![]() ![]() Note that the radius r and the angular velocity are perpendicular to each other so that their product is equal to the magnitude of their vector product. Note that the angular component is proportional to the rate of change of area so that The velocity is in the plane of the ellipse and can be divided into radial and angular components: So the time rate of change of the area swept out is Integrating over r from the focus outward gives Law of Areas: Considering the area of an elliptical orbit, an infinitesemal area element can be expressed as The center of mass will be at one focus of each ellipse. Setting these two expressions equal to each other gives an expression for the angular momentum Lįor a bound binary orbit, each object will follow an elliptical orbit about the center of mass of the system. For e<1 this is the form of an ellipse and can be expressed in terms of the semimajor axis a. This form for the equation of a conic section expresses it in terms of the angular momentum L, the eccentricity e, and the angle θ measured from the perihelion. ![]() The final step in showing that this is the expression for a conic section orbit is to define e=D/GMμ so that the radius r becomes Using the final expression for angular momentum L from above, this can be put in the form To introduce the radius vector r into this expression the following procedure with a vector identity is followed. Where we can argue that the vector constant of integration D must be in the plane of the orbit since the other two quantities lie in that plane. Now integrating the two differential quantities gives ![]() The rule for differentiation of a product is used along with the fact that dL/dt=0 (conservation of angular momentum). Where the facts that the scalar product of two unit vectors is equal to 1 and the scalar product of the unit vector and its derivative is equal to 0 are used in addition to the vector identity.Ĭontinuing a strategy directed toward finding an expression for radius vector r in terms of the angular momentum, the above expression is modified so that it is an equality of two derivatives. As a strategy for obtaining an expression for r, the following evaluation is made: The acceleration of the reduced mass is given by Note that one term drops out of the vector product expression because it contains the vector product of the unit vector and itself and is therefore zero. In this expression, the unit vectors in the r direction have been introduced because both the magnitude of r and its direction can change. The angular momentum L of the system can be expressed as The determination that the motion of both masses in a bound binary system execute elliptical orbits about a focus point at the center of mass of the system requires the analysis of the force and angular momentum of the system. The magnitude of r is the same as the relative distance r in the development above. The orbit of one of the masses in a binary system can be described as the motion of the reduced mass around a point at relative distance r where the total mass is placed. The conservation of angular momentum leads to both Kepler's Law of Orbits and Law of Areas. This establishes the fact that the angular momentum L is conserved for any system of two orbiting masses acted upon by a central force. Using the expression for angular momentum L obtained above: This requires the expression for the derivative of a product of functions. This is done by showing that the derivative of the angular momentum is zero for the case where the force of attraction acts along the line between the two bodies. This provides the necessary framework for showing that angular momentum is conserved for an orbiting planet or a member of a binary star system. Starting with the individual angular momenta, the angular momentum of the system L can be expressed as follows: The angular momentum of the two body system can be expressed in terms of their relative velocity and the reduced mass of the system. Kepler's Laws depend upon the principle of conservation of angular momentum, and since these are inherently vector quantities, the angular momentum is expressed in terms of vector products. The individual vector coordinates of the masses can be expressed in terms of the reduced mass: The vector distance between the two masses isĪnd the motion of one of the masses relative to the other uses the concept of reduced mass: The motion of a binary system can be described relative to the center of mass of the system. Law of Orbits: two masses orbiting each other in bound orbits under the influence of the law of gravity will follow elliptical orbits about the center of mass of the two-body system. Kepler's Laws Developing Kepler's Law of Orbits ![]()
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