Orbit determination must take into account that the apparent celestial motion of the body is influenced by the observer's own motion. For instance, an observer on Earth tracking an asteroid must take into account the motion of the Earth around the
Sun, the rotation of the Earth, and the observer's local latitude and longitude, as these affect the apparent position of the body. A key observation is that (to a close approximation) all objects move in orbits that are
conic sections, with the attracting body (such as the Sun or the Earth) in the
prime focus, and that the orbit lies in a fixed plane.
Vectors drawn from the attracting body to the body at different points in time will all lie in the
orbital plane. If the position and velocity relative to the observer are available (as is the case with radar observations), these observational data can be adjusted by the known position and velocity of the observer relative to the attracting body at the times of observation. This yields the position and velocity with respect to the attracting body. If two such observations are available, along with the time difference between them, the orbit can be determined using Lambert's method, invented in the 18th century. See
Lambert's problem for details. Even if no distance information is available, an orbit can still be determined if three or more observations of the body's right ascension and declination have been made.
Gauss's method, made famous in his 1801 "recovery" of the first
lost minor planet,
Ceres, has been subsequently polished. One use is in the determination of asteroid masses via the
dynamic method. In this procedure Gauss's method is used twice, both before and after a close interaction between two asteroids. After both orbits have been determined the mass of one or both of the asteroids can be worked out.
Orbit determination from a state vector The basic orbit determination task is to determine the classical
orbital elements or
Keplerian elements, a, e, i, \Omega, \omega, \nu, from the
orbital state vectors [\vec{r}, \vec{v}], of an orbiting body with respect to the
reference frame of its central body. The central bodies are the sources of the gravitational forces, like the Sun, Earth, Moon and other planets. The orbiting bodies, on the other hand, include planets around the Sun, artificial satellites around the Earth, and spacecraft around planets. Newton's
laws of motion will explain the trajectory of an orbiting body, known as
Keplerian orbit. The steps of orbit determination from one state vector are summarized as follows: \vec{h}: specific angular momentum of the orbiting body. --> • Compute the
specific angular momentum \vec{h} of the orbiting body from its state vector: \vec{h} = \vec{r} \times \vec{v} = \left| \vec{h} \right| \vec{k} = h\vec{k}, where \vec{k} is the unit vector of the z-axis of the orbital plane. The specific angular momentum is a constant vector for an orbiting body, with its direction perpendicular to the orbital plane of the orbiting body. • Compute the
ascending node vector \vec{n} from \vec{h}, with \vec{K} representing the unit vector of the Z-axis of the reference plane, which is perpendicular to the reference plane of the central body: \vec{n} = \vec{K} \times \vec{h}. The ascending node vector is a vector pointing from the central body to the
ascending node of the orbital plane of the orbiting body. Since the line of ascending node is the line of intersection between the orbital plane and the reference plane, it is perpendicular to both the normal vectors of the reference plane (\vec{K}) and the orbital plane (\vec{k} or \vec{h}). Therefore, the ascending node vector can be defined by the
cross product of these two vectors. • Compute the
eccentricity vector \vec{e} of the orbit. The eccentricity vector has the magnitude of the
eccentricity, e, of the orbit, and points to the direction of the
periapsis of the orbit. This direction is often defined as the x-axis of the orbital plane and has a unit vector \vec{i}. According to the law of motion, it can be expressed as: \begin{align} \vec{e} &= {\vec{v}\times\vec{h}\over{\mu}} - {\vec{r}\over{\left|\vec{r}\right|}} = e \vec{i}\\ &= \left ( {{\left |\vec{v} \right |}^2 \over {\mu} }- {1 \over{\left|\vec{r}\right|}} \right ) \vec{r} - {\vec{r} \cdot \vec{v} \over{\mu}} \vec{v} \\ &= \frac{1}{\mu} \left[ \left( {{\left |\vec{v} \right |}^2 }- {\mu \over{\left|\vec{r}\right|}} \right ) \vec{r} - {(\vec{r} \cdot \vec{v})} \vec{v} \right] \end{align} e = \left| \vec{e} \right| where \mu = GM is the
standard gravitational parameter for the central body of mass M, and G is the
universal gravitational constant. • Compute the
semi-latus rectum p of the orbit, and its semi-major axis a (if it is not a
parabolic orbit, where e = 1 and a is undefined or defined as infinity): p = \frac{h^2}{\mu} = a (1-e^2) a = \frac{p}{1-e^2}, (if e \ne 1). • Compute the
inclination i of the orbital plane with respect to the reference plane: \begin{align} \cos(i) &= \frac{\vec{K}\cdot\vec{h}}{h} = \frac{h_K}{h} \\ \Rightarrow i &= \arccos\left(\frac{\vec{K}\cdot\vec{h}}{h}\right), & i \in [0,180^\circ], \end{align} where h_K is the Z-coordinate of \vec{h} when it is projected to the reference frame. • Compute the
longitude of ascending node \Omega, which is the angle between the ascending line and the X-axis of the reference frame: \begin{align} \cos(\Omega) &= \frac{\vec{I}\cdot\vec{n}}{n} = \frac{n_I}{n} = \cos(360 -\Omega) \\ \Rightarrow \Omega &= \arccos\left(\frac{\vec{I}\cdot\vec{n}}{n}\right) = \Omega_0, \text{ or } \\ \Rightarrow \Omega &= 360^\circ - \Omega_0, \text{ if } n_J where n_I and n_J are the X- and Y- coordinates, respectively, of \vec{n}, in the reference frame. Notice that \cos(A)=\cos(-A)=\cos(360-A)=C, but \arccos(C) is defined only in [0,180] degrees. So \arccos(C) is ambiguous in that there are two angles, A and 360-A in [0,360], who have the same \cos value. It could actually return the angle A or 360 - A. Therefore, we have to make the judgment based on the sign of the Y-coordinate of the vector in the plane where the angle is measured. In this case, n_J can be used for such judgment. • Compute the
argument of periapsis \omega, which is the angle between the periapsis and the ascending line: \begin{align} \cos(\omega) &= \frac{\vec{n}\cdot\vec{e}}{n e} = \cos(360 -\omega) \\ \Rightarrow \omega &= \arccos\left(\frac{\vec{n}\cdot\vec{e}}{n e}\right) = \omega_0, \text{ or } \\ \Rightarrow \omega &= 360^\circ - \omega_0, \text{ if } e_K where e_K is the Z-coordinate of \vec{e} in the reference frame. • Compute the
true anomaly \nu at epoch, which is the angle between the position vector and the periapsis at the particular time ('epoch') of observation: \begin{align} \cos(\nu) &= \frac{\vec{e}\cdot\vec{r}}{e r} = \cos(360 -\nu) \\ \Rightarrow \nu &= \arccos\left(\frac{\vec{e}\cdot\vec{r}}{e r}\right) = \nu_0, \text{ or } \\ \Rightarrow \nu &= 360^\circ - \nu_0, \text{ if } \vec{r}\cdot\vec{v} The sign of \vec{r}\cdot\vec{v} can be used to check the quadrant of \nu and correct the \arccos angle, because it has the same sign as the
fly-path angle \phi. And, the sign of the fly-path angle is always positive when \nu \in [0,180^\circ], and negative when \nu \in [180^\circ,360^\circ]. Both are related by h = r v \sin(90-\phi) and \vec{r}\cdot\vec{v} = r v \cos(90-\phi) = h \tan(\phi). • Optionally, we may compute the
argument of latitude u=\omega+\nu at epoch, which is the angle between the position vector and the ascending line at the particular time: \begin{align} \cos(u) &= \frac{\vec{n}\cdot\vec{r}}{n r} = \cos(360 -u) \\ \Rightarrow u &= \arccos\left(\frac{\vec{n}\cdot\vec{r}}{n r}\right) = u_0, \text{ or } \\ \Rightarrow u &= 360^\circ - u_0, \text{ if } r_K where r_K is the Z-coordinate of \vec{r} in the reference frame. ==References==