Spacecraft Trajectories You Can Get There from Here! John F Santarius Lecture 9 Resources from Space NEEP 533/ Geology 533 / Astronomy 533 / EMA 601 University of Wisconsin

JFS 1999 Newton’s Laws of Motion The fundamental laws of mechanical motion were first formulated by Sir Isaac Newton ( ), and were published in his Philosophia Naturalis Principia Mathematica. Calculus, invented independently by Newton and Gottfried Leibniz ( ), plus Newton's laws of motion are the mathematical tools needed to understand rocket motion.

JFS 1999 Newton’s Laws of Motion 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. 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. To every action there is an equal and opposite reaction. To every action there is an equal and opposite reaction. 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. 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.

JFS 1999 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. 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. G=6.67× m 3 s -2 kg -1 is the gravitational constant.

JFS 1999 Kepler’s Laws of Planetary Motion The planets move in ellipses with the sun at one focus. 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. 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. T 2  a 3 The square of the period of revolution is proportional to the cube of the semimajor axis. T 2  a 3 Conic sections

JFS 1999 Essential Orbital Dynamics Circular velocity Circular velocity Escape velocity Escape velocity Energy of a vehicle following a conic section (a  semimajor axis) Energy of a vehicle following a conic section (a  semimajor axis) G=6.67× m 3 s -2 kg -1

Hohmann’s Minimum-Energy Interplanetary Transfer JFS 1999 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.

JFS 1999 Calculating Hohmann Transfers Kepler's third law, T  a 3/2, 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). Kepler's third law, T  a 3/2, 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, call the travel time for an Earth-Mars trip T and the semimajor axis of the Hohmann ellipse a. For example, call the travel time for an Earth-Mars trip T and the semimajor axis of the Hohmann ellipse a. a = (1 AU AU)/2 = 1.25 AU T = 0.5 (a / 1 AU) 3/2 years = ~0.7 years = ~8.4 months = ~0.7 years = ~8.4 months

JFS 1999 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 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 a rocket and its exhaust leads to Conservation of momentum for a rocket and its exhaust leads to

High Exhaust Velocity Gives Large Payloads or Fast Travel 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. 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. JFS 1999 Chemical rocket

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. 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. 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. JFS 1999 Starting from an Earth-Jupiter Hohmann trajectory and performing a Jupiter flyby at one Jovian radius, as shown above, the hyperbolic excess velocity v h is approximately 5.6 km/s and the angular change in direction is about 160 o. Starting from an Earth-Jupiter Hohmann trajectory and performing a Jupiter flyby at one Jovian radius, as shown above, the hyperbolic excess velocity v h is approximately 5.6 km/s and the angular change in direction is about 160 o. VhVh VhVh Motion in planet’s frame of reference

Efficient Solar-System Travel Requires High-Exhaust-Velocity, Low-Thrust Propulsion JFS 1999 Electric power can be used to drive high-exhaust-velocity plasma or ion thrusters, or fusion plasmas can be directly exhausted. Electric power can be used to drive high-exhaust-velocity plasma or ion thrusters, or fusion plasmas can be directly exhausted. – Allows fast trip times or large payload fractions for long- range missions. Uses relatively small amounts of propellant, reducing total mass. Uses relatively small amounts of propellant, reducing total mass. Fusion rocket (   specific power )

How Do Separately Powered Systems Differ from Chemical Rockets? Propellant not the power source. Propellant not the power source. High exhaust velocity (  10 5 m/s). High exhaust velocity (  10 5 m/s). Low thrust (  m/s}  Earth gravity). Low thrust (  m/s}  Earth gravity). Thrusters typically operate for a large fraction of the mission duration. Thrusters typically operate for a large fraction of the mission duration. High-exhaust-velocity trajectories are fundamentally different from chemical-rocket trajectories. High-exhaust-velocity trajectories are fundamentally different from chemical-rocket trajectories. JFS 1999

Chemical rocket trajectory (minimum energy) JFS 1999 Taking Full Advantage of High Exhaust Velocity Requires Optimizing Trajectories Fusion rocket trajectory (variable acceleration) Note: Trajectories are schematic, not calculated.

JFS 1999   mission power-on time M w  power plant mass M l  payload mass M p  propellant mass M 0  total mass = M w + M l + M p M  propellant flow rate = M p /  F  thrust = M v ex P w  thrust power = ½ M v ex 2  [kW/kg]  specific power = P w / M w v ch  characteristic velocity = (2  ) ½ Useful Propulsion Definitions

Rocket Equation for Separately Powered Systems Explicitly including the power-plant mass through the characteristic velocity modifies the rocket equation: Explicitly including the power-plant mass through the characteristic velocity modifies the rocket equation: JFS 1999 U/V ch = U  mission energy requirement v ch  characteristic velocity = (2  ) ½

Fusion Propulsion Would Enable Efficient Solar-System Travel Comparison of trip times and payload fractions for chemical and fusion rockets JFS 1999