Dynamic method
Encyclopedia
The dynamic method is a procedure for the determination of the mass
es of asteroid
s. The procedure gets its name from its use of the Newtonian
laws of the dynamics, or motion, of asteroids as they move around the solar system. The procedure works by taking multiple position measurements to determine the gravitational deflection caused when two or more asteroids move past each other. The method relies on the fact that the large number of known asteroids means they will occasionally move past one another at very close distances. If at least one of the two interacting bodies is large enough, its gravitational influence on the other can reveal its mass. The accuracy of the determined mass is limited by the precision and timing of the appropriate astrometric
observations being made to determine the gravitational deflection caused by a given interaction.
Because the method relies on detecting the amount of gravitational deflection induced during an interaction, the procedure works best for objects which will produce a large deflection in their interactions with other objects. This means that the procedure works best for large objects, but it can also be effectively applied to objects which have repeated close interactions with each other such as when the two objects are in orbital resonance
with one another. Regardless of the mass of the interacting objects, the amount of deflection will be greater if the objects approach nearer to each other and it will also be greater if the objects pass slowly, allowing more time for gravity to perturb the orbits of the two objects. For large enough asteroids this distance can be as large as ~0.1 AU, for less massive asteroids the conditions of the interaction would need to be correspondingly better.
between oppositely charged objects (so that the force if attractive rather than repulsive). When rewritten in the more familiar notation used in celestial mechanics deflection angle can be related to the eccentricity of the hyperbolic orbit of the smaller object relative to the larger one by the following formula:
Here
is the angle between the asymptote
s of the hyperbolic orbit of the small object relative to the large one, and
is the eccentricity of this orbit (which must be greater than 1 for a hyperbolic orbit).
A more sophisticated description using matrices
can be achieved by separating the observed objects position on the sky as a function of time into the sum of two components: one which is a result of the relative motion of the objects themselves, and the other the motion induced by the gravitational influence of the two bodies. The relative contributions of the two terms in the best fit of this equation onto the actual observations of the objects yields the objects masses.
Mass
Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...
es of asteroid
Asteroid
Asteroids are a class of small Solar System bodies in orbit around the Sun. They have also been called planetoids, especially the larger ones...
s. The procedure gets its name from its use of the Newtonian
Classical mechanics
In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...
laws of the dynamics, or motion, of asteroids as they move around the solar system. The procedure works by taking multiple position measurements to determine the gravitational deflection caused when two or more asteroids move past each other. The method relies on the fact that the large number of known asteroids means they will occasionally move past one another at very close distances. If at least one of the two interacting bodies is large enough, its gravitational influence on the other can reveal its mass. The accuracy of the determined mass is limited by the precision and timing of the appropriate astrometric
Astrometry
Astrometry is the branch of astronomy that involves precise measurements of the positions and movements of stars and other celestial bodies. The information obtained by astrometric measurements provides information on the kinematics and physical origin of our Solar System and our Galaxy, the Milky...
observations being made to determine the gravitational deflection caused by a given interaction.
Because the method relies on detecting the amount of gravitational deflection induced during an interaction, the procedure works best for objects which will produce a large deflection in their interactions with other objects. This means that the procedure works best for large objects, but it can also be effectively applied to objects which have repeated close interactions with each other such as when the two objects are in orbital resonance
Orbital resonance
In celestial mechanics, an orbital resonance occurs when two orbiting bodies exert a regular, periodic gravitational influence on each other, usually due to their orbital periods being related by a ratio of two small integers. Orbital resonances greatly enhance the mutual gravitational influence of...
with one another. Regardless of the mass of the interacting objects, the amount of deflection will be greater if the objects approach nearer to each other and it will also be greater if the objects pass slowly, allowing more time for gravity to perturb the orbits of the two objects. For large enough asteroids this distance can be as large as ~0.1 AU, for less massive asteroids the conditions of the interaction would need to be correspondingly better.
Mathematical analysis
The simplest way to describe the deflection of the asteroids is in the case where one object is significantly more massive than the other. In this case the equations of motion are the same as for that of Rutherford scatteringRutherford scattering
In physics, Rutherford scattering is a phenomenon that was explained by Ernest Rutherford in 1911, and led to the development of the Rutherford model of the atom, and eventually to the Bohr model. It is now exploited by the materials analytical technique Rutherford backscattering...
between oppositely charged objects (so that the force if attractive rather than repulsive). When rewritten in the more familiar notation used in celestial mechanics deflection angle can be related to the eccentricity of the hyperbolic orbit of the smaller object relative to the larger one by the following formula:
Here
is the angle between the asymptoteAsymptote
In analytic geometry, an asymptote of a curve is a line such that the distance between the curve and the line approaches zero as they tend to infinity. Some sources include the requirement that the curve may not cross the line infinitely often, but this is unusual for modern authors...
s of the hyperbolic orbit of the small object relative to the large one, and
is the eccentricity of this orbit (which must be greater than 1 for a hyperbolic orbit).A more sophisticated description using matrices
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
can be achieved by separating the observed objects position on the sky as a function of time into the sum of two components: one which is a result of the relative motion of the objects themselves, and the other the motion induced by the gravitational influence of the two bodies. The relative contributions of the two terms in the best fit of this equation onto the actual observations of the objects yields the objects masses.