TR19 - Classical Elements from Position and Velocity - 1
David Reain Memorial Observatory
Founded 1988
154 Fairwood Drive, Keswick, ON, L4P 3Y3, 905-476-9467,
Henri M. van Bemmel B.Sc.(Hons.), B. Ed. - Director
21 October 2000
(revised 23 August 2002)
(revised 01 December 2002)
Technical Response 19
Issue
Classical Elements from Position and Velocity
Original Author of Request
A. Ayers, MGCI
Response
I Background
When spacecraft are operated, there are two vectors, which completely describe their motion this is known as the state vector. The position vector and the velocity vector . These vectors are very useful for mission planners when they make course corrections, as it is the velocity vector that is being modified. However, the consequences to the orbit of the spacecraft are less apparent. It is therefore useful to have a method of converting the orbit described by and to one described by the six orbital elements as explained by van Bemmel (2000).
Spacecraft operators must use the elements to compute long-range predictions of the destination, especially when objects are quite far apart. However, these values change over time since the solar systems has many objects inside its boundary all of which exert a small amount of influence on any object. Navigation people determine the location of the spacecraft by its position in the sky and solving a complicated process known as the method of Gauss that yields the orbital elements from at least 3 and preferably more positions and times. The speed is found by assessing how much the radio signal is Doppler shifted from what would be expected for a stationary object. So it is important to understand that the orbital elements are only valid for a few days or weeks around the date when they were found (computed from observations). When the location of the object is required with some precision then this window of accuracy narrows considerably. The location of a spacecraft is computed practically every time the spacecraft communicates with the Earth.
When course corrections are effected the velocity is changed in x, y and z by an amount that would give the desired values to the orbital elements. Thus is important for spacecraft operators to have the capability to switch back and forth between the two systems. The x, y, z format is useful for computations and the orbital element version is much more effective at helping astronomers and spacecraft dynamists to visualize the orbit.
Furthermore, it is important that the student of this material remember that spacecraft unlike those of fiction have very little fuel onboard and so their manoeuvres must be done with care using gravity’s ability to curve travel paths to the optimum. Mistakes can be catastrophic as the spacecraft may lack the fuel to effect an appropriate correction. In addition, spacecraft are often in proximity of planets and their satellites making the spectre of error substantially more grim for the mission. The reader is reminded of the demise of the two Mars probes, Mars Climate Orbiter and Mars Polar Lander in 2001. Work on space probes tends to be either wildly successful or disappointing with an almost total loss of the carefully planned mission.
II Frame of Reference
When coordinates are given, it is essential to indicate their frame of reference or these values will have little meaning. It is common to refer state vectors to the mean celestial equator and by definition; orbital elements are referred to the fundamental plane. Thus before a transformation from state vector to orbital elements can be effected an initial transformation between equatorial and ecliptic coordinates must be made. This is effected in the following manner.
Given a vector and its velocity in 3D then,
To find and the following computations are made
where is the obliquity of the ecliptic for the instant of the measurement. This value hovers around 23.5 degrees, but the orbital elements are sensitive to changes in this value so it needs to be sourced for the appropriate date.
The components for the velocity vector are transformed in the same manner. However, to conform to the standard units they must be divided by the Gaussian gravitational constant in these units. This gravitational constant is determined in astronomical units where distance is AU, speed AU day—1 and mass is in solar masses.
The value for the number is
k = 0.01720209895
III The Conic Parameters
Two of the orbital elements are parameters of the ellipse that defines the orbit. These are found by the following. The eccentricity vector is given by,
[1]
where:
[2]
[3]
[4]
and the eccentricity (e) is given my the magnitude of the vector hence
[5]
The value of the semi major axis (a) is given from
[6]
where is the modified mass or the product of G m1 m2 in consistent units with v and r.
III The Orientation Angles
The orbit has been given its size and shape from the work in section II, but the orientation of the orbit in space is equally important if proper predictions are going to be made about the future position of an object. There are three angles which perform this function. The computation of these is given below.
First the angular momentum vector is computed
[7]
Then the dot product of the unit fundamental z vector with will give the an opportunity to compute the inclination angle (i).
[8]
and [9]
then [10]
A similar situation exists for the longitude of the ascending node (). Recall that
[11]
Then by taking the dot product with the unit fundamental vector we get
[12]
hence
[13]
The argument of the perifocus(apse) is given by
[14]
The longitude of the perifocus () is given by
= + [14a]
This gives two conic parameters and three orientation angles. The only parameter left is the actual position on the orbit of the object.
All of this is simply an extension of high school geometry. The orbit is a plane so it has only two degrees of freedom and these are the eccentricity and the semi major axis. They control the shape and size of the ellipse respectively. So now this disc of an orbit needs to be oriented in space. It can rotate around any one of the three principal axes so three vectors are needed to fix its orientation in space. The vector h is by definition normal to the plane. The eccentricity vector points to the perifocal passage location. Finally the N vector, must be in the plane of the orbit since it takes two vectors to define a plane. However, choosing any random vector would not be helpful so the vector N is engineered so that it points in the direction of the ascending node. The longitude of the ascending node is then simply the angle between the e and N vectors a concept extracted from a senior algebra class.
IV The Mean Anomaly
Now that the only consideration is where on a given orbit is an object located. The mathematics can be reduced to two dimensions, since every unperturbed orbit is planar. Consider the diagram in Fig.1. Let x’ and y’ be the x and y axes referred to the orbital instead of the fundamental plane.
y’
B’
orbit
B
E
K C G x’
auxiliary circle
Fig.1 Elliptical Orbit Geometry
From Fig.1 the following new variables are defined
K – geometric centre of orbit
C – Dynamical centre of orbit
B – location of object
B’ – projection of B in the y direction on the auxiliary circle
E – Eccentric Anomaly
- True Anomaly (this is the Greek letter "nu", NOT a “vee”)
The vector CB is the range vector . The distance KC = ae.
The distance KG = a cos E. The distance GB’ = a sin E.
Recalling the general polar equation of a conic we have
= r(1+e cos )[15]
where is the true anomaly and,
[15a]
and
[16]
Then standard geometry gives,
x’ = a (cos E – e)[17]
y’ = b sin E
so it follows that
[18]
where the semi-minor axis can be found by
[19]
Thus E is determined and can be placed in the correct quadrant. The mean anomaly is defined by,
M = E – e sin E (E in rads)[20]
The Mean Longitude (L) is given by the addition of
L = M + [21]
It is recommended that these equations be committed to a spreadsheet so that test data can be used to verify that the constants used are suitable, before putting them into a computer program. There is a lot of non-intuitive material here.
V Sources
Boulet, D., Methods of Orbit Determination for the Microcomputer, Willman-Bell (1991), pg. 135ff.
Blumberg, R., and Boksenberg, A., Astronomical Almanac, US Government Printing (1994)
Seidelmann, P.K., Explanatory Supplement to the Astronomical Almanac, University Science Books, 1992
van Bemmel, H.M., Orbital Elements (Elliptical) TR19, DRMO Press (2000)
VI Distribution
Ayers, A., MGCI, 23 October 2000
AP Physics C 203, October 2002
DRMO Files, Hardcopy