleration is negative so radius decreases and velocity increases with time. For the Mars capture segment, DV = 10.1011 km/sec. Therefore, the engine must produce a combined for the two spiral trajectories of 17.1135 km/sec.
Figure
10. Proposed Interplanetary
Figure 11. Earth Escape Spiral Trajectory Summary
Figure 12. Mars Capture Spiral Trajectory Summary
To complete the estimation for the total DV for the mission, we must consider the velocity requirements from patched conics. This requires a departure from the typical formulation since we are talking about low-thrust propulsion systems. The patched conics methodology for an impulsive interplanetary mission is included in the Appendix. For the case of impulsive maneuvers, the engine is assumed to impart the necessary escape velocity (twice the circular velocity) instantaneously. However, for low thrust devices, the vehicles velocity is always the circular velocity (or very close to it). For this reason, the total velocity required to escape and achieve a given hyperbolic excess velocity is given by Hunter as:
Equation 19
Where VC
is the circular velocity of the satellite, and 
is the hyperbolic
excess velocity (with respect to Earth).
This analogous to the expressions shown in the
Appendix as the equations for DV1
and DV2. For the current analysis, it is assumed that
the satellite escapes the gravitational well of the attracting body exactly
when the sphere of influence is crossed:
Equation 20
The new expression for DV is substituted into the patched conics scheme resulting in the following:
DV1 =
3.601 km/sec
DV2 = 2.922 km/sec
DV1+DV2 = 6.523 km/sec
The additional DV is added to the values obtained for spiral out from earth and the spiral into Mars circular orbit to give:
DVTotal = 7.0124 km/sec + 10.1011 km/sec + 6.523 km/sec = 23.637 km/sec.
The patched conics method applied to the same problem yielded DVTotal = 5.6617 km/sec, a significantly lower value.
It is obvious that the impulsive maneuver case requires much less DV. These results would be amplified if a more sophisticated analysis was considered. For example, the gravity losses, as well as losses due to solar radiation pressure and atmospheric drag would certainly require more DV from the ion thruster. Another potential drawback from using the continuous thrust system would be the time to complete the mission. The time required for the escape spiral is approximately 0.2174 years while the mars capture spiral takes about 0.3313 years. If the coast phase is included the total trip time comes out to be 1.2620 years. This can is compared to 0.7092 years for the impulsive maneuver mission design. However, it is important to remember that the low-thrust mission would require much less fuel, which translates into increased payload capability or reduced launch costs.
7. Conclusion
The maturation of enabling technologies in low-thrust propulsion has lead to many exciting alternatives for mission designers. The use of more efficient, better performing spacecraft can translate into launch cost savings or increased payload availability. In some cases, the low-thrust option is the only option since it provides a gently “push” instead of a high-thrust “kick”. However, the use of low-thrust systems presents a number of new challenges to mission design. One must take into account the amount of time that might be required for a mission. One must also be cognizant of gravitational losses and perturbations that are characteristic of low-thrust devices. By understanding the fundamental physics, and accepting or mitigating the known drawbacks, low thrust propulsion will likely become a more popular means of space propulsion that it is today.
8. Bibliography
Print Sources
1.
Sutton, G. P., and Biblarz,
O., Rocket Propulsion Elements, 7th ed., John Wiley and Sons, Inc.,
2.
Kantha, L., ASEN 5053 Class
Notes,
3. Humble,
R. W., Henry, G. N., Larson, W. J., Space
Propulsion Analysis and Design,
4. Hunter,
M. W., Thrust Into Space, Holt,
Rinehart, and Winston, Inc.,
5. Cornelisse, J., W., Shroyer, H. F. R, and Wakker, K. F., Rocket Propulsion and Spaceflight Dynamics, Pitman Publishing Ltd., London, 1979.
6. Brewer,
G. R., Ion Propulsion: Technology and
Applications, Gordon and Breach Science Publishers,
7. Deutsch,
R., Orbital Dynamics of Space Vehicles,
Prentice-Hall,
8. Stuhlinger, E., Ion
Propulsion for Space Flight,
Internet Sources
11. http://alfven.princeton.edu/papers/Encyclopedia.pdf
12. http://www.daviddarling.info/encyclopedia
13. http://www.waynesthisandthat.com/ep2.htm
9. Appendix:
Patched Conics Method for Earth-Mars Mission
For a Hohmann transfer going from Earth parking orbit to a circularized Mars orbit, the Earth escape velocity is equivalent to the velocity on the transfer ellipse at periapse:
= 32.7294 km/sec.
where aH is the semi-major axis of the transfer ellipse and RE is the orbital radius of the Earth. Next, the Earth’s orbital velocity is calculated as:
= 29.7847 km/sec.
The hyperbolic excess velocity is simply the difference between these two:
= 2.944 km/sec.
Next, the DV required for this maneuver is found by:
= 3.569 km/sec.
Here, rE and hE are the Earth’s equatorial radius and the Earth-altitude of the satellite, respectively. When the satellite approaches Mars, it has a velocity (relative to Mars) of:
= 21.4805 km/sec.
The orbital velocity of Mars can be calculated as:
= 24.1294 km/sec.
The Mars hyperbolic excess velocity can then be found by subtracting:
= 2.6489 km/sec.
Finally, to establish a circular orbit about
Mars another DV must be applied. The magnitude of this DV is given by:
= 2.0926.
Therefore the total DV for the interplanetary transfer is DV1 + DV2 = 5.6617 km/sec.