ASEN 5070 Statistical Orbit determination I Fall 2012 Professor George H. Born

ASEN 5070 Statistical Orbit determination I Fall 2012 Professor George H. Born Professor Jeffrey S. Parker Lecture 2: Background

Announcements • Monday is Labor Day! • D2L issue • CAETE issue • Quiz - Now 24 hours per quiz (1pm – 1pm) • Homework 1 • Office Hours

Quiz Results

Office Hours Changes • Had to shift Eduardo’s office hours because there was a distinct lack of awesomeness on Tuesdays.

Today’s Lecture • Astrodynamics background • Review orbital elements a bit more • Talk about perturbations in dynamical model • J2, Drag, SRP, etc. • Talk about partials • Coordinate Frames and Time Systems • Coding hints and tricks (mostly next Tuesday) • LaTex: intro • MATLAB: ways to speed up your code • Python: intro • Notes about laptops and phones, etc.

Astrodynamics Background • Representing a satellite’s state • Cartesian Coordinates • x, y, z, vx, vy, vz in some coordinate frame • Spherical Elements • Lat, Lon, Alt, V, FPA, FPAz in some coordinate frame (or similar set) • Keplerian Orbital Elements • a, e, i, Ω, ω, ν in some coordinate frame (or similar set) • When are each of these useful?

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

Astrodynamics Background • Keplerian Orbital Elements • Orientation of ellipse requires a reference frame • Typically “Earth Mean Equator and Equinox of J2000.0” (EME2000, or just J2000). • Or Earth Mean Ecliptic of J2000. • The obliquity of the Earth’s spin axis is the angle between the equatorial and ecliptic planes. • ~23.5 deg at present.

Astrodynamics Background • Keplerian Orbital Elements • Orientation of ellipse requires a reference frame • Typically “Earth Mean Equator and Equinox of J2000.0” (EME2000, or just J2000). • Or Earth Mean Ecliptic of J2000. • i = inclination • Ω = Right ascension of ascending node (= longitude of ascending node in an inertial J2000 coordinate frame)

Astrodynamics Background • Shape and Size • a, e, rp, ra, Period • Orientation • i, Ω, ω • Position • f/υ, E, M, (t-tp) • Advantages • Visualization. • In a 2-body world, they don’t change with time (except the position). • In the real world, do they change?

Astrodynamics Background • Shape and Size • a, e, rp, ra, Period • Orientation • i, Ω, ω • Position • f/υ, E, M, (t-tp) • Advantages • Visualization. • In a 2-body world, they don’t change with time (except the position). • In the real world, do they change? Earth’s orbit Moon’s orbit Yes, but usually not much, and we can use perturbation theory to model the variations.

Coordinate Frames • Ascending node is the point where the Sun crosses the equator moving from the southern hemisphere to the northern hemisphere: vernal equinox (~ March 21) • The descending node is autumnal equinox (~Sept 21)

Coordinate Frames • Choose X-axis (coinciding with vernal equinox) as inertial direction; Z-axis coincident with Earth angular velocity vector (ωe), period of rotation = 86164 sec, “sidereal” period • GMST=αG = ωe(t – t0) + αG0

Coordinate Frames • (XYZ) represents a nonrotating coordinate system with X directed to the vernal equinox, and origin coinciding with Earth center (geometric center of the spherical Earth, or more precisely, the Earth center of mass) • In reality, the location of the equinoxes change with time (use the equinox of a particular date as reference, e.g., January 1, 2000, 12:00 or more specifically, mean equator and vernal equinox of J2000) • (xyz) is an Earth-fixed frame (ECF) and rotates with it, with x coincident with the intersection of the Greenwich meridian and the equator

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

Effects of Small Variations • I’d like us to think about the effects of small variations in coordinates, and how these impact future states. Example: Propagating a state in the presence of NO forces Final State: (xf, yf, zf, vxf, vyf, vzf) Initial State: (x0, y0, z0, vx0, vy0, vz0)

Effects of Small Variations • What happens if we perturb the value of x0? Force model: 0 Initial State: (x0+Δx, y0, z0, vx0, vy0, vz0) Final State: (xf, yf, zf, vxf, vyf, vzf) Initial State: (x0, y0, z0, vx0, vy0, vz0)

Effects of Small Variations • What happens if we perturb the value of x0? Force model: 0 Final State: (xf+Δx, yf, zf, vxf, vyf, vzf) Initial State: (x0+Δx, y0, z0, vx0, vy0, vz0) Final State: (xf, yf, zf, vxf, vyf, vzf) Initial State: (x0, y0, z0, vx0, vy0, vz0)

Effects of Small Variations • What happens if we perturb the position? Force model: 0 Final State: (xf+Δx, yf+Δy, zf+Δz, vxf, vyf, vzf) Initial State: (x0+Δx, y0+Δy, z0+Δz, vx0, vy0, vz0) Initial State: (x0, y0, z0, vx0, vy0, vz0)

Effects of Small Variations • What happens if we perturb the value of vx0? Force model: 0 Initial State: (x0, y0, z0, vx0-Δvx, vy0, vz0) Final State: (xf, yf, zf, vxf, vyf, vzf) Initial State: (x0, y0, z0, vx0, vy0, vz0)

Effects of Small Variations • What happens if we perturb the value of vx0? Force model: 0 Final State: (xf+tΔvx, yf, zf, vxf+Δvx, vyf, vzf) Initial State: (x0, y0, z0, vx0+Δvx, vy0, vz0) Final State: (xf, yf, zf, vxf, vyf, vzf) Initial State: (x0, y0, z0, vx0, vy0, vz0)

Effects of Small Variations • What happens if we perturb the position and velocity? Force model: 0

Effects of Small Variations • We could have arrived at this easily enough from the equations of motion. Force model: 0

Effects of Small Variations • This becomes more challenging with nonlinear dynamics Force model: two-body

Effects of Small Variations • This becomes more challenging with nonlinear dynamics Final State: (xf, yf, zf, vxf, vyf, vzf) Force model: two-body Initial State: (x0, y0, z0, vx0, vy0, vz0) The partial of one Cartesian parameter wrt the partial of another Cartesian parameter is ugly.

Effects of Small Variations • This becomes more challenging with nonlinear dynamics Final State: (xf, yf, zf, vxf, vyf, vzf) Force model: two-body

Effects of Small Variations • Represent variations as functions of Keplerian orbital elements Force model: two-body Final State: Then, what is ?

Effects of Small Variations • Represent variations as functions of Keplerian orbital elements Force model: two-body Final State:

Effects of Small Variations • Represent variations as functions of Keplerian orbital elements Force model: two-body If Final State: Then, what is ?

Effects of Small Variations • Represent variations as functions of Keplerian orbital elements Force model: two-body If Final State: (Remember, af=a0)

Effects of Small Variations • Represent variations as functions of Keplerian orbital elements Force model: two-body If Final State:

Effects of Small Variations • The point is that we can relate small perturbations from one element to another easier using Keplerian orbital element than Cartesian. • Other brain teasers

Effects of Small Variations • How does Rf vary if V0 is increased? Force model: two-body A: Increases B: Decreases C: Stays the same D: Not enough information Final State (Rf, Vf) Initial State (R0, V0)

Effects of Small Variations • How does Rf vary if R0 is increased? Force model: two-body Hint: A: Increases B: Decreases C: Stays the same D: Not enough information Final State (Rf, Vf) Initial State (R0, V0)

Intermission • Any Questions? • (quick break)

LaTex • The Not-So-Short Guide to LaTex

Everyday Orbital Motion • Ballistic motion is orbital motion • Solid Earth prevents a body in ballistic motion from reaching perigee • A body dropped from rest, at the equator, is shown • Perigee: 11.2 km • Eccentricity: 0.9965

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, f 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 external mass 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: • 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

Earth J2 (Degree 2 Zonal Harmonic) • J2 represents a dominant characteristic of the shape of the planet • Positive J2: oblate spheroid • Negative J2: prolate spheroid • Scientific controversy in 1735: was Earth oblate or prolate? Oblate spheroid Prolate spheroid

Resolution of Controversy • In 1735, one view of the shape of the Earth was based on work of Newton, who had argued for the oblate shape (centrifugal forces) • Another view was based on measurements of the length of 1° of latitude in France, supported a prolate spheroid • French Academy of Sciences funded two expeditions to make measurements of 1° of latitude near the Arctic Circle (northern Scandinavia) and near the equator (now Ecuador) • It took ~10 years for the equator team to complete, so the first results were from Scandinavia, and equator verified it: the Earth was an oblate spheroid, J2 is +

Vanguard • Determination of Earth gravity coefficients resulted from Vanguard-I (NRL project) • First network of tracking stations, known as Minitrack, was deployed to support objectives: “determine atmospheric density and the shape of the Earth” • To achieve objectives, all basic elements of orbit determination were involved and a state of the art IBM 704 computer was used to determine the orbit

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

J2 and Orbit Design • As altitude increases, J2 perturbation diminishes (from a great distance the Earth is equivalent to a point mass) • Use J2 perturbation in orbit design, e.g., solar synchronous satellite • If dΩ/dt = +360°/365.25 days, the line of nodes will keep a fixed (in an average sense) orientation with respect to the Earth-Sun direction • Must be retrograde; for 600 km altitude, i=98°

Perturbations from Spherical Harmonics • Mean Ω, ω, M exhibit secular variation (caused by even degree Jn) • Mean a, e, i are constant • Odd degree Jn cause long period perturbations (period of argument of perigee motion) • All harmonic coefficients cause short period perturbations (period is 1, ½, 1/3, etc multiple of the orbital period) • m≠0 harmonic coefficients cause m-daily perturbations (i.e., 1, ½, 1/3, etc multiple of one day) • Special category: resonant perturbations (e.g., geosynchronous, GPS, …)

Secular Variations • Secular variations of Ω (positive J2) • 0° < i <90° : dΩ/dt < 0 • i = 90°: dΩ/dt = 0 • 90° < i < 180°: dΩ/dt > 0 • Secular variations of ω (positive J2) • i=63.4° or 116.6°, dω/dt = 0 (critical i) • See Table 2.3.3 for more details • Secular variations produced by all even-degree zonal harmonics

ASEN 5070 Statistical Orbit determination I Fall 2012 Professor George H. Born