Basic Orbital Mechanics

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Basic Orbital Mechanics

• Jeff Crum
• 23 Aug 01

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Introduction

• Orbital mechanics is the study of the motions of
  artificial satellites and space vehicles moving
  under the influence of forces such as gravity,
  atmospheric drag, thrust, etc.
• Modern offshoot of celestial mechanics, the study
  of the motions of the moon, planets, and stars

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Training Topics

• Kepler and Newton
• Orbital Elements
• Ground Traces
• Orbit Types
• Orbit Perturbations

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Planetary motion

• The problem of accurately describing the motions
  of planets has challenged observers for centuries
• Early theories held that the Earth was the center
  of the universe and all other heavenly bodies
  traveled in perfect circles about the Earth
• In 1543, Nicolaus Copernicus published his
  heliocentric theory that postulated that the Sun
  was the center of the universe, and that all
  planetsincluding the Earth, revolved about it in
  perfect circles

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Kepler

• Johannes Kepler agreed with Copernicus
  revolutionary and highly controversial theory

• Using thousands of astronomical observations from
  the Danish astronomer Tycho Brahe, Kepler tried
  in vain to fit the motion of the planet Mars to a
  circular orbit
• Kepler finally discovered the answer planets
  travel not in circles, but ellipses about the sun

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Ellipse
GEOMETRIC CENTER
Semi-Minor Axis (b)
FOCUS
FOCUS
Semi-Major Axis (a)
Focal Length (c)
MAJOR AXIS
MINOR AXIS

• Based on the results of his observations, Kepler
  published his three laws of planetary motion

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Keplers First Law

• The orbit of each planet is an ellipse, with the
  Sun at one focus

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Keplers Second Law

• The line joining the planet to the Sun sweeps out
  equal areas in equal times

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Keplers Third Law

• The square of a planets orbital period is in
  direct proportion to the cube of the semi-major
  axis
• Orbital period (P) time required to make one
  complete revolution around the sun
• P2 ? a3
• For example, Mercury, the closest planet to the
  Sun, completes an orbit in 88 days
• Pluto, the furthest planet from the Sun,
  completes an orbit every 248 years

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Newton

• Building upon the work of Kepler and others,
  Isaac Newton put forward his laws of motion and
  formulated his law of universal gravitation

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Newtons Laws of Motion

• First Law A body at rest remains at rest, and a
  body in motion continues to move at a constant
  velocity unless acted upon by an external force
• Second Law A force (F) acting on a body gives it
  an acceleration (a) which is in the direction of
  the force and has a magnitude inversely
  proportional to the mass of the body (m)
• F ma
• Third Law Whenever a body exerts a force on
  another body, the latter exerts a force of equal
  magnitude and opposite direction on the former

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Newton in simpler terms

• Objects in motion want to travel in straight
  lines at constant speed (Newtons first law)
• But
• Force of gravity causes the path to curve

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Newton in simpler terms

• Amount of curve depends on initial speed and
  direction

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Newton in simpler terms

• Satellites must have a balance of

Speed Gravity No orbit Orbit No Orbit
(too slow)
(too fast)
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Satellite orbits

• The initial speed and direction of an orbiting
  satellite creates an ellipse with the Earth at
  one focus

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Satellite orbits

• Apogee highest altitude, lowest speed
• Perigee lowest altitude, fastest speed

Perigee
Apogee
Note In generic terms, periapsis is the point in
an orbit closest to the primary focus, and
apoapsis is the farthest point. These terms are
usually modified to apply to the body being
orbited (e.g., perihelion/aphelion for the Sun,
perigee/apogee for the Earth, perijove/apojove
for Jupiter, etc.)
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Satellite Orbits
Orbit size determines satellite period, or time
to make one orbit
DMSP 510 miles 110 minutes
GPS 10900 miles 11 hrs, 58 min
DSP 22300 miles 23 hrs, 56 min
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Satellite Orbits

• Orbital element set
• Semi-Major Axis.. Size
• Eccentricity. Shape
• Inclination.. Tilt
• Right Ascension of the
• Ascending Node.. Direction
• Argument of Perigee. Rotation
• True Anomaly Position
• Epoch Time. Time Stamp

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Semi-Major Axis (Size)
Semi-Major Axis (a)
FOCUS
APOGEE
PERIGEE
Focal Length (c)
Focal Length (c)
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Eccentricity (Shape)
a
e .75
?
?
c
e .45
?
?
c
e 0
?
c
Eccentricity (e) c/a
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Inclination (Tilt)
Inclination Angle 0? to lt 90 ?
Equatorial Plane
Ascending Node
Inclined Prograde Orbit
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Inclination (Tilt)
Inclination Angle gt90? to lt 180 ?
Equatorial Plane
Inclined Retrograde Orbit
Ascending Node
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Right Ascension of Ascending Node (Direction)
Inclined Prograde Orbit
0? First Point of Aries
0?
180?
Ascending Node
Line of Nodes
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Right Ascension of Ascending Node (Direction)
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GPS constellation has 6 orbit planes, all
inclined at 55
1
6
Each orbit plane is spaced 60 from the previous
plane
2
4
3
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Argument of Perigee (Rotation)
Argument of Perigee is the angular distance
between the ascending node and the point of
perigee
Perigee
Equatorial Plane
Ascending Node
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True Anomaly
True Anomaly is the angular distance between
perigee and some point in the orbit. Usually
this is used to describe where an SV is in the
orbit at a given time.
Equatorial Plane
Perigee
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Epoch Time

• The epoch time is an exact specification of the
  date and time at which a given Keplerian element
  set is valid

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Ground Trace

• A ground trace is the projection of a satellites
  orbit onto the earths surface

• Note that the highest northern and southern
  latitudes reached by a satellites ground trace
  is equal to the satellites orbital inclination
• If inclination gt 90, then highest latitude of
  the ground trace is 180 minus the orbit
  inclination

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Ground Trace

• Consider an orbit of a satellite to lie in a
  plane that passes through the center of a
  theoretically spherical Earth
• The trace of this plane on the surface of a
  non-rotating Earth is a great circle
• If the Earth did not rotate, the satellite would
  retrace the same ground over and over

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Ground Trace

• If we consider a rotating Earth, the orbital
  plane of a satellite remains fixed in space as
  the Earth turns under it
• Effect of Earths rotation is to displace the
  ground trace on each successive revolution of the
  satellite
• Ground trace displaced by the number of degrees
  the Earth rotates during on orbital period
• This displacement is called nodal regression

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Nodal Regression

• Earths rotation causes the ground trace to
  regress westward for each successive orbit
• Earth rotates approximately 15 per hour
• Nodal regression (orbit period in hours) 15
• A satellite in a polar orbit has the potential to
  overfly all the Earths surface
• If the time required for one complete rotation of
  the Earth is an exact multiple of the satellites
  period, then eventually the satellite will
  retrace exactly the same path as it did on some
  previous revolution

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GROUND TRACESWestward Regression
Pictures and animation courtesy of Capt Troy
Endicott, Det 1 533 TRS, Schriever AFB, AETC
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Orbit Types

• Satellites use a wide variety of orbits to
  fulfill their missions
• Factors determining orbit type
• Mission requirements
• Booster capability and cost
• Satellite and orbital mechanics

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Orbit Types

• For a spacecraft to achieve orbit, it must be
  launched to an elevation above the Earths
  atmosphere and accelerated to orbital velocity
• The most energy efficient orbit is a direct, low
  inclination orbit
• To achieve this orbit, the spacecraft is launched
  in an eastward direction from a site near the
  Earths equator
• The rotational speed of the Earth contributes to
  the spacecrafts final orbital speed
• e.g. A due east launch from the Cape (28.5 deg
  north latitude) results in a free ride of 915
  mph

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Orbit Types

• Launching a spacecraft in a direction other than
  east, or from a site far from the equator,
  results in an orbit of higher inclination
• High inclination orbits are less able to take
  advantage of the initial speed provided by the
  Earths rotation
• Launch vehicle must provide greater energy to
  attain orbital velocity

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Launch Azimuth
The desired orbit inclination determines the
azimuth of launch.
Vandenberg AFB
Patrick AFB
Safety constrains possible launch azimuths
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Orbit Types

• Low Earth Orbit (LEO)
• Sun Synchronous
• Polar
• Medium Earth Orbit (MEO)
• Geosynchronous/Geostationary Earth Orbit (GEO)
• Highly inclined orbits
• Molniya

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Low Earth Orbits

• Up to 520 miles
• Common missions
• Manned (shuttle)
• Reconnaissance
• Communications

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LEO
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Polar Orbits

• Inclination of 90 degrees
• Missions
• Mapping
• Surveillance

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Sun Synchronous

• 460-520 miles
• Near-polar inclination
• Orbital plane precesses with the same period as
  the Earths solar period
• Satellite crosses perigee at the same local time
  every orbit
• Common missions
• Earth sensing (LANDSAT)
• Weather (DMSP/NOAA)

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Sun Synchronous
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Medium Earth Orbit

• Also called semi-synchronous
• 10,900 miles high inclination
• Missions
• Navigation (GPS, GLONASS)

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MEO
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Geosynchronous and Geostationary

• Geosynchronous
• Altitude 22,300 miles
• Period 24 hours
• Any inclination
• Does not have to be circular
• Geostationary
• Altitude 22,300 miles
• Period 24 hours
• Inclination near zero
• Eccentricity nearly zero

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GEO

• GEO spacecraft appear to hang motionless above
  one position on the Earths surface
• Missions
• Communications
• Weather

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GEO
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Highly Elliptical Orbits

• 63.4-116.6 degree inclination
• 200-23,800 mile altitude
• Missions
• Comm relay (Molniya)

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Molniya Orbits

• Many Russian cities are at high northern
  latitudes where it is impractical to use GEO
  satellites for telecommunications
• GEO satellites appear either low on the horizon
  or are not visible at all
• Molniya orbit has a 12 hour period at high
  eccentricity and inclination (63.4)
• Satellite spends most of its time near apogee, so
  for approximately 11 hours of each orbit the
  satellite is above the horizon for high northern
  latitudes

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Molniya Orbit
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GROUND TRACESHighly Elliptical Orbit
Pictures and animation courtesy of Capt Troy
Endicott, Det 1 533 TRS, Schriever AFB, AETC
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Orbit Perturbations

• Any disturbance in the regular motion of a
  satellite resulting from a force other than those
  causing regular motion
• Non-spherical earth
• Atmospheric drag
• Sun/moon gravity
• Space environment

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Summary

• TBD

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Backup Charts
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