To Orbit…and Beyond (Intro to Orbital Mechanics) Scott Schoneman 6 November 03

Agenda Some brief history - a clockwork universe? The Basics What is really going on in orbit: Is it really zero-G? Motion around a single body Orbital elements Ground tracks Perturbations J2 and gravity models Drag “Third bodies”

Why is this important? The physics of orbit mechanics makes launching spacecraft difficult and complex: It’s difficult to get there! (with current technology) Orbit mechanics touches the design of essentially all spacecraft systems: Power (shadows? Distance from Sun?) Thermal ( “ ) Attitude Control (disturbance environment) Propulsion systems (launch, orbit maneuvers - indirectly affects structures) Radiation environment (electronic design) All of the above can affect software Practical problems: Where will the satellite be & when can I talk to it? When will it see/not see it’s mission target? How do I get it to see its mission target or ground stations (attitude, propulsion maneuvers)?

Earth-Centered & Sun-Centered The Universe must be perfect! All motion must be based on spheres and circles (Aristotle) Ptolemy (c. 150 AD) worked out a system of “epicycles”, “eccentrics” and “equants” based on circles Fit observations for many centuries Copernicus (1543) published his sun-centered universe “Mathematical description only” Described retrograde motion well, but still used circles and epicycles to fit observational details

Observations & Ellipses Tycho Brahe (1546 - 1601): Foremost observer of his day Most accurate and detailed observations performed up to that time Johannes Kepler (1571 - 1630) Used Tycho’s observations in attempt to fit his sun- centered system of spheres separated by regular polyhedra Could not fit the observations to systems of circles and spheres Resorted to other shapes, eventually settling on the ellipse

Kepler’s Laws Kepler made the leap to generalize 3 laws for planetary motion: 1) Planets move in an ellipse, with the sun at a focus 2) The motion of a planet “sweeps out” area at a constant rate (thus the speed is not constant) 3) Period2 is proportional to (average distance)3 “The harmony of the worlds” “My aim in this is to show that the celestial machine is ...... a clockwork” Note that these were purely EMPIRICAL laws - there’s no “physics” behind them.

Halley and Newton Edmond Halley (1656 - 1742) sought to predict the motion of comets, but couldn’t fit modern observations with older comet theories Suspected inverse-square law for force, but sought Newton’s help Helped Newton (technically & financially) publish “Principia” Isaac Newton (1643 - 1727) proved inverse-square law yields elliptical motion Published “Principia” in 1687, bringing together gravity on Earth and in space (between the Sun, planets, and comets) into a single mathematical understanding Also developed differential and integral calculus, derived Kepler’s three laws, founded discipline of fluid mechanics, etc.

(Newton is close enough for most engineering purposes) Albert Einstein Showed that Newton was all wrong (or at least not quite right), but we won’t talk about that. (Newton is close enough for most engineering purposes)

The Basics and Two-Body Motion

Illustration from “Principia” Newton’s Mountain “The knack to flying lies in knowing how to throw yourself at the ground and miss.” (paraphrased) - Douglas Adams Orbit is not “Zero-G” - There IS gravity in space - Lots of it What’s really going on: You are in FREE-FALL You are always being pulled towards the Earth (or other central body) If you have enough “sideways” speed, you will miss the Earth as it curves away from beneath you. Illustration from “Principia”

Gravitational Force Newton’s 2nd Law: Newton’s Law Of Universal Gravitation (assuming point masses or spheres): Putting these together:

Gravitational Force - Simplified (Two Bodies, No Vectors) Newton’s 2nd Law: Newton’s law of universal gravitation (assuming point masses or spheres): Putting these together:

The Gravitational Constant “G” is one of the less-precisely known numbers in physics It’s very small You need to first know the mass and measure the force in order to solve for it You will almost always see the combination of “GM” together Usually called m Can be easily measured for astronomical bodies (watching orbital periods) µ (km3/sec2)

Conic Sections Newton actually proved that the inverse-square law meant motion on a “conic section” http://ccins.camosun.bc.ca/~jbritton/jbconics.htm

Conic Sections - Characteristics

Ellipse Geometry Most Common Orbits are Defined by the Ellipse: a = “semi-major axis” e = eccentricity = e / c = ( ra - rp )/ ( ra + rp ) Periapsis = rp , closest point to central body (perigee, perihelion) Apoapsis = ra , farthest point from central body (apogee, aphelion)

The Classical Orbital Elements (aka Keplerian Elements) Also need a timestamp (time datum)

State Vectors A state vector is a complete description of the spacecraft’s position and velocity, with a timestamp Examples Position (x, y, z) and Velocity (x, y, z) Classical Elements are also a kind of state vector Other kinds of elements NORAD Two-Line-Elements (TLE’s) (Classical Elements with a particular way of interpreting perturbations) Latitude, Longitude, Altitude and Velocity Mathematically conversion possible between any of these

Orbit Types LEO (Low Earth Orbit): Any orbit with an altitude less than about 1000 km Could be any inclination: polar, equatorial, etc Very close to circular (eccentricity = 0), otherwise they’d hit the Earth Examples: ORBCOMM, Earth-observing satellites, Space Shuttle, Space Station MEO (Medium Earth Orbit): Between LEO and GEO Examples: GPS satellites, Molniya (Russian) communications satellites GEO (Geosynchronous): Orbit with period equal to Earth’s rotation period Altitude 35786 km, Usually targeted for eccentricity, inclination = 0 Examples: Most communications satellite missions - TDRSS, Weather Satellites HEO (High Earth Orbit): Higher than GEO Example: Chandra X-ray Observatory, Apollo to the Moon Interplanetary Used to transfer between planets: the Sun is the central body Typically large eccentricities to do the transfer

Ground Tracks Ground Tracks project the spacecraft position onto the Earth’s (or other body’s) surface (altitude information is lost) Most useful for LEO satellites, though it applies to other types of missions Gives a quick picture view of where the spacecraft is located, and what geographical coverage it provides

Example Ground Tracks LEO sun-synchronous ground track

Example Ground Tracks Some general orbit information can be gleaned from ground tracks Inclination is the highest (or lowest) latitude reached Orbit period can be estimated from the spacing (in longitude) between orbits By showing the “visible swath”, you can estimate altitude, and directly see what the spacecraft can see on the ground Example: swath

Geosynchronous and Molniya Orbit Ground Tracks GEO ground track is a point (or may trace out a very small, closed path) Molniya ground track “hovers” over Northern latitudes for most of the time, at one of two longitudes

Perturbations: Reality is More Complicated Than Two Body Motion The International Sun/Earth Explorer 3 (ISEE-3) satellite, launched in 1978, was designed to monitor the interaction between the solar wind, a stream of charged gas emanating from the Sun, and the Earth's magnetic field. Because of this mission, the spacecraft was not equipped with a camera system, but with numerous instruments to measure the characteristics and composition of the solar wind. When the decision was made to change the trajectory for a comet intercept, the spacecraft was renamed the International Cometary Explorer.

Orbit Perturbations J2 and other non-spherical gravity effects Earth is an “Oblate Spheriod” Not a Sphere Atmospheric Drag “Third” bodies Other effects Solar Radiation pressure Relativity

J2 Effects - Plots J2-orbit rotation rates are a function of: semi-major axis inclination eccentricity (Regresses West) (Regresses East)

Applications of J2 Effects Sun-synchronous Orbits The regression of nodes matches the Sun’s longitude motion (360 deg/365 days = 0.9863 deg/day) Keep passing over locations at same time of day, same lighting conditions Useful for Earth observation “Frozen Orbits” At the right inclination, the Rotation of Apsides is zero Used for Molniya high-eccentricity communications satellites

Atmospheric Drag Along with J2, dominant perturbation for LEO satellites Can usually be completely neglected for anything higher than LEO Primary effects: Lowering semi-major axis Decreasing eccentricity, if orbit is elliptical In other words, apogee is decreased much more than perigee, though both are affected to some extent For circular orbits, it’s an evenly-distributed spiral

Atmospheric Drag Effects are calculated using the same equation used for aircraft: To find acceleration, divide by m m / CDA : “Ballistic Coefficient” For circular orbits, rate of decay can be expressed simply as: As with aircraft, determining CD to high accuracy can be tricky Unlike aircraft, determining r is even trickier Density is calculated using various atmospheric models US Standard atmosphere 1976- approximate, no variations Harris-Priester - dates to 1960’s, incorporates variations from Solar Activity, atmospheric swelling on the day-side Jacchia-Roberts - various models in late 1960’s, 1970’s. Updated models from Harris Priester, incorporates effects due to Earth magnetic activity MSIS-86, MSISE-90 - Incorporates seasonal & geographic variations, constituent chemistry model Many others exist Even the most complex models have a hard time predicting density to < 10% error Solar activity and geomagnetic activity very uncertain and have drastic effects For typical LEO altitudes, Solar activity can change density nearly 2 orders of magnitude A lot of upper atmospheric dynamics not well understood

Applications of Drag Aerobraking / aerocapture Instead of using a rocket, dip into the atmosphere Lower existing orbit: aerobraking Brake into orbit: aerocapture Aerobraking to control orbit first demonstrated with Magellan mission to Venus Used extensively by Mars Global Surveyor Of course, all landing missions to bodies with an atmosphere use drag to slow down from orbital speed (Shuttle, Apollo return to Earth, Mars/Venus landers)

Third-Body Effects Gravity from additional objects complicates matters greatly No explicit solution exists like the ellipse does for the 2-body problem Third body effects for Earth-orbiters are primarily due to the Sun and Moon Affects GEOs more than LEOs Points where the gravity and orbital motion “cancel” each other are called the Lagrange points Sun-Earth L1 has been the destination for several Sun-science missions (ISEE-3 (1980s), SOHO, Genesis, others planned)

Lagrange Points Application Genesis Mission: NASA/JPL Mission to collect solar wind samples from outside Earth’s magnetosphere Launched: 8 August 2001 Returning: Sept 2004

Third-Body Effects: Slingshot A way of taking orbital energy from one body ( a planet ) and giving it to another ( a spacecraft ) Used extensively for outer planet missions (Pioneer 10/11, Voyager, Galileo, Cassini) Analogous to Hitting a Baseball: Same Speed, Different Direction planet’s orbit velocity spacecraft incoming to planet hyperbolic flyby (relative to planet) spacecraft departing planet departing sun-centric velocity incoming sun-centric velocity Analogous to hitting a baseball The incoming and departing speeds, relative to the bat, are the same Relative to the ground, the speed and Kinetic Energy has greatly increased Example: Velocity relative to planet was turned Same speed, different direction (relative to planet) Exiting velocity now adds directly to the velocity relative to the Sun, instead of being a component 90 degrees from it Total velocity relative to Sun was increased