Basic Orbital Mechanics for the Operation of the ISS
Since the beginning of humankind, people have been scanning the skies to follow the celestial
stars. Celestial mechanics is the study of the motion of the planets, moons, and stars in the sky.
Johannes Kepler’s laws of planetary motion were first written in 1609 to describe the orbits of
celestial bodies around the sun. Almost a century later, Sir Isaac Newton (1642-1727) set down
the roots for orbital mechanics, which involve detailed mathematical descriptions of orbits as
well as the ability to change the altitude, velocity and periods of manmade bodies in orbit. It is
on the shoulders of Kepler’s and Newton’s work that modern day orbital mechanics is built.
What is an Orbit?
An orbit is the movement of a body, such as a planet or spacecraft, around another usually much
larger object, such as the sun or earth. Kepler found that the planetary orbits around the sun are
elliptical. Newton found that orbiting bodies can follow the shape of the conical shapes cut from
a right circular conical cone as pictured below.
Exhibit 1. Conical Shapes
The circle is a special case of an ellipse. The circle is cut parallel to the circular base of the cone,
while the ellipse is not parallel to the base. The ellipse is the shape of orbits of planets and
spacecraft, such as the ISS. If you wanted to fly by a planet or escape from it, the shape you
would use is a hyperbola. In the real world the parabola does not exist, because it involves flying
an infinite distance from a planet. In algebra, the circle, ellipse, parabola, and hyperbola all have
mathematical formulas that describe them. Do you know what they are?
How to Describe an Elliptical Orbit
There are two assumptions that you must keep in mind when describing elliptical orbits. The
first is that pictures are not drawn to scale. This is because the differences in the magnitude of
the distances can be impossible to depict. For example, the ISS travels only 160 nautical miles
above the surface of the earth. The diameter of the earth is 6887.8 nautical miles. If a drawing
was to scale, the ISS would appear orbiting at the surface of the earth. The other assumption is
that the directions of the orbits are counterclockwise.
An elliptical orbit can be described by three measurements. They are the lengths of its major and
minor axis, and its eccentricity. The length of the major axis is the furthest distance between two
points of the orbit that goes through the center of the orbit. The length of the minor axis is the
closest distance between two points of the orbit that goes through the center of the orbit. In
orbital mechanics instead of the size of the entire length of the major axis, the value of half the
axis called the semi-major axis is given. For example, if the major axis is 10 nautical miles the
length of the semi-major axis is 5 nautical miles. One nautical mile is approximately equal to
6076.12 feet. Nautical miles are used by nations to measure air and sea travel. There are 5280
feet in a standard mile.
Exhibit 2. Major and Minor Axis of an Ellipse
Major Axis
Minor Axis
Eccentricity measures the shape of the orbit. A circle has the eccentricity value of 0. Ellipses
have eccentricity values between 0 and 1. The closer an eccentricity value is toward zero the
more the ellipse looks like a circle.
Exhibit 3. Eccentricity of an Ellipse
Circle eccentricity = 0
Ellipse eccentricity of 0.88
Ellipse eccentricity of 0.99
How to Mathematically Describe the Location of the ISS in Orbit
There are six quantities that mathematically describe an orbit and a spacecraft’s location in that
orbit around Earth. These orbital elements are semi-major axis (a), eccentricity (e), inclination
(i), argument of Perigee (little omega w), time of perigee passage (T), and the right ascension of
the ascending node (big omega Ω).
Semi-major axis (a) determines the size of the orbit. For an object orbiting the Earth, the
closest point of the orbit to Earth on the major axis is called perigee. The farthest point of the
orbit from the Earth is called the apogee. The line drawn through the center of the orbit from the
perigee to the apogee is the major axis and half of it is called the semi-major axis that is represented by
the letter “a”.
Exhibit 4. Semi-Major Axis of an Ellipse
perigee
apogee
semi-major axis “a”
Eccentricity (e) determines the shape of the orbit. For an ellipse the eccentricity values are between 0
and 1. An eccentricity of 0 is a circle and an eccentricity of 1 is a parabola.
Inclination (i) determines the angle between the equatorial plane (the plane that goes through the equator
of Earth) and the orbital plane (the plane that the orbit is within). The ISS travels at an inclination of 51.6
degrees to the equatorial plane of Earth.
Exhibit 5. Inclination
Argument of perigee (w) is the angle between the ascending node and the perigee of the orbit.
When a spacecraft is in orbit around the Earth there is a point in the orbit where it crosses Earth’s
equator from the Southern to the Northern hemisphere. That point is called the ascending node.
The descending node is where the spacecraft in orbit around the Earth crosses from the Northern
to the Southern hemispheres. A line drawn between the nodes is called the line of nodes. The
figure below shows the argument of perigee (w) formed between the ascending node and the
point of perigee of a spacecraft. This measurement determines the placement of the orbit in
relation to the Earth. If the argument of perigee is large, then Earth is situated toward the
descending node of the orbit. If the argument of perigee is small then Earth is situated toward
the ascending node of the orbit. This can be seen by placing a hula hoop around a ball. The
argument of perigee tells you how the ball or Earth is situated within the hula hoop or orbit.
Exhibit 6. Argument of Perigee
Ascending Node
Time of perigee passage (T) is the time elapsed since the spacecraft has passed the point of
Perigee. This time is used to locate the spacecraft in an orbit when the velocity of the spacecraft
is known. For example, if a spacecraft is travelling at 1.77 nautical miles per minute and “T” is 5
minutes, then multiplying 1.77 times 5 gives you the distance the spacecraft is from perigee,
which in this example is 8.85 nautical miles.
Exhibit 7. Time of Perigee Passage
Perigee
The right ascension of the ascending node (Ω) locates where the orbit crosses Earth’s equator.
This measurement is essential to pin point the orbit in a 3 dimensional space around the Earth.
The measurement is found by measuring the angle between the x axis of the Earth’s plane and
the orbit’s ascending node. This can be demonstrated by a hula hoop to represent the orbit of a
spacecraft and a ball to represent Earth. The hula hoop may have a given inclination around
Earth but also the placement of the hula hoop in relation to the equatorial plane is important to
describe. The orbit may be encompassing just a small section of Earth, which would give a
small value for Ω or it may be toward the back of Earth giving a larger Ω value. The figure
below demonstrates this measurement.
Exhibit 8. The Right Ascension of the Ascending Node
With the accurate measurement of the above mentioned 6 quantities a description of an orbit and
where a spacecraft is in that orbit can be determined.
What do Burns do for Spacecraft?
A burn is the ignition of a rocket on the ISS. Rockets are used to change the altitude, velocity, or
orientation of the ISS’s orbit. The affect of burns on the ISS depends on the direction that the ISS
is moving in relation to the direction of the ignited rocket. The types of burns are postigrade,
retrograde, radial in, and radial out.
Postigrade burn occurs along the direction that the spacecraft is moving. The burn therefore
increases the kinetic energy of the spacecraft. However, the assumption is made that the burn is
instantaneous (even though it might take a few minutes to occur). This assumption means that
the potential energy of the spacecraft does not change at the burn point. The spacecraft remains
at the same altitude, since the spacecraft does not travel unless time elapses. With the increase in
kinetic energy comes a corresponding increase in mechanical energy. When mechanical energy
is increased the orbit becomes larger. This occurs at every point in the orbit except at the point
of the burn. The change in altitude of the orbit is directly proportional to the increase in velocity.
Postigrade burns give the maximum increase in velocity resulting in the maximum increase in
orbital altitude.
Retrograde burns have the opposite effect as postigrade burns. The burn occurs along the
opposite direction that the spacecraft is moving. The burn therefore decreases the kinetic energy
of the spacecraft. The assumption that the burn is instantaneous is still made, which means the
potential energy of the spacecraft does not change at the burn point. With the decrease in kinetic
energy comes a corresponding decrease in mechanical energy. When mechanical energy is
decreased the orbit becomes smaller. This occurs at every point in the orbit except at the point of
the burn. The change in altitude of the orbit is directly proportional to the decrease in velocity.
Retrograde burns give the maximum decrease in orbital altitude.
Exhibit 9. Posigrade Burn
Posigrade Burn
Initial Orbit
Final Orbit
Exhibit 10. Retrograde Burn
Retrograde Burn
Radial In Burns occurs perpendicular to the spacecraft’s direction and in toward the Earth. It
does not significantly change the orbiting velocity of the spacecraft, unless it is very large.
However it does change the eccentricity and shifts the perigee and apogee of the orbit. A radial
in burn that occurs at perigee will shift the point of perigee ahead of the spacecraft so that the
spacecraft after the burn is heading toward its new perigee. A radial in burn also increases the
eccentricity of the orbit.
Exhibit 11. Radial In Burn
Radial Out Burns occur perpendicular to the velocity of the spacecraft and away from the
Earth. It does not significantly change the velocity of the spacecraft, unless it is very large.
However it does change the eccentricity and the shifts the perigee and apogee of the orbit. A
radial out burn that occurs at perigee will shift the point of perigee behind the spacecraft so that
the spacecraft after the burn is heading toward its new apogee. A radial out burn also decreases
the eccentricity of the orbit.
As of July 6, 2010 the following is the orbital information of the ISS:
ISS Orbit (as of this morning, 7:32am EDT [= epoch])
Mean altitude – 352.8 km
Apogee height – 359.6 km
Perigee height – 345.9 km
Period -- 91.59 min.
Inclination (to Equator) -- 51.65 deg
Eccentricity -- 0.0010135
Solar Beta Angle -- 63.5 deg (magnitude decreasing)
Orbits per 24-hr. day -- 15.72
Mean altitude loss in the last 24 hours – 158 m
Revolutions since FGB/Zarya launch (Nov. 98) – 66,556
From website http://www.spaceref.com/news/viewsr.html?pid=34399