Lambert's problem

Lambert's problem
In celestial mechanics, Lambert's problem is concerned with the determination of an orbit from two position vectors and the time of flight, posed in the 18th century by Johann Heinrich Lambert and formally solved with mathematical proof by Joseph-Louis Lagrange. It has important applications in the areas of rendezvous, targeting, guidance, and preliminary orbit determination.
Suppose a body under the influence of a central gravitational force is observed to travel from point P1 on its conic trajectory, to a point P2 in a time T. The time of flight is related to other variables by Lambert's theorem, which states:
The transfer time of a body moving between two points on a conic trajectory is a function only of the sum of the distances of the two points from the origin of the force, the linear distance between the points, and the semimajor axis of the conic.
Stated another way, Lambert's problem is the boundary value problem for the differential equation
of the two-body problem when the mass of one body is infinitesimal; this subset of the two-body problem is known as the Kepler orbit.
The precise formulation of Lambert's problem is as follows:
Two different times and two position vectors are given.
Find the solution satisfying the differential equation above for which
Initial geometrical analysis
The three points
, the centre of attraction the point corresponding to vector the point corresponding to vector ,
form a triangle in the plane defined by the vectors and as illustrated in figure 1. The distance between the points and is , the distance between the points and is and the distance between the points and is . The value is positive or negative depending on which of the points and that is furthest away from the point . The geometrical problem to solve is to find all ellipses that go through the points and and have a focus at the point
The points , and define a hyperbola going through the point with foci at the points and . The point is either on the left or on the right branch of the hyperbola depending on the sign of . The semi-major axis of this hyperbola is and the eccentricity is . This hyperbola is illustrated in figure 2.
Relative the usual canonical coordinate system defined by the major and minor axis of the hyperbola its equation is
with
For any point on the same branch of the hyperbola as the difference between the distances to point and to point is
For any point on the other branch of the hyperbola corresponding relation is
i.e.
But this means that the points and both are on the ellipse having the focal points and and the semi-major axis
The ellipse corresponding to an arbitrary selected point is displayed in figure 3.
Solution for an assumed elliptic transfer orbit
First one separates the cases of having the orbital pole in the direction or in the direction . In the first case the transfer angle for the first passage through will be in the interval and in the second case it will be in the interval . Then will continue to pass through every orbital revolution.
In case is zero, i.e. and have opposite directions, all orbital planes containing corresponding line are equally adequate and the transfer angle for the first passage through will be .
For any with the triangle formed by , and are as in figure 1 with
and the semi-major axis (with sign!) of the hyperbola discussed above is
The eccentricity (with sign!) for the hyperbola is
and the semi-minor axis is
The coordinates of the point relative the canonical coordinate system for the hyperbola are (note that has the sign of )
where
Using the y-coordinate of the point on the other branch of the hyperbola as free parameter the x-coordinate of is (note that has the sign of )
The semi-major axis of the ellipse passing through the points and having the foci and is
The distance between the foci is
and the eccentricity is consequently
The true anomaly at point depends on the direction of motion, i.e. if is positive or negative. In both cases one has that
where
is the unit vector in the direction from to expressed in the canonical coordinates.
If is positive then
If is negative then
With
semi-major axis
eccentricity
initial true anomaly
being known functions of the parameter y the time for the true anomaly to increase with the amount is also a known function of y. If is in the range that can be obtained with an elliptic Kepler orbit corresponding y value can then be found using an iterative algorithm.
In the special case that (or very close) and the hyperbola with two branches deteriorates into one single line orthogonal to the line between and with the equation
Equations and are then replaced with
is replaced by
and is replaced by
Numerical example
Assume the following values for an Earth centered Kepler orbit
r1 = 10000 km
r2 = 16000 km
α = 100°
These are the numerical values that correspond to figures 1, 2, and 3.
Selecting the parameter y as 30000 km one gets a transfer time of 3072 seconds assuming the gravitational constant to be = 398603 km3/s2. Corresponding orbital elements are
semi-major axis = 23001 km
eccentricity = 0.566613
true anomaly at time t1 = −7.577°
true anomaly at time t2 = 92.423°
This y-value corresponds to Figure 3.
With
r1 = 10000 km
r2 = 16000 km
α = 260°
one gets the same ellipse with the opposite direction of motion, i.e.
true anomaly at time t1 = 7.577°
true anomaly at time t2 = 267.577° = 360° − 92.423°
and a transfer time of 31645 seconds.
The radial and tangential velocity components can then be computed with the formulas (see the Kepler orbit article)
The transfer times from P1 to P2 for other values of y are displayed in Figure 4.
Practical applications
The most typical use of this algorithm to solve Lambert's problem is certainly for the design of interplanetary missions. A spacecraft traveling from the Earth to for example Mars can in first approximation be considered to follow a heliocentric elliptic Kepler orbit from the position of the Earth at the time of launch to the position of Mars at the time of arrival. By comparing the initial and the final velocity vector of this heliocentric Kepler orbit with corresponding velocity vectors for the Earth and Mars a quite good estimate of the required launch energy and of the maneuvers needed for the capture at Mars can be obtained. This approach is often used in conjunction with the patched conic approximation.
This is also a method for orbit determination. If two positions of a spacecraft at different times are known with good precision (for example by GPS fix) the complete orbit can be derived with this algorithm, i.e. an interpolation and an extrapolation of these two position fixes is obtained.
Parametrization of the transfer trajectories
It is possible to parametrize all possible orbits passing through the two points and using a single parameter .
The semi-latus rectum is given by
The eccentricity vector is given by
where is the normal to the orbit. Two special values of exists
The extremal :
The that produces a parabola:
Open source code
From MATLAB central
PyKEP a Python library for space flight mechanics and astrodynamics (contains a Lambert's solver, implemented in C++ and exposed to python via boost python)
References
External links
Lambert's theorem through an affine lens. Paper by Alain Albouy containing a modern discussion of Lambert's problem and a historical timeline.
Revisiting Lambert's Problem. Paper by Dario Izzo containing an algorithm for providing an accurate guess for the householder iterative method that is as accurate as Gooding's Procedure while computationally more efficient.
Lambert's Theorem - A Complete Series Solution. Paper by James D. Thorne with a direct algebraic solution based on hypergeometric series reversion of all hyperbolic and elliptic cases of the Lambert Problem.
Orbits
Conic sections
Astrodynamics