Lecture 2.

Keplerian orbital elements

GENERAL DEFINITION

The elements of an orbit are the parameters

needed to specify that orbit uniquely.

The problem of satellite motion in space contains

three degrees of freedom (the 3 coordinates).

Therefore, each particular orbit is fully defined by

six quantities - the initial values of satellite’s

position and velocity (as 3 derivatives from 3

coordinates). For this reason, all sets of orbital

elements contain exactly six parameters.

Orbital elements specify the satellite’s position at a certain time

called the epoch. The elements are only accurate for a limited

period around the epoch.

EPOCH

In astronomy, an epoch is a moment in

time used as a reference point for some time-

varying astronomical quantity.

The semi-major axis fixes the size of satellite’s orbit. It is one half of the

major axis, and thus runs from the centre through a focus to the edge of the

ellipse; essentially, it is the radius of an orbit at the orbit's two most distant

points. For the special case of a circle, the semi-major axis is the radius.

SEMI-MAJOR AXIS

ECCENTRICITY

The eccentricity (ε) fixes the shape of satellite’s orbit. The eccentricity

tells you how flat the orbit is. This parameter indicates the deviation of

the orbit from a perfect circle. A circular orbit has an eccentricity of 0,

while a highly eccentric orbit is closer to (but always less than) 1.

INCLINATION

The orbit ellipse lies in a plane, and this plane forms an angle with

the plane of the equator. This angle is called the orbital inclination.

Inclination defines the orientation of the orbital plane.

LONGITUDE OF THE ASCENDING NODE

The satellite orbit cuts the equatorial plane at two points: the first,

called the descending node, where the satellite passes from the

northern hemisphere to the southern hemisphere, and the second,

called the ascending node, where the satellite passes from the

southern hemisphere to the northern hemisphere.

LONGITUDE OF THE ASCENDING NODE

The longitude of the ascending node is longitude of a point where

the satellite crosses the equatorial plane headed from South to North.

Two numbers orient the orbital plane in space. The first number is

inclination. The second one is longitude of the ascending node.

Satellite’s ground track is the path on the surface of the Earth

that lies directly below its orbit.

The ground track of a satellite can take a number of different

forms, depending on the values of the orbital elements,

parameters which define the size, shape, and orientation of the

satellite's orbit.

ARGUMENT OF PERIGEE

The argument of perigee (ω) is the angle formed between the two

directions to the point of perigee and the point of ascending node. So,

if the perigee would occur at the ascending node, the argument of

perigee would be 0.

TRUE ANNOMALY

The true anomaly (υ) tells you where the satellite is in its orbital path.

True anomaly is the angle measured in the direction of motion from

perigee to the satellite's position at epoch time.

TWO-LINE ELEMENT FORMAT

The North American Aerospace Defense Command (NORAD) developed

the Two-Line Element (TLE) format for transmitting satellite orbital

elements. This is a structured format intended to be very compact. TLE set

consists of two 69-character lines of data which can be used to determine

the position and velocity of the associated satellite. The only valid

characters are the numbers 0-9, the capital letters A-Z, the space, and the

plus and minus signs—no other characters are valid.

1 NNNNNC NNNNNAAA NNNNN.NNNNNNNN +.NNNNNNNN +NNNNN-N +NNNNN-N N NNNNN

2 NNNNN NNN.NNNN NNN.NNNN NNNNNNN NNN.NNNN NNN.NNNN NN.NNNNNNNNNNNNNN

International Space Station

1 25544U 98067A 04236.56031392 .00020137 00000-0 16538-3 0 5135

2 25544 51.6335 341.7760 0007976 126.2523 325.9359 15.70406856328903

Field Column Description

1.1 01 Line Number of Element Data

1.2 03-07 Satellite Number

1.3 08 Classification

1.4 10-11 International Designator (Last two digits of launch year)

1.5 12-14 International Designator (Launch number of the year)

1.6 15-17 International Designator (Piece of the launch)

1.7 19-20 Epoch Year (Last two digits of year)

1.8 21-32 Epoch (Day of the year and fractional portion of the day)

1.9 34-43 First Time Derivative of the Mean Motion

1.10 45-52 Second Time Derivative of Mean Motion (decimal point assumed)

1.11 54-61 BSTAR drag term (decimal point assumed)

1.12 63 Ephemeris type

1.13 65-68 Element number

1.14 69 Checksum (Modulo 10)

Table 1. Two-Line Element Set Format Definition, Line 1

Field Column Description

2.1 01 Line Number of Element Data

2.2 03-07 Satellite Number

2.3 09-16 Inclination [Degrees]

2.4 18-25 Right Ascension of the Ascending Node [Degrees]

2.5 27-33 Eccentricity (decimal point assumed)

2.6 35-42 Argument of Perigee [Degrees]

2.7 44-51 Mean Anomaly [Degrees]

2.8 53-63 Mean Motion [Revs per day]

2.9 64-68 Revolution number at epoch [Revs]

2.10 69 Checksum (Modulo 10)

Table 2. Two-Line Element Set Format Definition, Line 2