A Brief Introduction to Astrodynamics

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A Brief Introduction to Astrodynamics

• Shaun Gorman
• Iowa State University
• Ames, Iowa

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Topics Discussed

• Coordinate Systems
• Orbital Geometry
• Classical Orbital Elements
• Classes of Orbits
• Two-line Element Sets

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Coordinate Systems

• Heliocentric-Ecliptic Coordinate System
• Geocentric-Equatorial Coordinate System
• Right Ascension-Declination System
• Perifocal Coordinate System

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Heliocentric-Ecliptic Coordinate System

• Origin at the center of the sun.
• X-Y plane coincides with the earths plane of
  revolution
• X axis points in the direction of the vernal
  equinox
• Z axis points in the direction of the suns north
  pole

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Geocentric-Equatorial Coordinate System

• Also called Earth Centered Inertial or ECI
• Origin at the center of the earth
• X-Y plane coincides with the earths equator
• X axis points in the direction of the vernal
  equinox
• Z axis points in the direction of the north pole
• I, J and K unit vectors lie along the X, Y and Z
  axes

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Right Ascension-Declination System
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Perifocal Coordinate System

• Origin at the center of the earth
• P-Q plane coincides with the satellites orbit
  plane
• P axis points in the direction of the vernal
  equinox
• Q axis is 90o from the P axis in the direction of
  satellite motion
• W axis is normal to the satellite orbit

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ECI Coordinate Systems

• Several different types of ECI coordinate
  systems.
• Fixed
• J2000
• B1950
• TEME of Epoch
• TEME of Date

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ECI Coordinate Types
Classical Orbital Elements (COE) Uses the traditional osculating Keplerian orbital elements to specify the shape and size of an orbit.
Cartesian Uses the initial X, Y and Z position and velocity components of the satellite.
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Fixed

• X is fixed at 0 deg longitude, Y is fixed at 90
  deg longitude, and Z is directed toward the north
  pole.
• Only Cartesian type of coordinates can be used.

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J2000

• X points toward the mean vernal equinox and Z
  points along the mean rotation axis of the Earth
  on 1 Jan 2000 at 120000.00 TDB, which
  corresponds to JD 2451545.0 TDB.
• Can use either Cartesian or COE.

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B1950

• X points toward the mean vernal equinox and Z
  points along the mean rotation axis of the Earth
  at the beginning of the Besselian year 1950 (when
  the longitude of the mean Sun is 280.0 deg
  measured from the mean equinox) and corresponds
  to 31 December 1949 220907.2 or JD 2433282.423.
• Can use either Cartesian or COE.

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TEME of Epoch

• X points toward the mean vernal equinox and Z
  points along the true rotation axis of the
  Coordinate Epoch.
• Can use either Cartesian or COE.

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TEME of Date

• X points toward the mean vernal equinox and Z
  points along the true rotation axis of the Orbit
  Epoch.
• Can use either Cartesian or COE.

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Orbital Geometry

• Apoapsis- farthest point in an orbit
• Periapsis- nearest point in an orbit
• Line of Nodes - The point where the vehicle
  crosses the equator
• Radius - distance from the center of the Earth to
  the orbit

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Orbital Geometry
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Classical Orbital Elements

• a - Semi-major Axis-a constant defining the size
  of the orbit
• e Eccentricity-a constant defining the shape of
  the orbit (0circular, Less than 1elliptical)
• i Inclination-the angle between the equator and
  the orbit plane
• W - Right Ascension of the Ascending Node-the
  angle between vernal equinox and the point where
  the orbit crosses the equatorial plane
• w - Argument of Perigee-the angle between the
  ascending node and the orbit's point of closest
  approach to the earth (perigee)
• v - True Anomaly-the angle between perigee and
  the vehicle (in the orbit plane)

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C.O.E. (continued)
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Vector Re-fresher

• Before we start lets go over some basic vector
  math

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Determining Orbital Elements

• Lets say that a ground station on the earth is
  able to provide the position and velocity of a
  satellite by providing us with vectors r and v.

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Conversion from Cartesian to COE

• Given the position and velocity vectors r
  and v
• Determine the six classical orbital elements e,
  a, i, W, w and v

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Setting up a coordinate system

• We will use the geocentric equatorial coordinate
  system.
• The I axis points towards the vernal equinox.
• The J axis is 90o to the east in the equatorial
  plane.
• The K axis points directly through the north pole.

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Determining Orbital Elements

• The expression, which is called
  specific angular momentum, must be held constant
  due the law of conservation of angular momentum.
• Thus

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Determining Orbital Elements

• An important thing to remember is that h is a
  vector perpendicular to the plane of the orbit.
  The node vector is defined as.
• Thus

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Determining Eccentricity

• The eccentricity vector is just a function of the
  gravitational parameter m and the r and v vectors
• For the Earth

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Determining Semi-major Axis

• The equation for the semi-major is a function of
  the velocity and radius vectors along with the
  gravitational parameter m
• If e1, ainf.

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Determining Inclination

• Since the inclination is the angle between K and
  h, the inclination can be found using the
  formula
• Inclination is always between zero and pi.

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Determining RAAN

• Since the Right Ascension of the Ascending Node
  is the angle between I and n, the inclination can
  be found using the formula
• RAAN is always between pi and two pi.

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Determining Argument of Perigee

• Since the Argument of Perigee is the angle
  between n and e, the inclination can be found
  using the formula
• Argument of Perigee is always between zero and
  pi.

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Determining True Anomaly

• Since the True Anomaly is the angle between e and
  r, the inclination can be found using the
  formula

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Classes Of Orbits

• Types of rotation
• Prograde
• Retrograde
• Polar
• Types Of Orbital Geometry
• Elliptical
• Circular
• Parabolic
• Hyperbolic

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Prograde

• The Prograde or direct orbit moves in direction
  of Earth's rotation
• 0oltilt90o

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Retrograde

• The retrograde or indirect moves against the
  direction of Earth's rotation
• 90oltilt180o

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Polar

• Direct orbit over north and south pole
• i90o

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Elliptical

• Eccentricity, 0ltelt1
• Semi-major Axis, rpltaltra
• Semiparameter, rpltplt2rp

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Circular

• Eccentricity, e0
• Semi-major Axis, ar
• Semiparameter, pr

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Parabolic

• Eccentricity, e1
• Semi-major Axis, ainf
• Semiparameter, p2rp

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Hyperbolic

• Eccentricity, egt1
• Semi-major Axis, alt0
• Semiparameter, pgt2rp

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Two-line Element Sets

• One of the most commonly used methods of
  communicating orbital parameters is the Two-line
  element sets generated by NORAD. It is important
  to note that TLEs were developed for use only
  with the MSGP-4 propagator. Using TLEs with any
  other propagator may invalidate some of the
  built-in assumptions.
• These elements contain most of the same elements
  as the classical orbital elements, along with
  some additional parameters for identification
  purposes and for use in modeling perturbations in
  the MSGP-4 propagator.

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TLEs

• TLEs contain 12 different variables
• Six for the Classical Orbital Elements
• Four actual C.O.E.s e, i, W and w
• Two variables that can be used in place of
  C.O.E. M, Mean motion and n, mean anomaly
• Three to describe the effects of perturbations on
  satellite motion Bstar, and
• Two for identification purposes
• One for the time when this data was observed

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TLE format

• The following is an example of a Two-line Element
  set.
• This Format looks rather intimidating and is read
  the following way

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TLE Classical Orbital Elements

• The two-line element sets provide four of the
  classical orbital elements e, i, W and w.
• Instead of true anomaly the TLE gives the mean
  anomaly because it can be calculated at future
  time easier.
• This is also true for the substitution of mean
  motion for semi-major axis which will be
  explained on the next slide.

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Mean Motion to Semi-major Axis

• n15.5911407 revolutions/day
• n5612.81065 degrees/day
• 1 day107.088278 TU
• n52.4129 degrees/TU
• 1 radian57.2957795 degrees
• n.9147782342 radians/TU
• a
• a1.061180 ER
• 1 ER6378.1363 km
• a6768.357 km

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TLE Perturbations Effects

• The three perturbation effects in the TLEs are
  mean motion rate, mean motion acceleration and B
  a drag parameter
• The ballistic coefficient, BC, can be found from
  B

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TLE Identification Purposes

• The Satellite number
• The International Designation tells us the year
  of the satellite launch, launch number of year
  and section
• For this satellite is 86017A, that means it was
  the 17th launch of 1986 an it was the A section.

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TLE Time

• The epoch is what time the values were recorded
• The Time give was 93352.53502934
• This Translates to the 352nd day of 1993 which
  was December 18.
• To find the Hours, minutes and seconds just take
  the remainder divide by 24 to get the hours, take
  the remainder of that divide by 60 to get the
  minutes and take the remainder of that divide by
  60 to get the seconds
• This should translate to 12 h. 50 min. and 26.535
  sec.