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

CCARColorado Center for

Astrodynamics Research

University of ColoradoBoulder 1

ASEN 5070Statistical Orbit determination I

Fall 2012

Professor George H. BornProfessor Jeffrey S. Parker

Lecture 3: Astro and Coding




Homework 1

Another change in office hours

Yep, D2L

HW 2 will be posted after this class

Announcements




Quiz Results

79% of responses were correct.

Answer in mid-lecture.




Quiz Results





Quiz Results






Quiz Results





Quiz Results






Review solutions for HW1

Show HW2

Homeworks




Coordinate Frames and Time Systems

Homework details◦ Cartesian to Keplerian conversions◦ When elements aren’t well-defined.

Integrators

Coding hints and tricks◦ LaTex: intro◦ MATLAB: ways to speed up your code◦ Python: intro

Today’s Lecture




Inertial: fixed orientation in space◦ 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 Frames




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.

Is the Earth-centered J2000 coordinate system inertial?

Coordinate Systems




ICRF International Celestial Reference Frame, a realization of the ICR

System.

Defined by IAU (International Astronomical Union) Tied to the observations of a selection of 212 well-known quasars and

other distant bright radio objects.◦ Each is known to within 0.5 milliarcsec

Fixed as well as possible to the observable universe. Motion of quasars is averaged out.

◦ Coordinate axes known to within 0.02 milliarcsec Quasi-inertial reference frame (rotates a little) Center: Barycenter of the Solar System

Useful Coordinate Systems




ICRF2 Second International Celestial Reference Frame, consistent with

the first but with better observational data.

Defined by IAU in 2009. Tied to the observations of a selection of 295 well-known quasars

and other distant bright radio objects (97 of which are in ICRF1).◦ Each is known to within 0.1 milliarcsec

Fixed as well as possible to the observable universe. Motion of quasars is averaged out.

◦ Coordinate axes known to within 0.01 milliarcsec Quasi-inertial reference frame (rotates a little) Center: Barycenter of the Solar System





EME2000 / J2000 / ECI

Earth-centered Mean Equator and Equinox of J2000◦ Center = Earth◦ Frame = Inertial (very similar to ICRF)

X = Vernal Equinox at 1/1/2000 12:00:00 TT (Terrestrial Time)

Z = Spin axis of Earth at same time Y = Completes right-handed coordinate frame





EMO2000

Earth-centered Mean Orbit and Equinox of J2000◦ Center = Earth◦ Frame = Inertial

X = Vernal Equinox at 1/1/2000 12:00:00 TT (Terrestrial Time)

Z = Orbit normal vector at same time Y = Completes right-handed coordinate frame

◦ This differs from EME2000 by ~23.4393 degrees.





Note that J2000 is very similar to ICRF and ICRF2◦ The pole of the J2000 frame differs from the ICRF pole by ~18

milliarcsec◦ The right ascension of the J2000 x-axis differs from the ICRF by

78 milliarcsec

JPL’s DE405 / DE421 ephemerides are defined to be consistent with the ICRF, but are usually referred to as “EME2000.” They are very similar, but not actually the same.





ECF / ECEF / Earth Fixed / International Terrestrial Reference Frame (ITRF)

Earth-centered Earth Fixed◦ Center = Earth◦ Frame = Rotating and osculating (including

precession, nutation, etc) X = Osculating vector from center of Earth toward

the equator along the Prime Meridian Z = Osculating spin-axis vector Y = Completes right-handed coordinate frame





The angular velocity vector ωE is not constant in direction or magnitude◦ Direction: polar motion

Chandler period: 430 days Solar period: 365 days

◦ Magnitude: related to length of day (LOD)

Components of ωE depend on observations; difficult to predict over long periods

Earth Rotation





Principal Axis Frames

Planet-centered Rotating System◦ Center = Planet◦ Frame:

X = Points in the direction of the minimum moment of inertia, i.e., the prime meridian principal axis.

Z = Points in the direction of maximum moment of inertia (for Earth and Moon, this is the North Pole principal axis).

Y = Completes right-handed coordinate frame





IAU Systems

Center: Planet Frame: Either inertial or fixed

Z = Points in the direction of the spin axis of the body. Note: by convention, all z-axes point in the solar system

North direction (same hemisphere as Earth’s North). Low-degree polynomial approximations are used to

compute the pole vector for most planets wrt ICRF. Longitude defined relative to a fixed surface feature

for rigid bodies.





Example:◦ Lat and Lon of Greenwich, England, shown in EME2000.◦ Greenwich defined in IAU Earth frame to be at a

constant lat and lon at the J2000 epoch.





Synodic Coordinate Systems

Earth-Moon, Sun-Earth/Moon, Jupiter-Europa, etc◦ Center = Barycenter of two masses◦ Frame:

X = Points from larger mass to the smaller mass. Z = Points in the direction of angular momentum. Y = Completes right-handed coordinate frame





Converting from ECI to ECF

Coordinate System Transformations

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




Question: How do you quantify the passage of time?

Time Systems





Year Month Day Second Pendulums Atoms

Time Systems





Year Month Day Second Pendulums Atoms

Time Systems

What are some issues with each of these?

GravityEarthquakesSnooze alarms




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




The duration of time required to traverse from one perihelion to the next.

The duration of time it takes for the Sun to occult a very distant object twice.

Time Systems: The Year

(exaggerated)

These vary from year to year.

Why?




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

Time Systems: The Year




Quiz Results




Same variations in definitions exist for the month, but the variations are more significant.

Time Systems: The Month




Civil day: 86400 SI seconds (+/- 1 for leap second on UTC time system)

Mean Solar Day: 86400 mean solar seconds◦ Average time it takes for the Sun-Earth line to rotate 360 degrees◦ True Solar Days vary by up to 30 seconds, depending on where the

Earth is in its orbit.

Sidereal Day: 86164.1 SI seconds◦ Time it takes the Earth to rotate 360 degrees relative to the

(precessing) Vernal Equinox

Stellar Day: 0.008 seconds longer than the Sidereal Day◦ Time it takes the Earth to rotate 360 degrees relative to distant stars

Time Systems: The Day




From 1000 AD to 1960 AD, the “second” was defined to be 1/86400 of a mean solar day.

Now it is defined using atomic transitions – some of the most consistent measurable durations of time available.◦ One SI second = the duration of 9,192,631,770 periods of the

radiation corresponding to the transition between the two hyperfine levels of the ground state of the Cesium 133 atom.

◦ The atom should be at rest at 0K.

Time Systems: The Second




Time Systems: Time Scales




TAI = The Temps Atomique International◦ International Atomic Time

Continuous time scale resulting from the statistical analysis of a large number of atomic clocks operating around the world.◦ Performed by the Bureau International des Poids et Mesures

(BIPM)

Time Systems: TAI

TAI




UT1 = Universal Time Represents the daily rotation of the Earth Independent of the observing site (its longitude, etc) Continuous time scale, but unpredictable Computed using a combination of VLBI, quasars, lunar laser

ranging, satellite laser ranging, GPS, others

Time Systems: UT1

UT1




UTC = Coordinated Universal Time Civil timekeeping, available from radio broadcast signals. Equal to TAI in 1958, reset in 1972 such that TAI-UTC=10 sec Since 1972, leap seconds keep |UT1-UTC| < 0.9 sec In June, 2012, the 25th leap second was added such that TAI-

UTC=35 sec

Time Systems: UTC

UTC




Time Systems: UTC




Time Systems: UTC

What causes these variations?




Quiz Results




Quiz Results




TT = Terrestrial Time Described as the proper time of a clock located on the geoid. Actually defined as a coordinate time scale. In effect, TT describes the geoid (mean sea level) in terms of a

particular level of gravitational time dilation relative to a notional observer located at infinitely high altitude.

Time Systems: TT

TT TT-TAI=

~32.184 sec




TDB = Barycentric Dynamical Time JPL’s “ET” = TDB. Also known as Teph. There are other

definitions of “Ephemeris Time” (complicated history) Independent variable in the equations of motion governing

the motion of bodies in the solar system.

Time Systems: TDB

TDB TDB-TAI=

~32.184 sec+relativistic




Long story short

In astrodynamics, when we integrate the equations of motion of a satellite, we’re using the time system “TDB” or ~”ET”.

Clocks run at different rates, based on relativity. The civil system is not a continuous time system. We won’t worry about the fine details in this class, but in

reality spacecraft navigators do need to worry about the details.◦ Fortunately, most navigators don’t; rather, they permit one or

two specialists to worry about the details.◦ Whew.

Time Systems: Summary




Questions on Coordinate or Time Systems?

Quick Break

Next topics:◦ Cartesian to Keplerian conversions.

◦ Integration

◦ Coding Tips and Tricks

Questions




Shape:◦ a = semi-major axis◦ e = eccentricity

Orientation:◦ i = inclination◦ Ω = right ascension of ascending node◦ ω = argument of periapse

Position:◦ ν = true anomaly

Keplerian Orbital Elements







What if i=0?








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”








What if e=0?








What if e=0?

◦ If orbit is circular, e = 0 and ω is undefined. In that case we can use the “Argument of Latitude”








What if i=0 and e=0?








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”








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”





Quiz Results




Quiz Results




Handout offers one conversion. I’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. Also good to check the angular momentum vector.

Cartesian to Keplerian Conversion




An “Integrator” in the context of astrodynamics is something that propagates the state of an object forward (or backward) in time.

Say we have a satellite in a realistic force model (not just 2-body) at some state “X”.

We have models to describe the accelerations on it. Where will the satellite go in the future?

Integration

Earth




We know the current state and time. We know the derivative of the state.

Integration

Earth Accelerations are the equations of motion (two-body, J2, Drag, etc)

2nd order ODE may be integrated as system of 1st order ODEs.




Many techniques to predict the motion of an object.

Simplest: 1st order Taylor expansion: Euler’s Method

Integration

X(0)

X(10)X(20)





Euler’s Method with predictor/corrector using trapezoidal rule

Integration

X(0)

X(10)X(20)





Runge-Kutta (RK4)

Integration





Varying time-step / adaptive time-step methods

◦ Perform a 4th and a 5th-order approximation. Check the difference.◦ If smaller than tolerance, keep the 5th order state and move on.◦ If not smaller than tolerance, reduce the time-step and repeat.

Small time-steps when trajectory and/or force model changes rapidly.

Integration





Symplectic integrators◦ Rather than focusing on maintaining position accuracy, these integrators

focus on conserving energy.◦ Tend to drift along-track

◦ In a conservative force-field, a satellite’s specific energy should be constant.

Integration




Quiz Results




(Show course website’s MATLAB integrator handout)

Integration




Questions on Integrators?

Next topics:◦ Coding Tips and Tricks

Questions




MATLAB!

Coding Tips and Tricks

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