NEEP533 Course Notes (Spring 1999)
Resources from Space

Lecture #9: You Can Get There from Here!

Title: Spacecraft Trajectories

February 8, 1999

 Click here to see the PowerPoint viewgraphs shown in class.

Next: Lecture 10: Evolution of the Moon as a Planet
Previous: Lecture 8: Origin of the Solar System--Asteroids and Comets
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, and most of the present discussion treats this type of trajectory. Rockets launched through atmospheres face additional complications, such as air friction and winds.

 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 (1643-1727), and were published in his Philosophia Naturalis Principia Mathematica. They are:
Sir Isaac Newton 
  1. Every body continues in its state of rest or of uniform motion in a straight line except insofar as it is compelled to change that state by an external impressed force. 
  2. The rate of change of momentum of the body is proportional to the impressed force and takes place in the direction in which the force acts. 
  3. To every action there is an equal and opposite reaction. 
  4. More compactly, dp / dt = F 

Isaac Newton on German Postage Stamp 
Calculus, invented independently by Newton and Gottfried Leibniz (1646-1716), 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 m1 m2 er / r2,

where G=6.67 x 10-11 m3 s-2 kg-1, m1 and m2 are the interacting masses (kg), r is the distance (m) between them, and er 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 (1546-1601) observations, from which Johannes Kepler [Note: Web link in German] (1571-1630) 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 then-known planets into a framework based on the five regular platonic solids. The laws are:

  1. The planets move in ellipses with the sun at one focus.
  2. Areas swept out by the radius vector from the sun to a planet in equal times are equal.
  3. The square of the period of revolution is proportional to the cube of the semimajor axis. That is,

  4. T2 = const x a3

Conic sections

In a central-force gravitational potential, 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 is the semi-major axis.   
Special cases for conic sections are shown at right. E is the (constant) energy of a body on its trajectory.   
Conic section examples  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 semi-major axis   

Hohmann's minimum-energy, interplanetary transfer trajectory

The minimum-energy transfer between circular orbits is an elliptical trajectory called the Hohmann trajectory. It is shown at right for the Earth-Mars case, where the minimum total delta-v expended is 5.6 km/s. The values of the energy per unit mass on the circular orbit and Hohmann trajectory are shown, along with the velocities at perihelion (closest to Sun) and aphelion (farthest from Sun) on the Hohmann trajectory and the circular velocity in Earth or Mars orbit. The differences between these velocities are the required delta-v values in the rocket equation. 

 Kepler's third law, T2 = const x a3, can be used to calculate the time required to traverse a Hohmann trajectory by raising to the 3/2 power the ratio of the semimajor axis of the elliptical Hohmann orbit to the circular radius of the Earth's orbit and dividing by two (for one-way travel). For example, calling the travel time for an Earth-Mars trip T and the semimajor axis of the Hohmann ellipse a, 

 a = (1 AU + 1.5 AU)/2 = 1.25 AU 
T = 0.5 (a / 1 AU)^3/2 years = 0.7 years = about 8.4 months 


Mars Global Surveyor

The Mars Global Surveyor mission arrived at Mars on September 11, 1997 and illustrates that the simple formulas given above are capable of estimating some important trajectory parameters within a few tens of percent. Selected parameters are shown in the following table


Parameter Hohmann trajectory Mars Global Surveyor 
Travel time 8.4 months 10 months 
Velocity near Mars 22 km/s 24 km/s 
Velocity difference with Mars 2.9 km/s 2.5 km/s 
The differences between the simple calculations and the actual values are due to somewhat a different trajectory than Hohmann for Mars Global Surveyor and neglected effects--such as the gravity of Mars. The Mars Global Surveyor Web page contains color viewgraphs summarizing the mission trajectory, aerobraking, and further information on the instruments and other aspects of the mission.

Lagrange points

The Lagrange (sometimes called Libration) points are positions of equilibrium for a body in a two-body 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 Earth-Moon or Sun-Earth Lagrange points are shown below.  Lagrange Points 

Rocket equation

Conservation of momentum leads to the so-called 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 nuclear-thermal rockets. First derived by Konstantin Tsiolkowsky in 1895 for straight-line 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.  Payload fraction versus exhaust velocity 

Gravity assist

Gravity assists enable or facilitate many missions. A spacecraft arrives within the sphere of influence of a body with a so-called 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 trade-off of momentum is a small change in the planet's velocity and a very large delta-v for the spacecraft. Starting from an Earth-Jupiter Hohmann trajectory and performing a Jupiter flyby at one Jovian radius, as shown at right, the hyperbolic excess velocity vh is approximately 5.6 km/s and the angular change in direction is about 160o Jupiter Flyby 

Galileo: a gravity-assist example

High-exhaust-velocity, low-thrust trajectories

The simplest high-exhaust-velocity analysis splits rocket masses into three categories:
  1. Power plant and thruster system mass, Mw.
  2. Payload mass, Ml. (Note that this includes all structure and other rocket mass that would be treated separately in a more sophisticated definition.)
  3. Propellant mass, Mp.

Useful definitions and relations

Mission power-on time tau 
Total mass  M0=Mw+Ml+Mp 
Empty mass  Me=Mw+Ml 
Specific power [kW/kg]   
Propellant flow rate  
Thrust power  

High-exhaust-velocity rocket equation

Assume constant exhaust velocity, vex, which greatly simplifies the analysis. The empty (final) mass in the Tsiolkovsky rocket equation now becomes Mw+Ml, so

 where u measures the energy expended in a manner analogous to delta-v. After some messy but straightforward algebra, we get the high-exhaust-velocity rocket equation:

 Note that a chemical rocket effectively has Mw=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 power-on 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 high-exhaust-velocity rocket becomes

which is plotted below.

Payload fraction versus mission delta-v and exhaust velocity

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 Hohmann-trajectory energy. The payload capacity of a fixed-velocity rocket vanishes at u=0.81 vch, where vex=0.5 vch. Substituting these values into the rocket equation gives


Example: 9 month Earth-Mars trajectory 
(alpha=0.1 kW/kg, alpha tau=2x109 J/kg.) NB: When the distance travelled is factored into the analysis, only u>10 values turn out to be realistic. 
Payload fraction versus mission delta-v and exhaust velocity for an Earth-Mars 9-month mission with alpha=0.1 kW/kg 

Trade-off between payload fraction and trip time for selected missions using a hypothetical fusion rocket.

Earth-Mars case Earth-Mars and Earth-Jupiter cases

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 central-force 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.

Useful references


Low-thrust trajectory analysis

History of orbital mechanics

The quest to understand the motions of heavenly bodies profoundly influenced human thinking.  These works all make interesting reading, partly for the perspective they give on the scientific process as often nearer to a random walk than a straight line.  It should be illegal to use the word "paradigm" without having read Kuhn's classic work.

Related University of Wisconsin courses

Worldwide web

Example questions

  1. What is the travel time from Earth to Saturn on a Hohmann trajectory?
  2. If a spacecraft is on an escape trajectory, what conic section does it follow?
  3. A satellite is in a circular, geosynchronous orbit around the Earth (mass = 6 x 1024 kg). Assume that the terrestrial day is exactly 24 h and neglect the satellite's mass. Calculate the satellite's
    1. Geocentric radius (km)
    2. Velocity (m/s)
  4. Explain how high-exhaust-velocity, separately powered systems, even with their low thrust levels, can facilitate development of the Solar System.

Next:  Lecture 10: Evolution of the Moon as a Planet
Previous: Lecture 8: Origin of the Solar System--Asteroids and Comets
Up: Resources from Space syllabus

Dr. John F Santarius

Fusion Technology Institute,
University of Wisconsin-Madison
1500 Engineering Dr.
Madison, WI 53706

415 Engineering Research Building
e-mail:; ph: 608/263-1694; fax: 608/263-4499
Last modified: September 22, 1997

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