Introduction to Orbital Mechanics

What Is an Orbit?

•Johannes Kepler discovered in 1600s that planet orbits form

ellipses, not circles.

•Satellites (natural or human-made) orbit Earth as an ellipse.

•Elliptical orbits remain fixed in space, and Earth spins under a

fixed satellite orbit.

A closed path around which a planet or satellite travels.

Graphic obtained from

Astronautics Primer by Jerry

Sellers.

What Is an Ellipse?

• An ellipse is the two-dimensional shape that is produced by a

plane fully intersecting a cone.

• Note that a plane intersecting the cone at a angle perpendicular to

the cone’s center line will form a special ellipse called a circle.

What Is an Ellipse?

A B

Ellipse has two focii instead of a

center

Sum of distances from focii is

constant

A+B = constant

Circle is simply an ellipse with both focii located at the same spot.

Circle is a set of points fixed (constant

distance) from a center point (focus)

A = constant

A

•Satellites orbit Earth with one focus at Earth’s center.

•The other focus is an empty point, which may or may not be within

Earth’s boundaries.

What Is an Ellipse?

• a defines ½ the major axis length

• b defines ½ the minor axis length

• c is the distance from the center of the ellipse to either focal point

• For a circle, a and b are equal to the radius, and both focal points are co-located at the center of the ellipse

Diverse Orbits

Basic Orbits

How Are Orbits Described?

Orbits are described by a set of parameters called

orbital elements (i.e., Keplerian elements).

The Keplerian element set consists of 6 parameters

(plus a time stamp):

• Two of these describe the size and shape of an orbit

• Three of these describe the orientation of the orbit in space

• One of these describes the location of the satellite

within the orbit

Eccentricity (e)

Eccentricity describes the roundness of an orbit. It describes the

shape of the ellipse in terms of how wide it is.

Semi-major axis, a

Semi-minor axis, b

Calculate the eccentricity of a circle.

Eccentricity can vary from 10 e

𝑒 = 1 −𝑏2

𝑎2

Eccentricity

This value is between 0 and 1 (for “closed” orbits).

Eccentricity of 0 means the orbit is circular.

An eccentricity of 1 or

greater means the orbit is not

closed. Such would be used

for interplanetary missions.

Satellites in these types of

orbits do not come back to

their starting point.

EccentricityValues between 0 and 1 mean the orbit is elliptical.

e = .74

e = .60

e = .4

e = 0

Beyond Eccentricity

Orbits may have the

same eccentricity

(e) but may be

different sizes.

There must be a

Keplerian element

which describes the

size of an orbit.

Semi-Major AxisMajor axis, 2a

Semi-major axis, a

Semi-major axis

a describes the

size of the

ellipse. It is half

of the largest

diameter (the

major axis) of the

orbit.Center

of

ellipse

The semi-major axis originates from the center of the orbit, but we

are located on Earth. This makes semi-major axes difficult for us to

visualize from our reference point.

Important Points on the Orbit

Perigee

Apogee

Apogee defines the point in an orbit that is farthest from Earth.

Perigee describes the point in an orbit that is closest to Earth.

“gee” suffix means Earthe.g. apoapsisand periapsis.Apogee altitude

Perigee

altitude

Apogee altitude is the distance between the surface of the Earth and

apogee.

Perigee altitude is the distance between the surface of the Earth and

perigee.

Apogee, Perigee, and Circular Orbits

• In circular orbit, apogee altitude and perigee altitude are the

same.

• Perfectly circular orbit has neither an apogee nor perigee and is

undefined.

• Perfectly circular orbits cannot be achieved.

• Generally circular orbits are described by their altitude.

•Semi-major axis rarely used to describe circular orbits.

Perigee

ApogeeApogee

altitudePerigee

altitude

For circular orbit

Apogee Altitude = Perigee

Altitude

Semi-Major Axis

(Altitude for circular orbits)

Semi-major axis is the only orbital parameter that determines the orbital period.

Translated as Kepler’s 3rd Law: The square of the period of a planet is

proportional to the cube of its mean distance from the Sun.

𝑇 = 𝑂𝑟𝑏𝑖𝑡𝑎𝑙 𝑃𝑒𝑟𝑖𝑜𝑑

𝑎 = 𝑆𝑒𝑚𝑖𝑀𝑎𝑗𝑜𝑟 𝐴𝑥𝑖𝑠𝜇 = 𝐺𝑟𝑎𝑣𝑖𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑙

𝑃𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟G = Universal Gravitation Constant(6.67x10-11 m3/kg*s2)𝑀 = 𝑀𝑎𝑠𝑠 𝑜𝑓 𝑐𝑒𝑛𝑡𝑟𝑎𝑙 𝑏𝑜𝑑𝑦

𝑇 = 2𝜋 ×𝑎32

𝜇= 2𝜋 ×

𝑎32

𝐺𝑀

Semi-Major Axis

Let’s Have a Race

𝑇 = 2𝜋 ×𝑎32

𝜇= 2𝜋 ×

𝑎32

𝐺𝑀

Semi-Major Axis

• These orbits all have

the same semi-major

axis (a), but their

eccentricities (e) and

their orientations

around Earth are

different.

• Observe the orbital

periods.

Describing the Orientation of the

Orbit in Space Orbits may have

identical sizes and

shapes (a and e), yet

they can vary in their

orientation in space.

Three additional

Keplerian elements

define this orientation:

• Inclination

• Right ascension of

the ascending node

• Argument of

perigee

Inclination (i)Inclination is the angle between the Earth’s equatorial plane and the

plane of the orbit. It describes the tilt of the orbit.

i = 5o

i = 25o

i = 45o

i = 75o

???

Which satellite will

complete one orbit first?

We interrupt our regularly scheduled presentation on inclination to

bring you important information regarding ground traces!

If a long string with a magic marker tied to the end of it were

hung from a satellite, the path which the magic marker would

trace over the ground is the ground trace. A ground trace is a

projection of the satellite’s orbit onto the Earth.The satellite

appears to move

westward on

(most)

conventional

orbits because the

Earth is rotating

eastward.

(More on this

later!)

Ground TracesAfter a full day, the ground trace of a satellite with an approximate 90

minute orbital period would look like this. Because the Earth is continually

rotating below the orbit of the satellite, the ground trace eventually spans

all longitudes.

Back to Inclination Inclination determines the

northern and southern latitude

limits over which the satellite

orbits. For example, a satellite

with a 45o inclination will have a

ground trace ranging from 45o

north to 45o south.

You can determine

the inclination of

an orbit simply by

examining its

ground trace.

Inclination

An orbit with an

inclination of 0

degrees is called an

equatorial orbit.

An orbit with an

inclination of 90

degrees is called a

polar orbit.

Inclination

A satellite in an

equatorial orbit will

pass directly over the

equator.

A satellite in polar

orbit will pass

over the entire

Earth.

What Do Ground Traces Reveal?

• Inclination is determined simply by noting the northern and

southern latitude limits of the ground trace.

• Orbital period can be determined using a simple calculation.

Based on what we have already learned about orbital parameters, we

can determine both inclination and orbital period from a ground trace.

1st pass, 0 degrees

longitude

2nd pass, 25 degrees

west longitude

Determining a Satellite’s Orbital

Period from its Ground Trace1. Recall that the orbit of a satellite remains fixed in space,

and the Earth rotates underneath it.

2. The westward regression of the ground trace is due to

the rotation of the Earth.

3. Determine how many minutes it takes for the Earth to

rotate one degree:

1440 minutes/360 degrees = 4min/degree

4. Determine how many degrees per pass the satellite’s

orbit regresses on consecutive orbits (equatorial crossing

is a common reference point). We’ll use 25 degrees as an

example.

5. How long did it take the Earth to rotate this many

degrees? That’s the period of the satellite.

25degrees * 4min/degree = 100 minutes

Right Ascension of the Ascending

Node (RAAN, W )Satellites may have identical eccentricities, semi-major axes, and

inclinations (e, a, and i) yet may still be oriented differently in space –

they can be “rotated” or “twisted” about the Earth in various ways.

Each satellite here starts

out above a different

longitude on the Earth.

However, longitude

can’t be used as a

reference point because

the Earth will rotate

underneath the orbits,

changing the reference

longitude on each

satellite pass.

RAAN

Right ascension of the

ascending node is the

angle measured along the

equatorial plane between

a vector pointing to a

fixed reference point in

space (the first point of

Aries, also known as the

vernal equinox) and the

point on the orbit where

the orbital motion is

from south to north

across the equator (this

point is called the

ascending node).

W = 0o

W = 30o

W = 60o

W = 90o

Argument of Perigee (w)

It is measured as

the angle from the

ascending node to

the perigee point

in the direction of

the satellite’s

motion.

Orbits may have the same e, a, I, and W, yet may still have different

orientations around the Earth. The location of their perigee point can

vary within the orbital plane.

w= 0o

w = 90o

w = 180o

w = 270o

Argument of perigee

describes the orientation

of the orbit within the

orbital plane (where is

apogee and where is

perigee?).

True Anomaly (u)After an orbit and its orientation have been thoroughly described,

there must be a way to describe the satellite’s position within an orbit

at any instant. True anomaly is the angle

between the perigee point

and the satellite’s location

(measured in the direction

of the satellite’s motion).

This value is constantly

changing as the satellite

moves in its orbit.

True anomaly is 0 degrees

at perigee, 180 degrees at

apogee.

Keplerian Elements in ReviewThe Keplerian element set consists of 6 parameters:

Two of these describe the size and shape of an orbit:

Three of these describe the orientation of the orbit in space:

•Eccentricity (e)

•Semi-major axis (a)

•Inclination (i)

•Right ascension of the ascending node (W)

•Argument of perigee (w)

One of these describes the location of the satellite within the orbit:

•True anomaly (u)

A time stamp, referred to as an “epoch,” must also be included when

providing a Keplerian element set. This is so that it is known WHEN this set

of values was accurate for the satellite or when the “snapshot” of the orbit

was taken.

Kepler’s LawsKepler’s 1st Law: Satellites will travel around Earth in elliptical paths with the

center of Earth at one of the foci.

Translated, this means

the speed of a satellite

changes as the

distance between it

and Earth changes. At

perigee a satellite is

moving its fastest; at

apogee, it is moving

its slowest.

Kepler’s 3rd Law: The period of an orbit (T) is related to its semi-major axis

(a) by: T2 = 4p2

m* a3

Kepler’s 2nd Law: A line drawn between Earth and a satellite will sweep out

equal areas during equal time periods anywhere along the orbit.

Time1

Time1

Special Orbit Types

The Keplerian element set chosen for any

given satellite is highly dependent on its

mission. Certain orbits are better suited

for certain missions.

LEO (Low Earth Orbit)•No specified minimum altitude

•Relatively close to the Earth (several hundred km)

•Short orbital periods ~90 minutes

•Many revolutions per day

•Limited swath areas

•What can the satellite view on Earth’s surface?

•All manned space missions (except lunar missions)

were LEO

•Many Earth-observing satellites

•Weather and imagery

•Why is this?

LEO (Low Earth Orbit)

Image is to scale showing International Space

Station height of orbit ~ 350 km

GEO (Geostationary)What’s in a name?

• Geostationary satellite remains over one

location on Earth

• Achieved by placing the satellite in a special

orbit where period exactly equals one day

• Altitude: roughly 36,000 km (22,200 miles)

• Inclination is exactly zero degrees

GEO (Geostationary)

GEO (Geostationary)

• GEO satellite ONLY exists directly above equator

AKA sub-satellite longitude

• Geostationary satellite can see ~70 degrees north

and south of the equator

• Geostationary satellites mainly used for

communications or “permanent relay station” in

space

GEO• Only one altitude with a period of 24 hours

• All geostationary orbits are in a “ring” around the Earth

• The ring is called the geostationary belt

• Geostationary belt is a limited resource

• When a “Geobird” dies, it

• Must be removed from its slot in the geobelt

• Must make room for another satellite

• Is usually boosted to a slightly higher orbit

GEO• Difficult to orbit exactly 24-hour period and zero inclination

• Orbits typically have slight inclination

• Satellites drift slightly north and south of equator

• Slight east or west drift due to imperfect period

• Small orbit-adjustment burns performed (called station-keeps)

• Satellites with 24 hour period and non-zero inclination are

called geosynchronous

• Geostationary and geosynchronous often interchanged

Real Geobelt

•Ground traces projected out to geostationary altitude

•Large inclinations (figure 8) run out of station-keeping fuel

•Sine wave orbits are being drifted to new location

•Orbit color participation in data sharing program

GEO

A Short Lesson in Urban Navigation

How can you tell what direction is south if you’re lost

in the middle of an urban area in the United States

with no compass or GPS receiver? It is too cloudy to

see the sun, and there is no moss growing anywhere!

Think about what you have learned about orbits.

Q.

Just look for a building/house with a TV satellite dish.

Since geostationary satellites can only “hover” above

the equator, all dishes in the northern hemisphere that

are communicating with geostationary satellites must

be pointing toward the south.

A.

Molniya (Moly)Using geostationary satellites for communications posed severe

problems for Russia since so much of their land mass is near or

north of 70 degrees in latitude.

To overcome this problem, they created a type of orbit, a Molniya

orbit, to allow for long-term communications over their northern land

mass.

Molniya

•Highly inclined and highly elliptical orbit

•High inclination covers northern Russia

•High eccentricity

-- Large apogee altitude

-- Very slow velocity at apogee

•If apogee is over Russia, then satellite hangs over

Russia (Kepler’s 2nd Law)

Molniya

MolniyaThe Molniya ground trace looks quite different from most conventional

ground traces. It clearly illustrates the “hang time” of the satellite over

the Russian Federation.

PolarBecause the inclination of a polar orbit is 90 degrees, a satellite in

polar orbit will eventually pass over every part of the world. This

makes polar orbits well-suited for satellites gathering information

about the Earth, such as weather satellites.

A special type of polar orbit called a Sun-synchronous orbit passes

over the same part of the Earth at roughly the same local time

every day. Why might this be useful?

ConstellationsA single satellite is often insufficient to perform a particular mission.

Groups of satellites in various orbits will work together to accomplish

the mission. Such groupings of satellites are called constellations. GPS

(Global Positioning System) is one such example.

Now That You Know the Basics

1. If Norway wanted to obtain satellite imagery of

all of its major urban areas, what type of orbit

would be appropriate?

2. Could researchers at McMurdo Station in

Antarctica use geostationary satellites for

communications?

Use your new understanding of orbital mechanics to

answer the following questions.

Now That You Know the Basics

1. If Norway wanted to obtain satellite imagery of

all of its major urban areas, what type of orbit

would be appropriate?

For the Norwegian satellites, the satellite should

have a high inclination (since Norway is in the

northern latitude region) and low altitude,

circular orbit. The inclination is approximately

70-80 degrees with an altitude of several hundred

km.

Now That You Know the Basics

2. Could researchers at McMurdo Station in

Antarctica use geostationary satellites for

communications?

No, because the latitude of Antarctica is too far

south. However, options do exist.

Now That You Know the Basics

2. Could researchers at McMurdo Station in

Antarctica use geostationary satellites for

communications?

Option #1

Old geostationary satellites that have acquired

significant inclination (i.e., >10 degrees) can often

provide continuous communications for >6 hours

a day when they are in the southern portion of

their figure 8 ground trace.

Now That You Know the Basics

2. Could researchers at McMurdo Station in

Antarctica use geostationary satellites for

communications?

Option #2

Researchers in Antarctica can also communicate

using low Earth orbiting communication

constellations such as Iridium.

Optional Analysis Tool

STK software can be used to explore, create, and

analyze orbits in greater detail.

References

Analytical Graphics, Inc. (AGI). (2010). Educational

resources. Retrieved from

http://www.stk.com/resources/academic-

resources/for-students/access-resources.aspx

National Aeronautics and Space Administration

(NASA). (2009). Basics of flight. Retrieved from

http://www2.jpl.nasa.gov/basics/bsf3-1.php