Newton

Osiris-REx is a NASA mission to asteroid Bennu launched 8 September 2016 and rendezvoused with the asteroid 3 December 2018. Consider the binary system of probe Osiris of mass around 1000 kg and Bennu of mass 7.329e10 kg,

radius 250m. The orbiter is to have altitude around 1 km and a period of 60 hr. Gravity near the surface is around 6ug.

The orbit of Osiris around Bennu

Period Semi Major Axis Altitude Kinetic Energy Escape V Velocity

8.838 hr 240 m 0 m 30.99 J .209 m/s .176 m/s

12 294.3 m 54.3 m 25.27 J .1823 m/s .159 m/s

16 356.6 m 116.6 m 20.85 J .1656 m/s .144 m/s

32 566 m 326 m 13.14 J .1315 m/s .115 m/s

64 898.4 m 658.4 m 8.263 J .1044 m/s .091 m/s

Halley's Comet

Halley's comet in orbit around the sun played a role in the development of celestial mechanics. The Kepler calculator presents features of its path, including dynamical descriptions of the comet at perihelion and aphelion, the forces on the body, and the size of its orbit. The variation in action over a section of orbit is calculated.

The parameters which define Halley's comet are shown. Solar mass is 1.99e30 kg. Comet mass is 2e14. The latus rectum is 1.73e11 m. The period is 75 years. The semi major axis is 2.668e12 m.

Halley's perihelion and aphelion are displayed. Path resolution is set to n= 2971 to set DT=1e6 sec. The start time (green asterisk) of the path is -2.97e8 sec and the stop time (red asterisk) is 2.67e9 sec. so that the path crosses perihelion twice. The x coordinate of the comet at perihelion t=0 sec is 8.81e10 m.

Analysis of the motion of the comet at perihelion. DT= 1e6 sec. The velocity increment over this time is estimated in the table of 6 as ax(t)*DT= 1.6e4. In the quantities drop down this is added to velocity v1 + adt = v2 = 8e3 m/s.

Lagrangian shows the kinetic (yellow) and potential (red) energies as functions of time over the path. Two crossings of perihelion can be seen. The interval over which the Lagrangian (auto scaled in green) is to be integrated is indicated in white in both graphics. This interval is well nested between the start and stop times. The location and duration of the variation imposed on the stationary path is indicated by the yellow tic in Newton and blue in Lagrangian.

The action of the path is 5.67e11 J sec. The action increases by 1.26e21 J sec upon variation.

Elementary mechanics shows that the work per unit time done by gravity on a body is the dot product F.V of the force on the body with its velocity. The comet is shown at a time when the gravitational force and the velocity are nearly aligned to simplify calculations. The time derivative of the kinetic energy is found in the bottom row, to be compared with the dot product in entry C . These quantities change drastically together during the path and can be positive or negative. The dot product is also estimated in the table of 6 as D*E, the product of the x coordinates only.

Item YT (yellow triangle) presents the area dA =2.41e21 m^2 of the triangle and dA/DT= 2.4e15 m^2/sec, the sectorial velocity, which by Kepler's second law is constant. The product of the sectorial velocity and the period is the area of the orbit. This is constructed as A*B = 5.699 e24 m^2 in the table of 6 for presentation in the bottom row. AS is well known, the area of an ellipse is the product of the semi axes and pi. The product of the semiaxes

is constructed as D*E=1.814e24 m^2 which differs from A*B by factor pi.

The extremes of Halley's orbit. are presented in Galileo. The velocity and acceleration vectors are shown in green and red respectively. The kinetic and potential energy, and constant negative total energy, are shown in the line graph. The size of the orbit is compared to that of Jupiter.

At perihelion Halley's comet has velocity 53.45e3 m/s and kinetic energy 2.857e23 J.

At aphelion Halley's comet has velocity 913.6 m/s and kinetic energy 8.347e19 J. Time at aphelion is one half period.

A time interval of duration 1.133e8 sec is selected for computation of the action. The sum of kinetic and potential energy totals to the negative total energy.

chanics was born

A small variation q is imposed on the path. The action increases by 2.09e7 J sec . The Lagrangian is shown auto-scaled on the left in green.

A small variation -q is imposed on the path. The action increases by 2.09e7 J sec. The Lagrangian is shown auto-scaled on the left in green.

Variation q is doubled, resulting in quadruple change in action.