ASEN 5070: Statistical Orbit Determination I Fall 2014 Professor Brandon A. Jones Lecture 2: Basics of Orbit Propagation

Announcements Monday is Labor Day! Homework 0 – You could already have it finished Homework 1 – Due September 5 Syllabus questions?

Today’s Lecture The orbit propagation problem Orbital elements Perturbed Satellite Motion Coordinate and Time Systems

Brief Review of Astrodynamics

Review of Astrodynamics What’s μ (other than a greek letter)?

Review of Astrodynamics What’s μ (other than a greek letter)? μ is the gravitational parameter of a massive body μ = GM

Review of Astrodynamics What’s μ? μ is the gravitational parameter of a massive body μ = GM What’s G? What’s M?

Review of Astrodynamics What’s μ? μ is the gravitational parameter of a massive body μ = GM What’s G? Universal Gravitational Constant What’s M? The mass of the body

Review of Astrodynamics What’s μ? μ = GM G = 6.67384 ± 0.00080 × 10-20 km3/kg/s2 MEarth ~ 5.97219 × 1024 kg or 5.9736 × 1024 kg or 5.9726 × 1024 kg Use a value and cite where you found it! μEarth = 398,600.4415 ± 0.0008 km3/s2 (Tapley, Schutz, and Born, 2004) How do we measure the value of μEarth?

Review of Astrodynamics Problem of Two Bodies µ = G(M1 + M2) XYZ is nonrotating, with zero acceleration; an inertial reference frame

Integrals of Motion Center of mass of two bodies moves in straight line with constant velocity Angular momentum per unit mass (h) is constant, h = r x V = constant, where V is velocity of M2 with respect to M1, V= dr/dt Consequence: motion is planar Energy per unit mass (scalar) is constant

Orbit Plane in Space

Equations of Motion in the Orbit Plane The uθ component yields: which is simply h = constant

Solution of ur Equations of Motion The solution of the ur equation is (as function of θ instead of t): where e and ω are constants of integration.

The Conic Equation Constants of integration: e and ω e = ( 1 + 2 ξ h2/µ2 )1/2 ω corresponds to θ where r is minima Let f = θ – ω, then r = p/(1 + e cos f) which is “conic equation” from analytical geometry (e is conic “eccentricity”, p is “semi-latus rectum” or “semi-parameter”, and f is the “true anomaly”) Conclude that motion of M2 with respect to M1 is a “conic section” Circle (e=0), ellipse (0<e<1), parabola (e=1), hyperbola (e>1)

Types of Orbital Motion

Orbit Elements

Six Orbit Elements The six orbit elements (or Kepler elements) are constant in the problem of two bodies (two gravitationally attracting spheres, or point masses) Define shape of the orbit a: semimajor axis e: eccentricity Define the orientation of the orbit in space i: inclination Ω: angle defining location of ascending node (AN) : angle from AN to perifocus; argument of perifocus Reference time: tp: time of perifocus (or mean anomaly at specified time)

Orbit Orientation i - Inclination Ω - RAAN ω – Arg. of Perigee

Astrodynamics Background Keplerian Orbital Elements Consider an ellipse Periapse/perifocus/periapsis Perigee, perihelion r = radius rp = radius of periapse ra = radius of apoapse a = semi-major axis e = eccentricity = (ra-rp)/(ra+rp) rp = a(1-e) ra = a(1+e) ω = argument of periapse f/υ = true anomaly

Satellite State Representations Cartesian Coordinates x, y, z, vx, vy, vz in some coordinate frame Keplerian Orbital Elements a, e, i, Ω, ω, ν (or similar set) Topocentric Elements Right ascension, declination, radius, and time rates of each When are each of these useful?

Keplerian Orbital Elements Shape: a = semi-major axis e = eccentricity Orientation: i = inclination Ω = right ascension of ascending node ω = argument of periapse Position: ν = true anomaly What if i=0? If orbit is equatorial, i = 0 and Ω is undefined. In that case we can use the “True Longitude of Periapsis”

Keplerian Orbital Elements Shape: a = semi-major axis e = eccentricity Orientation: i = inclination Ω = right ascension of ascending node ω = argument of periapse Position: ν = true anomaly What if e=0? If orbit is circular, e = 0 and ω is undefined. In that case we can use the “Argument of Latitude”

Keplerian Orbital Elements Shape: a = semi-major axis e = eccentricity Orientation: i = inclination Ω = right ascension of ascending node ω = argument of periapse Position: ν = true anomaly What if i=0 and e=0? If orbit is circular and equatorial, neither ω nor Ω are defined In that case we can use the “True Longitude”

Keplerian Orbital Elements Shape: a = semi-major axis e = eccentricity Orientation: i = inclination Ω = right ascension of ascending node ω = argument of periapse Position: ν = true anomaly M = mean anomaly Special Cases: If orbit is circular, e = 0 and ω is undefined. In that case we can use the “Argument of Latitude” ( u = ω+ν ) If orbit is equatorial, i = 0 and Ω is undefined. In that case we can use the “True Longitude of Periapsis” If orbit is circular and equatorial, neither ω nor Ω are defined In that case we can use the “True Longitude”

Cartesian to Keplerian Conversion Handout offers one conversion. We’ve coded up Vallado’s conversions ASEN 5050 implements these Check out the code RVtoKepler.m Check errors and/or special cases when i or e are very small!

Perturbed Satellite Motion

Perturbed Motion The 2-body problem provides us with a foundation of orbital motion In reality, other forces exist which arise from gravitational and nongravitational sources In the general equation of satellite motion, a is the perturbing force (causes the actual motion to deviate from exact 2- body)

Perturbed Motion: Planetary Mass Distribution Sphere of constant mass density is not an accurate representation for planets Define gravitational potential, U, such that the gravitational force is

Gravitational Potential The commonly used expression for the gravitational potential is given in terms of mass distribution coefficients Jn, Cnm, Snm n is degree, m is order Coordinates of evaluation point are given in spherical coordinates: r, geocentric latitude φ, longitude 

Gravity Coefficients The gravity coefficients (Jn, Cnm, Snm) are also known as Stokes Coefficients and Spherical Harmonic Coefficients Jn, Cn0: Gravitational potential represented in zones of latitude; referred to as zonal coefficients Cnm, Snm: If n=m, referred to as sectoral coefficients If n≠m, referred to as tesseral coefficients These parameters may be used for orbit design (take ASEN 5050 for more details!)

Shape of Earth: J2, J3 U.S. Vanguard satellite launched in 1958, used to determine J2 and J3 J2 represents most of the oblateness; J3 represents a pear shape J2 = 1.08264 x 10-3 J3 = - 2.5324 x 10-6

Atmospheric Drag Atmospheric drag is the dominant nongravitational force at low altitudes if the celestial body has an atmosphere Depending on nature of the satellite, lift force may exist Drag removes energy from the orbit and results in da/dt < 0, de/dt < 0 Orbital lifetime of satellite strongly influenced by drag From D. King-Hele, 1964, Theory of Satellite Orbits in an Atmosphere

Other Forces What are the other forces that can perturb a satellite’s motion? Solar Radiation Pressure (SRP) Thrusters N-body gravitation (Sun, Moon, etc.) Electromagnetic Solid and liquid body tides Relativistic Effects Reflected radiation (e.g., ERP) Coordinate system errors Spacecraft radiation

Coordinate and Time Frames

Coordinate Frames Define xyz reference frame (Earth centered, Earth fixed; ECEF or ECF), fixed in the solid (and rigid) Earth and rotates with it Longitude λ measured from Greenwich Meridian 0≤ λ < 360° E; or measure λ East (+) or West (-) Latitude (geocentric latitude) measured from equator (φ is North (+) or South (-)) At the poles, φ = + 90° N or φ = -90° S

Coordinate Systems and Time The transformation between ECI and ECF is required in the equations of motion Depends on the current time! Thanks to Einstein, we know that time is not simple…

Time Systems Countless systems exist to measure the passage of time. To varying degrees, each of the following types is important to the mission analyst: Atomic Time Unit of duration is defined based on an atomic clock. Universal Time Unit of duration is designed to represent a mean solar day as uniformly as possible. Sidereal Time Unit of duration is defined based on Earth’s rotation relative to distant stars. Dynamical Time Unit of duration is defined based on the orbital motion of the Solar System.

Time Systems: Time Scales

What are some issues with each of these? Time Systems Question: How do you quantify the passage of time? Year Month Day Second Pendulums Atoms What are some issues with each of these? Gravity Earthquakes Snooze alarms

Time Systems: The Year Definitions of a Year Julian Year: 365.25 days, where an SI “day” = 86400 SI “seconds”. Sidereal Year: 365.256 363 004 mean solar days Duration of time required for Earth to traverse one revolution about the sun, measured via distant star. Tropical Year: 365.242 19 days Duration of time for Sun’s ecliptic longitude to advance 360 deg. Shorter on account of Earth’s axial precession. Anomalistic Year: 365.259 636 days Perihelion to perihelion. Draconic Year: 365.620 075 883 days One ascending lunar node to the next (two lunar eclipse seasons) Full Moon Cycle, Lunar Year, Vague Year, Heliacal Year, Sothic Year, Gaussian Year, Besselian Year

Coordinate Systems and Time Equinox location is function of time Sun and Moon interact with Earth J2 to produce Precession of equinox (ψ) Nutation (ε) Newtonian time (independent variable of equations of motion) is represented by atomic time scales (dependent on Cesium Clock)

Precession / Nutation Precession Nutation (main term):

Earth Rotation and Time Sidereal rate of rotation: ~2π/86164 rad/day Variations exist in magnitude of ωE, from upper atmospheric winds, tides, etc. UT1 is used to represent such variations UTC is kept within 0.9 sec of UT1 (leap second) Polar motion and UT1 observed quantities Different time scales: GPS-Time, TAI, UTC, TDT Time is independent variable in satellite equations of motion; relates observations to equations of motion (TDT is usually taken to represent independent variable in equations of motion)

Coordinate Frames Inertial: fixed orientation in space Rotating Inertial coordinate frames are typically tied to hundreds of observations of quasars and other very distant near-fixed objects in the sky. Rotating Constant angular velocity: mean spin motion of a planet Osculating angular velocity: accurate spin motion of a planet

Coordinate Systems Coordinate Systems = Frame + Origin Inertial coordinate systems require that the system be non-accelerating. Inertial frame + non-accelerating origin “Inertial” coordinate systems are usually just non- rotating coordinate systems.

Coordinate System Transformations Converting from ECI to ECF P is the precession matrix (~50 arcsec/yr) N is the nutation matrix (main term is 9 arcsec with 18.6 yr period) S’ is sidereal rotation (depends on changes in angular velocity magnitude; UT1) W is polar motion Earth Orientation Parameters Caution: small effects may be important in particular application