NEEP602 Course Notes (Spring 1996)
Resources from Space
Lecture #28: You Can Get There from Here!
Title: Spacecraft Trajectories
April 1, 1996
Next: Lecture 29: Chemical Rockets
Previous: Lecture 27: Helium3 fusion
Up: Resources from Space syllabus
Spacecraft trajectory overview
Spacecraft today essentially all travel by being given an impulse that places them on a trajectory in which they coast from one point to another, perhaps with other impulses or gravity assists along the way. The gravity fields of the Sun and planets govern such trajectories. Rockets launched through atmospheres face additional complications, such as air friction and winds. Most of the present discussion treats this type of trajectory.Advanced propulsion systems and efficient travel throughout the Solar System will be required for human exploration, settlement, and accessing space resources. Rather than coasting, advanced systems will thrust for most of a trip, with higher exhaust velocities but lower thrust levels. These more complicated trajectories require advanced techniques for finding optimum solutions, but a reasonably good approximate method will be given here.
Newton's laws of motion
The fundamental laws of mechanical motion were first formulated by Sir Isaac Newton (16431727), and were published in his Philosophia Naturalis Principia Mathematica. They are:Calculus, invented independently by Newton and Gottfried Leibniz (16461716), plus Newton's laws of motion are the mathematical tools needed to understand rocket motion.
Newton's law of gravitation
To calculate the trajectories for planets, satellites, and space probes, the additional relation required is Newton's law of gravitation:Every particle of matter attracts every other particle of matter with a force directly proportional to the product of the masses and inversely proportional to the square of the distance between them. Symbolically, the force is
$$F = G m_{1} m_{2} e_{r} / r^{2},
where $G=6.67\; x\; 1011m3s2kg1,\; m$_{1} and $m$_{2} are the interacting masses, $r$ is the distance between them, and $e$_{r} is a unit vector pointing between them.
Kepler's laws of planetary motion
The discovery of the laws of planetary motion owed a great deal to Tycho Brahe's (15461601) observations, from which Johannes Kepler [Note: Web link in German] (15711630) concluded that the planets move in elliptical orbits around the Sun. First, however, Kepler spent many years trying to fit the orbits of the five thenknown planets into a framework based on the five regular platonic solids. The laws are: The planets move in ellipses with the sun at one focus.
 Areas swept out by the radius vector from the sun to a planet in equal times are equal.
 The square of the period of revolution is proportional to the cube of the semimajor axis. That is, $$T^{2} = const $$x a^{3}
Conic sections
In a centralforce gravitational potential, small bodies will follow conic sections. The equation for a conic section with the origin at one focus appears at top right, where e is eccentricity and $a$_{0} is the semimajor axis.  
Special cases for conic sections are shown at right. E is the (constant) energy of a body on its trajectory. 
Conic section examples 
Some important equations of orbital dynamics
Circular velocity  
Escape velocity  
Energy of a vehicle following a conic section, where a is the semimajor axis 
Clicking on the conicsection trajectory formulary at right will show the main equations needed to define the position of a body at a given time. See text by Roy for definitions. 
Lagrange points
The Lagrange (sometimes called Libration) points are positions of equilibrium for a body in a twobody system. The points L1, L2, and L3 lie on a straight line throught the other two bodies and are points of unstable equilibrium. That is, a small perturbation will cause the third body to drift away. The L4 and L5 points are at the third vertex of an equilateral triangle formed with the other two bodies; they are points of stable equilibrium. The approximate positions for the EarthMoon or SunEarth Lagrange points are shown below. 
Rocket equation
Conservation of momentum leads to the socalled rocket equation, which trades off exhaust velocity with payload fraction. Based on the assumption of short impulses with coast phases between them, it applies to chemical and nuclearthermal rockets. First derived by Konstantin Tsiolkowsky in 1895 for straightline rocket motion with constant exhaust velocity, it is also valid for elliptical trajectories with only initial and final impulses. Conservation of momentum for the rocket and its exhaust leads to
The rocket equation shows why high exhaust velocity has historically been a driving force for rocket design: payload fractions depend strongly upon the exhaust velocity, as shown at right. 
Hohmann minimumenergy trajectory
Gravity assist
Gravity assists enable or facilitate many missions. A spacecraft arrives within the sphere of influence of a body with a socalled hyperbolic excess velocity equal to the vector sum of its incoming velocity and the planet's velocity. In the planet's frame of reference, the direction of the spacecraft's velocity changes, but not its magnitude. In the spacecraft's frame of reference, the net result of this tradeoff of momentum is a small change in the planet's velocity and a very large deltav for the spacecraft. Starting from an EarthJupiter Hohmann trajectory and performing a Jupiter flyby at one Jovian radius, as shown at right, the hyperbolic excess velocity$v$_{h} is approximately 5.6 km/s and the angular change in direction is about 160^{o}. 
Galileo: a gravityassist example
Highexhaustvelocity, lowthrust trajectories
The simplest highexhaustvelocity analysis splits rocket masses into three categories: Power plant and thruster system mass, $M$_{w}.
 Payload mass, $M$_{l}. (Note that this includes all structure and other rocket mass that would be treated separately in a more sophisticated definition.)
 Propellant mass, $M$_{p}.
Useful definitions and relations
Mission poweron time  tau 
Total mass  $M$_{0}=M_{w}+M_{l}+M_{p} 
Empty mass  $M$_{e}=M_{w}+M_{l} 
Specific power [kW/kg]  
Propellant flow rate  
Thrust power  
Thrust 
Highexhaustvelocity rocket equation
Assume constant exhaust velocity, $v$_{ex}, which greatly simplifies the analysis. The empty (final) mass in the Tsiolkovsky rocket equation now becomes $M$_{w}+M_{l}, sowhere u measures the energy expended in a manner analogous to deltav. After some messy but straightforward algebra, we get the highexhaustvelocity rocket equation:
Note that a chemical rocket effectively has $M$_{w}=0 ==> alpha=infinity, and the Tsiolkovsky equation ensues. The quantity alpha*tau is the energy produced by the power and thrust system during a mission with poweron time tau divided by the mass of the propulsion system. It is called the specific energy of the power and thrust system. Relating the specific energy to a velocity through $E=mv2/2$ gives the definition of a very important quantity, the characteristic velocity: . The payload fraction for a highexhaustvelocity rocket becomes
,
which is plotted below.
Analyzing a trajectory using the characteristic velocity method requires an initial guess for tau plus some iterations. The minimum energy expended will always be more than the Hohmanntrajectory energy. The payload capacity of a fixedvelocity rocket vanishes at $u=0.81\; v$_{ch}, where $v$_{ex}=0.5 v_{ch}. Substituting these values into the rocket equation gives
Example: 9 month EarthMars trajectory (alpha=0.1 kW/kg, alpha tau=2x$109$ J/kg.) NB: When the distance travelled is factored into the analysis, only u>10 values turn out to be realistic. 
Tradeoff between payload fraction and trip time for selected missions.
Variable exhaust velocity and gravity
Variable exhaust velocity and gravity considerably complicate the problem. When the exhaust velocity is varied during the flight, variational principles are needed to calculate the optimum $v(t)$. The key result is that it is necessary to minimize . Even the simplest problem with gravity, the centralforce problem, is difficult and requires advanced techniques, such as Lagrangian dynamics and Lagrange multipliers. In general, trajectories must be found numerically, and finding the optimum in complex situations is an art.Examples of highexhaustvelocity trajectories
[ Still looking for examples of these on the Web. ]
Useful references
Astrodynamics
 Archie E. Roy, The Foundations of Astrodynamics (Adam Hilger, Bristol, 1988).
 Richard H. Battin, An Introduction to the Mathematics and Methods of Astrodynamics (AIAA, New York, 1987).
Lowthrust trajectory analysis
 Ernst Stuhlinger, Ion Propulsion for Space Flight (McGrawHill, New York, 1964).
 Krafft A. Ehricke, Space Flight: II. Dynamics (Van Nostrand, Princeton, 1962).
University of Wisconsin Courses
 EMA 550/Astron 550, Astrodynamics, is frequently taught during the Fall semester, and covers many of the topics discussed on this page plus celestial coordinate systems.
 EMA 601, Plasma Propulsion, was taught in Spring, 1993 and may be taught again in Spring, 1997. Half of the course covers lowthrust trajectories and the other half treats plasma and electric thrusters (subject of lecture 30 in this course).
Worldwide Web
Spacerelated, government, and other potentially useful Web sites
Questions
 What is the travel time from Earth to Saturn on a Hohmann trajectory?
 If a spacecraft is on an escape trajectory, what conic section does it follow?
 A satellite is in a circular, geosynchronous orbit around the Earth (mass = $6\; x\; 1024kg$). Assume that the terrestrial day is exactly 24 h and neglect the satellite's mass. Calculate the satellite's
 Geocentric radius (km)
 Velocity (m/s)
 Why do some rockets have more than one stage? Explain using Tsiolkovsky's rocket equation.
 Explain how highexhaustvelocity, separately powered systems, even with their low thrust levels, can facilitate development of the Solar System.
Next: Lecture 29: Chemical rockets
Previous: Lecture 27: Helium3 fusion
Up: Resources from Space syllabus
Dr. John F Santarius
Fusion Technology Institute,
University of WisconsinMadison
1500 Engineering Dr.
Madison, WI 53706
USA
415 Engineering Research Building
email: santarius@engr.wisc.edu; ph: 608/2631694; fax: 608/2634499
Last modified: April 12, 1996
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