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SP 110
Orbits
Orbits
Basic requirements
Orbital motion in the traditional sense requires two
components
1. Attractive force
2. Relative motion away from the common axis
Energy is easiest to use for the description of orbits
1. Attractive force – potential energy
2. Relative motion – kinetic energy
Orbits
Gravity is the attractive force for objects in orbit in
space, although other attractive forces allow
orbits such as electrically charged particles with
relative motion that can orbit. These include
subatomic particles, atoms and molecules, and
larger particles
Gravitational force – use gravitational potential
energy
Ep = -GMm/r G = gravitational constant, M and m are the
two masses, and r is the separation
Relative motion – use kinetic energy
Ek = 1/2mV2 m = one of the masses, V is the relative
motion
Orbits
The sum of the kinetic and potential energies is also
important
Etotal = Ep + Ek = -GMm/r + ½ mV2
If Etotal is negative, the negative component
(gravitational potential an attractive force) is
greater than the kinetic energy (relative motion),
and the two objects can be in orbit
If Etotal is positive, the kinetic energy (relative velocity)
component is larger than the potential energy
component (attractive force), and the two objects
are not bound and there is no repeated orbit
Orbits
Two restrictions on these orbits needed
to keep the discussion simple
1. Two-body orbits only
2. The off-axis relative motion must be
sufficient to keep the two bodies from
colliding
Orbits - Kepler
Johannes Kepler (1570-1630) first described planetary
orbits with three laws
1. The planets orbit the Sun in elliptical orbits with the
Sun at one focus
2. The planets sweep out equal areas in these ellipses in
equal times (conservation of angular motion)
3. The period of orbit squared equals the semimajor axis
(average separation) cubed
P2 = a3 P = period in Earth-years, a = semimajor axis in
Astronomical Units
Orbits - Kepler
Implications of the three Keplerain laws
1. The planets orbit the Sun in elliptical orbits with the Sun at one
focus

Each body orbits about the center of mass, but each body
follows an elliptical path around the other, with the other at one
focus
2. The planets sweep out equal areas in equal times



Angular momentum is conserved
Total orbital energy is constant
In the elliptical orbit, decreasing separation corresponds to
decreasing potential energy (less negative), and therefore faster
orbital speeds (increasing kinetic energy to maintain the
constant), and vice-versa for greater separation
3. The period of orbit squared equals the semimajor axis cubed

Orbital velocity decreases with increasing separation
Orbits - Kepler
Keplerain orbits
P2 = a3
Although the assumption
here has been that there
are only two bodies
involved in orbits, orbital
motion described as
“Keplerian” can involve
many bodies in orbit,
including the planets
solar system
Kepler’s P2=a3 law is limited
to objects in orbit
around the Sun (planets,
asteroids, comets, etc)
Orbits - Newton
Issac Newton (1643-1727)
developed the basic laws of
motion and the theory of
gravity, and the relationship
between mass and force
Newton’s equation relating period
to semimajor axis is
generalized for any two-body
orbit (satellites around Earth
or the Moon, stars around the
galaxy centers, etc.)
P2 = a3 [4π2/G(M+m)]
Orbits - Newton
Newton’s period-semimajor axis equation can be used for
any gravitational orbit that approximates a two-body
orbit
Example: Find the semimajor axis of a geosynchronous
orbit
P2 = a3 [4π2/G(M+m)]
a = [p2G(M+m)/4π2]1/3 p = 24 hours (86,400 sec),
MEarth = 5.97x1024 kg, G=6.67x10-11Nm2/kg2, m is
mass of satellite and is insignificant (compared to
Earth)
a = 3.54x107m = 35,400 km (22,500 mi)
Orbit Definitions
Orbits - Definitions
Basic ellipse
Longest axis = major axis
a = semi major axis = 1/2
major axis
Smallest axis = minor axis
(perpendicular to major
axis)
b = semi minor axis = 1/2
minor axis
Orbits - Definitions
Eccentricity – a measure of
the flatness of the orbit

e = [1- (b/a)2]1/2

e varies from zero to 1 for a
bound orbit and greater
than 1 for an unbound orbit
(e = 1 for a parabolic orbit)

e = 0 for a circular orbit

e = 0.999999999 = flat
(highly eccentric) orbit
Orbits - Definitions
Inclination angle – the angle between orbit
plane and reference plane, or between the
orbit plane and the equator

i = 0 to 360o (0 to 2π)
Orbits - Definitions

Inclination - the angular difference
between the orbital plane and a
reference plane

Nodes - the two intersecting
points between an orbit and a
reference plane

Descending node - point of
intersection of object in orbit
that is descending through the
reference plane

Ascending node - point of
intersection of object in orbit
that is ascending through the
reference plane

Line of nodes - the line
intersecting the two nodes,
(also the intersection of the
orbit plane and the reference
plane)
Orbits - Definitions

Periapsis (or periastron)
is the closest approach
distance for a 2-body
orbit

Perigee - closest
point in the orbit
between the Earth and
the object in orbit

Perihelion - closest
point in orbit of
planets to the Sun
Orbits - Definitions

Apoapsis (or apoastron)
is the farthest point in
orbit between two bodies
in a 2-body orbit

Apogee - farthest point
in orbit between an
object and the Earth

Aphelion - farthest
point in orbit between
an object and the Sun
Orbits - Definitions
Eccentricity – a measure of
the flatness of the orbit
e = [1- (b/a)2]1/2
If a = b, the orbit is circular,
thus (b/a)2 = 1, and
therefore e = [1- (1)2]1/2 = 0
Orbit and Position
Reference
Orbit Reference
The most common type of orbit position reference is
the geocentric equatorial reference coordinate
system that employs the primary reference plane
as the Earth’s equator
This is a Cartesian reference with the X and Y axis on
the equatorial plane, and the polar axis along the Z
axis
The X-axis is the primary axis pointing to the vernal
equinox – the position of the Sun at the definition
of spring (apparent passage through the plane of
the ecliptic)
Orbit Reference
Orbit Reference
To identify the position of a satellite in orbit, the
geocentric equatorial coordinates are used as the
positional reference system

The orbit and orbit plane is used to identify the
satellite position using a set of lengths and angles
known as the orbital elements
 Can also be done in Cartesian coordinates

A second coordinate system is used to identify an
observer’s position with respect to the geocentric
equatorial reference
 Provides relative angles to satellite from the
observer’s position
Orbit Reference
The seven orbital elements are:

a = semimajor axis

i = inclination angle

e = eccentricity

Ω = Right Ascension (or longitude) of the ascending node angle between X-axis (vernal equinox) and the ascending node

ω = Argument (or longitude) of the perigee - angle between the
ascending node and the perigee

ν = True anomaly - angle between perigee and position of
orbiting object

T = Time since perigee passage
Orbit Reference
Orbital Energy
Orbital Energy
Changing an orbit requires energy
An orbit has a fixed semimajor axis based on the sum of the
kinetic and potential energies
Changing the semimajor axis changes the average potential
energy and the constant
Therefore, it takes energy (thrust) to boost to a higher or lower
orbit

Higher orbit requires added energy (forward thrust =
increased speed)

Lower orbit requires lower energy (retrograde thrust =
decreased speed)
Orbital Energy
Based on the total energy, there are three types of orbits - bound,
unbound , and escape (neither bound nor unbound)
•Bound Orbits - Elliptical orbits - potential greater than kinetic energy
•Unbound Orbits - Hyperbolic orbits - potential less than kinetic
•Escape orbits - Parabolic orbits - potential equal to kinetic
Orbit
Circular
Elliptical
Escape
/Parabolic
Hyperbolic
Condition
Bound
Bound
Neither
Unbound
Total Energy Negative
(Ek + Ep)
(Ek < Ep)
Negative
(Ek < Ep)
Zero
(Ek = Ep)
Positive
(Ek > Ep)
Eccentricity
0≤ e <1
e=1
e>1
e=0
Orbit Types
Orbit Types
The various types of orbits can describe the orbital energy and the orbital
shape (eccentricity mostly), or by reference orbit orientation, orbital
period or planetary surface coverage for the orbiting satellites
Communication, remote sensing, and surveillance all require specific
orientation throughout the satellite operation. A number of the types and
uses for these orbits are given below.
1. Bound (elliptical) orbit
A bound orbit, which is also an elliptical orbit, has relative kinetic energy
less than the combined gravitational potential energy
2. Unbound (hyperbolic) orbit
A hyperbolic orbit is unbound, meaning that the relative kinetic energy is
greater than the combined gravitational energy
3. Escape (parabolic) orbit
A parabolic orbit is neither bound nor unbound since the relative kinetic
energy is exactly equal to the combined gravitational potential energy.
The parabolic orbit conditions are the same as escape velocity.
Orbit Types
4. Prograde orbit
A prograde orbit has an inclination angle less than 900
which follows the same direction as the Earth's or the
orbited planet’s rotation
5. Retrograde orbit
A retrograde orbit has an inclination greater than 900
which travels in reverse direction to Earth's rotation
6. Polar orbit
A polar orbit has an inclination of 900 which allows
world-wide coverage over a period of hours to days
depending on altitude. This is an orbit commonly
used for meteorological and surveillance satellites
Orbit Types
7. Geosynchronous orbit
A geosynchronous orbit has an orbital period equal to
the Earth's rotation period of 24 hours (23h56m4.09s).
The semimajor axis of this orbit is 42,164 km. The
inclination for this orbit is often but not required to be
0o
8. Geostationary orbit
A geostationary orbit is also geosynchronous, but has
an equatorial orbit (i = 00), with an eccentricity of zero.
This provides a fixed communications platform with
respect to the Earth, and is used for communications,
remote sensing (whether satellites, for example.), and
surveillance.
Orbit Types
9. Sun-synchronous orbit
A sun-synchronous satellite orbit maintains a constant orientation
between the Sun and Earth which is useful for several applications. The
most common are the remote sensing applications satellites (Landsat,
for example) and astronomical observation satellites (IRAS, for example)
These applications require either full back-illumination, or complete
shadowing from the Sun. To accomplish this, a nearly polar orbit is used.
If the orbit were polar, the orientation of the orbit would be fixed with
respect to the Sun, but not the Earth
This would show a 0.98o per day change in orientation as the Earth does
an make the Sun-satellite orientation seasonal, as the Earth is. Hence,
the spacecraft needs a -0.98o per day retrograde motion in the orbital
plane to counter the Earth's orbital motion around the Sun.
To do this, an orbital retrograde rotation can be made by using the
oblateness of the Earth to place a torque on the orbit and produce a
precession of the nodes (rotation of the orbital plane)
A range of altitudes and corresponding inclinations are available for this
type of orbit. Landsat satellites use a = 709 km and I = 98.2o
Orbit Types
10. Molniya orbit
Russia has traditionally used communication satellite orbits that
are highly-inclined with high eccentricity for their high-latitude
ground stations with the extended orbit (apogee) over the higher
latitudes
The Molniya's 12 hour orbit period also allows for
communications between the Asian and North American
continents, since it is one-half of the sidereal day (43,082 sec)
The inclination for this orbit is 63.4o, with a semimajor axis of
26,562 km, and with varying apogee and perigee values that
satisfy the desired period and semimajor axis
11. Tundra orbit
A tundra orbit is an eccentric, high-inclination (63o) orbit similar
to the Molniya orbit but with a period twice as long (one sidereal
day)
Like the Molniya orbit, this tundra orbit is used for
communications at latitudes far from the equator
Orbit Types
12. Parking orbit
A parking orbit is a temporary orbit commonly used
for spacecraft checkout operations before departure
from Earth
13. Graveyard orbit
A graveyard orbit is a permanent, higher-thannormal orbit used to remove defective or aging
spacecraft from the busy geostationary region
Orbit Types
14. Walking orbit
A walking orbit gets its name from the rotation or
precessional motion of the orbit due to the
asymmetrical shape of the planet. An example of this
is the precession of the sun-synchronous orbit due to
the Earth's oblate shape (larger equatorial diameter
than polar diameter because of its rapid rotation).
15. Halo orbit
A halo orbit is found not around a celestial object but
around either the Lagrange L1 or L2 stability regions
(objects within or close to the L1 and L2 regions are
not in a stable orbit)
Orbit Transfers
Transfer Orbits
To get from one planet to another, a spacecraft must
follow an elliptical, heliocentric orbit for most of its
flight path
As the spacecraft departs the planet of origin, the target
planet must be in correct position so that spacecraft
arrival is coincident with the target planet
This simple requirement allows planet-to-planet flights
only at specific times, and similarly, any return flights
only at times of correct alignment of the two planets
Transfer Orbits
Basic orbit transfers
1. Hohmann Transfer - the most energy efficient coplanar orbit transfer
has an arrival point 180o from the departure point
2. Spiral (low thrust) - this is a long-period spiral orbit from launch to
arrival that is best suited for high efficiency, low thrust propulsion
systems (ion engines are a good example)
3. Direct (high thrust) - the direct, short-period transfer requires
greater thrust to accelerate the spacecraft quickly to desired orbit
at the expense of greater transfer energy and a larger propulsion
system
4. Gravity assist augmented propulsion - the gravitational attraction of
a spacecraft in a close encounter with a large planet can be used
to boost the kinetic energy of the spacecraft with reference to the
Sun
This technique can also be used to slow spacecraft to get to a
closer heliocentric orbit
Transfer Orbits
Traditional orbit transfer definitions

Type I (direct) transfer - transfer angle is less than the
Hohmann transfer ellipse (<180o)

This is also known as a direct transfer orbit

Hohmann transfer - transfer angle of 180o

Type II transfer - greater than 180o and less than 360o
(spiral)

Type III transfer - greater than 360o and less than 540o
(spiral)

Type IV transfer - greater than 540o and less than 720o
(spiral)
Transfer Orbits
A sketch of the orbital paths of the three
common types of transfer orbits
Transfer Orbits
Hohmann transfer for interplanetary missions

The simple expression of a Hohmann transfer can be calculated
in Earth years since the transfer orbit (trajectory cruise) occurs
within the Sun's gravitational influence

The ellipse representing the transfer will have the perihelion at
the inner planet orbit and the aphelion at the larger orbit planet

This technique also is used for Earth-orbit satellite transfers from
a low-Earth orbit to higher orbits, an example being a boost from
low-Earth parking orbit to a more distant geostationary orbit

Note that the transfer ellipse touches both the larger and smaller
circular orbits. That means that the semimajor axis of the
transfer orbit is one-half of the two orbits added together
Transfer Orbits
Calculations for the Hohmann transfer to the planets
atransfer = ½ (aA + aB)
A period calculation uses the same third law of Kepler, but
only one-half since the second half represents a return path to
the departure planet
ptransfer = ½ (atransfer3/2)
Start with the semimajor axis values for the planets
aEarth = 1.00
aJupiter = 5.20AU
aNeptune = 30.11 AU
aVenus = 0.72 AU
aSaturn = 9.56 AU
aPluto = 39.55 AU
aMars = 1.52 AU
aUranus = 19.22 AU
Transfer Orbits
Earth-Venus
pHohmann = ½ [(aHohmann )3/2 ] = ½ [(aA + aB)/2]3/2
= ½ [(1 AU + 0.72 AU)/2]3/2 = 0.40 yr
Earth-Mars
PHohmann = ½ [(1 AU + 1.52 AU)/2]3/2 = 0.72 yr
Earth-Jupiter
PHohmann = ½ [(1 AU + 5.20 AU)/2]3/2 = 2.73 yr
Earth-Saturn
PHohmann = ½ [(1 AU + 9.56 AU)/2]3/2 = 6.07 yr
Earth-Uranus
PHohmann = ½ [(1 AU + 19.22 AU)/2]3/2 = 16.1 yr
Earth-Neptune
PHohmann = ½ [(1 AU + 30.11 AU)/2]3/2 = 30.7 yr
Earth-Pluto
PHohmann = ½ [(1 AU + 39.55 AU)/2]3/2 = 45.7 yr
Gravity Assist
Gravity Assist
Gravity assist
Augmenting spacecraft propulsion with a gravity-assisted
boost is a technique that exchanges momentum between an
orbiting planet and a spacecraft during a close flyby of a
moving planet
The gravity assist propulsion boost can be used to either
increase or decrease the spacecraft velocity relative to the
Sun which can provide significant ΔV changes that may not
be possible with conventional boosters
Missions to Mercury, for example, used or are using gravity
assist to augment propulsion to overcome the extremely
high gravitational pull from the Sun to reach the planet
Gravity Assist

Missions beyond Jupiter are also not possible without
gravity assist since boosters are not available that are
capable of taking interplanetary spacecraft beyond
Jupiter, depending on the spacecraft mass

Gravity assist was first used on the Mariner 10 mission
to Mercury, and has since been used for all missions to
the Giant Planets, with the exception of the Pioneer 10
which targeted only Jupiter
 Although the mission targeted Jupiter in a close
approach trajectory, the resulting boost was simply
a part of the observation objectives and not an
intentional boost to reach a more distant planet
Gravity Assist

Even the Galileo spacecraft that was launched to
Jupiter required a gravity assist because of its
huge mass

A close flyby of the Earth and Venus were used
in a sequence labeled VEEGA (Venus-Earth-Earth
Gravity Assist) since the launcher and upperstage booster were insufficient to get the
spacecraft to Jupiter

The Voyager II spacecraft was boosted to Jupiter,
then used gravity assists to get to Saturn,
Uranus, and Neptune, then outside of the solar
system
Gravity Assist
In graphic form, with three of the four Jovian planets' gravity
added, the solar system gravitational potential looks like
this, with colored arrows showing the energy to/from orbits
Gravity Assist
A plot of the escape velocity throughout the solar system
can be made by taking the square root of the absolute
value of the potential energy, as shown below
Gravity Assist
Voyager II spacecraft was boosted to Jupiter, with gravity
assists to get to Saturn, Uranus, and Neptune, then
outside the solar system which is depicted below
Synodic Period and
Interplanetary Launch
Opportunities
Launch Opportunities
The period that the position of a planet is repeated in the sky is
called the synodic period
The calculation of the synodic period is simple but has two forms.
One is for orbits inside the Earth's orbit, the other for outside.
To calculate the synodic period of Venus, for example, and the
launch opportunity cycle between Earth and Venus, use:
1/Psynodic = 1/PEarth - 1/PVenus
or Psynodic = 1/(1/PEarth - 1/PVenus ) = 1/(1 - 1/0.615) yr
= -1.60 yr = 584 days
Launch Opportunities
Similar calculations produce the synodic periods for the other
planets with respect to Earth. The same calculations can be
made for relative position periods of any two planets
The practical application of the synodic period is that it provides the
frequency of launch opportunities from one planet to another









Mercury
Venus
Mars
Jupiter
Saturn
Uranus
Neptune
Pluto
Moon
116 days
584 days (19.2 mo)
780 days (25.6 mo)
399 days (13.1 mo)
378 days
370 days
367 days
367 days
30 days
Launch Opportunities
Launch opportunities from Earth to the planets are dictated primarily
by the synodic period and the phase position between the two
planets

Since the Earth is in a different orbital plane than all other
planets, the alignment of the intersection of the planes (line of
nodes) and the phase between the planets is an important
consideration in launch timing

In addition, none of the planets including Earth are in circular
orbits, making the optimum departure and arrival points a
compromise between three primary variables. Those are:
1. Orbit phase between launch and arrival planets
2. Line of nodes of the two planetary orbit planes
3. Position in orbit of the two planets
Launch Opportunities
Finding the minimum time or a minimum in transfer
energy of an actual transfer is a compromise
from the ideal orbits - circular and coplanar –
using actual orbits

The actual planetary orbit variables generate a
range of solutions that are plotted for ease in
planning and interpretation

The plots are called pork chop plots because of
their shape and have two general best solutions
corresponding to the Type I and Type II
transfers

From missions to Mars, the Type I would be
optimal for manned missions because of their
shortest transfer period

The Type II solution would be optimal for cargo
or instrumentation missions since the energy is
lower than the Type I solutions
Launch Opportunities
On the right is a graph of
the Type I and Type II
flight limitations for
the 2005 opportunity
to Mars
The minimum energy is
shown in the closed
concentric shapes,
with the lowest for
Type II missions
centered near 12
km2/s2 that
corresponds to a
flight duration of
approximately 370
days
Launch Opportunities
A plot of the
future 2.1 year
flight
opportunities
for Mars is
shown on the
right for Type II
(cargo)
missions.
Flight energy
(C3 is total
energy in
km2/s2) is listed
in the top series
and time of
flight shown on
the bottom
Mission to Mars
Mission to Mars
A quick and dirty estimate of the flight time to Mars and
back using 2-dimensional orbits begins with a simple
Hohmann transfer calculation

Transfer orbit semimajor axis is simply the average of
the Earth and Mars orbits, or (1+ 1.52)/2 AU = 1.26 AU

The corresponding transfer period is 1.263/2 years =
1.41 years for the complete orbit

The actual transfer period is one-half of this, or 0.71
years (8.5 months) each way
Mission to Mars
A Hohmann transfer requires
that the target planet be at
a position 180o from the
point of spacecraft launch
from Earth, therefore the
launch can only take place
during the correct
alignment of Mars and
Earth for the spacecraft to
reach the target planet
Mars which repeats with
the 25.6 month (2.13 yr)
synodic period
Mission to Mars
Phasing and synodic period
To calculate the approximate time of flight to Mars and back and the
time needed for the two planets to realign correctly for the return
trip, begin by finding the mean motion between the two planets
in their orbit around the Sun in degrees/day, or similar units

The Earth's mean motion is simply 360o/year x 1 year/365.24days,
or 0.986 deg/day

For Mars, this is 360 degrees/1.88 years x 1 year/365.24 days or
0.524 deg/day

Next, calculate the transfer period flight time to Mars using the
circular orbit approximation. This was calculated earlier with a
Hohmann transfer period of 0.71 years, or 259.3 days
Mission to Mars
Phasing and synodic period

The next step is to find how far Mars will travel in its orbit during
the spacecraft transfer period from Earth to Mars. This is just
Mars' mean motion times the transfer period, or 259.3 days x
0.524 deg/day = 135.87o

Mars must arrive at the 180o position during the transfer orbit;
therefore, the spacecraft, which is launched from Earth at the 0o
point, would be behind Mars by 180o - 135.87o, or 44.13o
(alternatively, Mars would be 44.13o ahead of Earth)

This alignment can be approximated from the Earth's and Mars'
orbital elements, being careful to not confuse the 2-dimensional
exercise with the 3-dimensional orbital elements
Mission to Mars
Phase angles for
the Earth-MarsEarth
interplanetary
mission
Mission to Mars
Phasing and synodic period

To calculate the time needed for the phase difference for the two
planets to move into a correct alignment for the return trip, use
the difference in mean motion between the two planets

This relative motion is just the difference in the mean motion of
the two planets, or 0.986 deg/day - 0.524 deg/day = 0.462 deg/day

During the spacecraft transfer that took it to Mars, the Earth has
traveled 255.66o in its orbit. The difference in positions would
therefore be 255.66o - 180o, or 75.66o, with the Earth ahead of
Mars by this value
Mission to Mars
Phasing and synodic period

For the return trip, Mars must be ahead of the Earth by the same
75.66 degrees (the launch from Mars can be visualized at the 0o
position with the Earth-arrival 180o from that launch position,
which would begin -255.66o from that point, hence the angular
difference between the Earth and Mars would be 255.66o -180o, or
75.66o)

For an estimate of the time required for the Earth to catch up to
this position, the angle needed is just 360o minus the current
lead of 75.66o minus the lead angle of 75.66o

This is a total of 208.68o (360o - 75.66o - 75.66o). At a closing rate
of 0.462 deg/day, this makes the return realignment period =
208.68o/0.462o/day = 451.69 day = 1.24 years
Mission to Mars
The important number here is the turnaround or
realignment period of 1.24 years. With a transfer
period of 0.71 years (times two for return travel time),
the total mission time would be 2.7 years for this 2dimensional, circular orbit, Hohmann transfer
approximation
Although the transfer orbit period can be decreased by
increasing vehicle propulsion thrust, the realignment
period is a critical element that can be as long or
longer than the travel time to and from Mars
Mission to Mars
A nearly three year minimum mission period to Mars and
return, of which nearly 1½ years is in zero-g transit,
means that the critical human space flight exposure
problems must be researched and solved before
humans can reach Mars
If the total mission duration were reduced to a minimum
with an immediate return - a turnaround time of zero
duration using a high-thrust, Type I direct transfer
orbit - the crews would encounter reduced space
exposure, but could contribute little, if anything, to a
Mars exploration mission
Finis