Kepler`s laws of planetary motion can be formulated as follows: Since angular momentum is constant, the surface velocity must also be constant. This is exactly Kepler`s second law. As with Kepler`s first law, Newton showed that this was a natural consequence of his law of gravity. In classical mechanics, surface velocity (also called sector velocity or sector velocity) is a pseudovector whose length corresponds to the rate of change at which a surface is swept by a particle as it moves along a curve. [clarification needed] In the figure opposite, suppose that a particle moves along the blue curve. At some time t, the particle is at point B, and shortly thereafter, at time t + Δt, the particle has moved to point C. The area scanned by the particle is shaded green in the figure, bounded by the straight line segments AB and AC and the curve along which the particle moves. The size of the area velocity (i.e. the area velocity) is the area of this region divided by the time interval Δt at the limit at which Δt becomes extremely small. The direction of the vector is postulated perpendicular to the plane containing the position of the particle and the velocity vectors, according to a convention known as the right-handed ruler. In order to move to the transfer ellipse and then move away again, we need to know each circular velocity and the orbital transfer velocities in perihelion and aphelion. The required speed gain is simply the difference between the circular path speed and the elliptical orbital velocity at each point.
We can find the circular velocities from equation 13.7. To determine the velocities of the ellipse, we assert without proof (since this is beyond the scope of this course) that the total energy for an elliptical orbit varies the orbital radius and angular velocity of the planet in the elliptical orbit. This is shown in the animation: the planet moves faster when it approaches the sun, and then slower when it is farther from the sun. Kepler`s second law states that the blue sector has a constant area. But r → ′ ( t ) {displaystyle {vec {r}},`(t)} is the velocity vector v → ( t ) {displaystyle {vec {v}}(t)} of the moving particle, such that If you then express the angular velocity ω with respect to the orbital period T {displaystyle {T}} and rearrange it, you get Kepler`s third law: The prevailing opinion at Kepler`s time was, that all planetary orbits were circular. The data for Mars posed the greatest challenge to this view and ultimately encouraged Kepler to abandon the popular idea. Kepler`s first law states that each planet moves along an ellipse, with the sun in a focus of the ellipse. An ellipse is defined as the set of all points such that the sum of the distance from each point to two foci is a constant. Figure 13.16 shows an ellipse and describes an easy way to create it. Kepler`s second law can also be described as “The surface velocity of a planet rotating in an elliptical orbit around the sun remains constant, meaning that the angular momentum of a planet remains constant.” Since angular momentum is constant, all planetary motions are planar motions, which is a direct consequence of the central force. The speed is greatest where the satellite is closest to the great mass and least where it is farthest away – at periapsis or apoapsis. It is the conservation of angular momentum that determines this relationship.
But it can also be obtained from the conservation of energy, the kinetic energy must be greatest where the potential gravitational energy is the lowest (the most negative). The force and therefore the acceleration in the diagram is always directed towards M, and the velocity is always tangential to the trajectory at all points. The acceleration vector has a tangential component along the direction of velocity at the apex of the y-axis; As a result, the satellite accelerates. It is exactly the opposite that happens at the lower position. Now consider Figure 13.21. A small triangular area ΔAΔA is swept in time ΔtΔt. The velocity is along the path and makes an angle θθ with the radial direction. Therefore, the perpendicular velocity is given by vperp=vsinθvperp=vsinθ. The planet moves at a distance Δs=vΔtsinθΔs=vΔtsinθ, projected along the direction perpendicular to r.
Since the area of a triangle is half the base (r) multiplied by the height (Δs)(Δs), the area is given by a small displacement of ΔA=12rΔsΔA=12rΔs. If you replace ΔsΔs, multiplied by m in the numerator and denominator and rearrange, you get Kepler`s second law: “The ray vector attracted by the sun to the planet sweeps equal areas at equal time intervals.” The Cartesian velocity vector can then be calculated as v = μ a r ⟨ − sin E , 1 − ε 2 cos E ⟩ {displaystyle mathbf {v} ={frac {sqrt {mu a}}{r}}leftlangle -sin {E}, {sqrt {1-varepsilon ^{2}}}cos {E}rightrangle } , where μ {displaystyle mu } is the default severity setting. [24] For orbits to be circular, planets must move at a certain speed, which is extremely unlikely.