Interface Galileo

A graphing calculator in MATLAB App Designer for presentation of

celestial mechanics and the calculus of variations

DSBirkett Boston 2022

Central and inertial motion are combined with the parallelogram rule in Principia.

All paths are stationary and every small variation in the path results in a small increase in the action.

History

Celestial mechanics was born of astronomy in the 17th and 18th centuries. Kepler's three laws established the elliptical shape, the sizes, and the periods of the planetary orbits, and the remarkable fact that the planets sweep out equal areas in equal times in orbit around the sun. Newton applied Euclidian geometry and his calculus to show that these properties follow directly from the universal law of gravitation, which holds that masses are subject to a central force attraction of magnitude proportional to the inverse of the square of the distance between them.

Celestial mechanics has developed beyond these foundations. In the 19th century Lagrange and others introduced the calculus of variations. This generalization of Newton's ideas revealed that Kepler’s

orbits have another important property: a certain time integral called the action taken along any orbital path is stationary, as this abstract mathematical property is defined. The calculus of variations has found application in fields of physics far from mechanics.

Newton's composition of inertial and central motion with the parallelogram rule. A construction in Euclidean geometry on these differentials demonstrates Kepler's second law. From The Principia, Section II Proposition 1 Theorem 1

The Calculator

An interactive graphing calculator can be developed which facilitates presentation of the principles and mathematics of celestial mechanics. Such a calculator will display and animate Kepler’s orbits on all time scales. It will display Newton’s fundamental decomposition of these curved paths into infinitesimal parallelograms. The interface will show, graphically and numerically, that the action of each planetary path is stationary.

In calculus the differential increment is fundamental. Many quantities appearing on the screen of the calculator can be incremented by one of two differentials set by the operator, with all related quantities changing consistently everywhere on the screen.

In Kepler's problem, a planetary orbit is described by its eccentricity, semi major axis, total energy, angular momentum, period, latus rectum, among other quantities over many orders of magnitude. Also required for complete description of the binary system are the planet’s mass, and the strength of the central gravitational field, which depends of course on the solar or primary mass. Many subtle relationships obtain between these quantities: Kepler’s third law relating semi major axis and period is the most important of these. Because of these relationships an orbit can be uniquely specified or determined with a variety of combinations of several parameters. However a given orbit is uniquely specified, a satisfactory graphing calculator will then compute for display a consistent set of all orbital parameters. It will animate every feature of the motion of the planet on the screen, along with scales of time and length.

Newton used only the properties of triangles known to Euclid to prove that the conservation of angular momentum follows from the properties of a central force field, using a construction of parallelograms well known to students of the Principia. The graphing calculator can draw this construction of parallelograms superimposed on a path under consideration, and take the construction to its infinitesimal limit. In this way the calculator can serve to illustrate the limit process which is at the very foundation of Newton’s calculus. The calculator can display the triangles swept out along an orbit in equal differentials of time to show they are equal in accordance with Kepler's second law.

The calculus of variations, as an important generalization of differential and integral calculus, involves among other things finding properties of the paths of bodies subject to Newton's F = ma, and it is shown that these properties are equivalent to F = ma. In particular, the time integration of a quantity called the Lagrangian, the difference between the body kinetic energy and potential energy, yields an extremum for the unique path which satisfies F = ma. This is shown, following Newton, by considering a small arbitrary variation of the actual path and computing the new time integral of the Lagrangian , called the action, which assumes a minimum value for the actual path. The Keplerian orbits presented by the calculator satisfy F= ma and therefore paths possess always this extremal property, they are stationary.

The Kepler calculator is configured to present the properties of a stationary path directly. The operator can graphically impose an arbitrary small change q, or its opposite -q, on a path. The calculator will present in response a new calculated action, which will exceed infinitesimally the action of the actual path for any small change whatever imposed by the operator. In this way the calculator illustrates interactively that the action is stationary in the mathematical meaning of the word.

Unit Ellipse

The calculator presents a body of unit mass in a counterclockwise orbit of period 17.25 sec. The semi latus rectum is 1 m and the eccentricity is .7. The body is shown in magenta at t = .1875 sec past periapsis at t = 0 sec. Its velocity is 1.656 m/s and kinetic energy 1.371 J. The start time (green asterisk) is -2.16 sec and the stop time (red asterisk) is 19.4 sec so that the path crosses periapsis twice. In the lower right the dynamical variables (in y) are listed for four consecutive steps of dt = .0469 sec.

Variation of a Short Path

A section of the orbit 1 sec after periapsis is selected for variation. The Lagrangian will be integrated between the white ticks in graphic Newton. A small variation near the yellow tick is imposed within this interval. The current time is in magenta. With this variation the action increases by .00998 J sec from 1.83 J sec.

Parallelogram Rule

Continuous motion is decomposed into differential parallelograms in Principia. The graphics show the impulsive gravitational force in green on the left, and this force in blue applied to the body in magenta. The composition of inertial and central motions of duration DT by the parallelogram rule is shown. The table in the lower right shows in both dimensions the incremental change in velocity a*dt where a is the acceleration of the body. Also shown in blue and in the table are the azimuths of the gravitational impulse F bo and the central motion a*dt, which are equal by construction for sufficiently small DT.

Impulses of Central Gravitational Force

The calculator is configured to show the parallelogram rule applied to a 10000 sec interval in the orbit of Osiris. Impulses shows the gravitational force applied to the body at the start of each interval. Parallelogram shows the composition of inertial and central motions. The increment in velocity a*DT over DT is shown on the lower right.

This is estimated for the x component in table of 6 and the bottom row. These values converge in the limit DT small.

Probe Osiris: Conservation of Energy

Graphic Galileo shows the kinetic (yellow)and potential (red) energy of spacecraft Osiris during orbit around asteroid Bennu. The red and green vectors show the velocity and acceleration, respectively, of the body. The radius of the yellow circle is proportional to the magnitude of the velocity. The line chart shows that the sum of the energies is a constant negative total energy. Perihelion is set at the primary radius.

Geo-Synchronous Orbit

The orbital parameters for a 1000 kg satellite in geostationary orbit. The orbital period is one sidereal day. The force on the satellite is 222 N. The velocity in orbit is 3074 m/s. The earth surface is shown in comparison with the orbit.

Sectorial Velocity of Mercury

The sectorial velocity of Mercury is found to compute the area of its ellipse. DT= 4.33e4 sec. dA=5.88e19 m^2.

The sectorial velocity dA/DT = 1.36e15 m^2/sec. The product of the sectorial velocity and the period is constructed as A*B = 1.035e22 m^2 in the table of 6. The product of the semi axes is constructed as pi*D*E. A*B = pi*D*E

Force on Halley's Comet

Gravitational force 3.778e9 N on Halley's does work, reducing the comet kinetic energy. The time derivative of kinetic energy is estimated in the bottom row. It equals the dot product of force and velocity, as shown in elementary mechanics. As there is no y component of velocity, D*E estimates this product using only x components of force and velocity. These equalities obtain for all times in the path, for increasing and decreasing kinetic energy.

Inverse Square Law

Earth Escape Velocity

The Main Screen

A variety of graphics are required to present the many mathematical and physical ideas from celestial mechanics. The screen presents a pair of graphics at left in conjunction with several numerical tables at bottom and right which will completely describe the orbit, the path, and dynamical state of the planet or body of interest. These data are always collectively consistent, representing the current time. The position of the body at this time is always shown

in magenta.

The following geometric graphics are available in the two drop downs along the top row of the screen.

Newton. The orbital path and its variation, showing the paths as discrete point functions of time. This graphic shows the start and stop times of the path, and on the path the interval over which the Lagrangian is integrated. This graphic also shows the location and form of the small variation within the action interval, which is critical. The number of points representing the path can be arbitrarily large, and the time resolution DT is inversely proportional to this number.

Parallelograms. The path near the current time is decomposed into a differential parallelogram whose vertices are adjacent points given in Newton. This construction shows the continuous motion of the planet represented as intervals DT of inertial motion added to motion due to a central impulsive force with the parallelogram rule. In the limit DT small, the central impulsive force is continuous.

Impulses. The central impulsive force. The magnitude of this impulsive force and its direction changes in accordance with the inverse square law as the planet is shown in orbit around the primary mass at the focus.

Graphically the impulse is shown to act impulsively on the body when it approaches the start of the differential parallelogram.

Principia. The parallelograms are superimposed on the path from start time to stop time. Triangles of area swept out in one differential DT of time are shown near the current time. Following Principia, these triangles show graphically that equal areas are swept in equal times.

Galileo. The changing kinetic energy of the planet is shown during orbit, along with the velocity and acceleration vectors, This energy is shown as a circle of area proportional to the magnitude squared of the velocity. At the bottom this graphic shows that the negative potential energy and positive kinetic energy of the planet during orbit always sum to the constant total energy, which is negative for all elliptical orbits.

On each geometric graphic the yellow + indicate the scale of length, which is the same for all graphics, and can be changed by zooming in or out. The scale of length in meters can be posted in the table of 6 at the bottom of the screen as length_m.

Secular Graphics

Data on the motion of a body can be presented in the form of functions of time. The domain of this data will be either the entire interval from start time to stop time, or the interval within this interval over which the Lagrangian is integrated. These secular graphics are also found along the top row of the main screen. The landmarks in the geometric graphics are correlated by color with their counterparts in the secular graphics.

x_Variation. The dynamical variables position (magenta), velocity (red), and acceleration (green) of the planet between the start time and stop time during orbit are shown. Superimposed on these curves are the same curves after variation. The curves after variation will visibly differ from their originals only if the imposed variation is sufficiently large. The current time in magenta is shown in relation to the other times as specified and the variation window is shown in blue. x_Variation is used in conjunction with the table of 6 below it, where the variables x(t), v(t), a(t), are presented numerically as they depend upon the current time in magenta.

Lagrangian. The kinetic and potential energies and the Lagrangian over the time period of the path, and their variations. The Lagrangian is integrated to find the action and change in action after variation. The change in Lagrangian depends visibly upon the amplitude, form and location of the variation imposed by the operator. The Lagrangian is auto-scaled so that its form near the disturbance can be seen even for extremely small variations, which are to be preferred for analytical reasons. Lagrangian is used in conjunction with the table of 6 below it, where the variables KE(t),PE(t), and the Lagrangian, are presented numerically as they depend upon the current time in magenta.

The kinetic energy (yellow) and potential energy (red) are presented in Lagrangian. Both functions display local minima or maxima at periapsis. The change in the Lagrangian after variation is shown auto-scaled in green. Times indicated by tic marks in Newton correspond by color to those in Lagrangian.

The characteristic x and y dynamical variables for an ellipse of ellipticity .7 are charted. The location and width window of the small variation to be imposed is in shown in blue. The magenta vertical shows the time. Green represents acceleration, red velocity, and magenta position. The Lagrangian is integrated between the white verticals. The start time (green *) and stop time (red *) are set so that two passages through periapsis are shown.

Charts and the table of 6

At the bottom of the main screen a table of 6 quantities requires selection and configuration of the quantities of importance to a particular problem. These quantities will typically include the current x-y dynamical variables of the body, or the numerous Keplerian parameters of the orbit under consideration. The kinetic energy and potential energy of the body can be displayed here, along with the Lagrangian. The changes in these quantities after small variation are also available. Data from tables on the right can also be transferred to the table of 6 for use. The table of 6 can be used in conjunction with secular graphic Charts.

Charts. This graphic displays as ordinates the three quantities selected below it in the table of 6, with abscissa the action interval. This data is generated when the time is swept over this interval using the large button at the lower right of the second graphic. The resolution of these graphics is also DT. Each quantity has its own scale, which can be adjusted in the variables drop down. After posting, the time in, the magenta vertical, in Charts continues to follow the time under differential control, so that both graphical and numerical data are presented together by the combination of Charts and table of 6.

Charts_OS This graphic is the same as Charts with all quantities presented on one scale for direct comparison.

Newton shows geometrically the interval between white ticks over which the Lagrangian is integrated. The region of variation, which is too small to be seen, is located by the yellow tick.

The kinetic energy (yellow) and potential energy (red) are shown in Lagrangian. The variation window is shown in blue and the auto-scaled change in Lagrangian is in green.

Charts is used to display the kinetic and potential energies, and change in action over the interval. The action and change in action appears numerically under Lagrange on the right.

The radius, x and y coordinates, of the body over the path appear on the right Charts. The Lagrangian is shown auto scaled in green in Charts_OS. Note that both Charts and Lagrangian present exactly the same Lagrangian.

Differential Control

Many quantities presented graphically and numerically in description of the body in orbit are subject to incremental change by one of two differentials. As true differentials these can be made arbitrarily small, securing for the calculator unlimited resolution. Differential control is effected primarily through the variables drop down.

At the top right the differential of parts per unit is set, and below this the differential of time dt is set. The time differential can be absolute, in seconds, or set as a convenient fraction of the period or interval of the orbit under consideration. The current time behind all quantities on display is increased or decreased by the time differential using the small button in the lower right of the second graphic. The differential of time dt is not the same as DT, the time duration of one of the parallelograms into which a path is decomposed, but dt can be set equal to DT to step along the parallelograms.

If the endpoints of the action integral are to be modified, or the location of the variation, the differential of time dt is automatically effected by variables drop down. Orbital parameters such as mass or period are changed incrementally with the parts per unit differential. Which differential is employed under the variables drop down depends of course on the nature of the quantity.

Many parameters must be carefully set to uniquely specify a Keplerian system and path. This is done most conveniently by calling up the parameters of one of a variety of reference orbits and paths with the drop down list in the extreme upper right. All parameters are then subject to differential control.

Also on the top right are graphics controls to zoom in and out on the path, and to place either the planet or the primary mass, which is always at a focus, at the center of the geometric screens.

Differentiation

Most quantities of interest on the screen are subject to incremental change under the variables drop down, with all other dependent quantities changing consistently under differential control. Any of these quantities can be placed as entry A or B (or any where else) in the table of six. The bottom row of the calculator can then construct dA/dB, the derivative of A with respect to B, as a quotient of differentials. The following examples illustrate the generality and unlimited resolution and configurability of differential control.

dF/dt in the bottom row is used to find the gravitational work done by gravity in accelerating Osiris in its orbit. The time derivative of the kinetic energy is computed 2.767e-5 J/s , to be compared with the dot product of force and velocity. The power is positive, indicating increasing kinetic energy. At the time in orbit shown there is no y component of velocity, so the dot product is estimated as a product D*E of x components.

The angular momentum of Osiris can be computed from the time rate of change of its azimuth

dphi/dt * r^2 * mass

which by the second law is constant. Agreement is shown here between the various estimates of angular momentum.

F=ma is illustrated with the table of 6 in conjunction with differentiation in the bottom row.

Variables

The variables drop down list with its buttons and fields on the right is where many quantities are to be set directly or changed incrementally, using the two control differentials. This includes the fundamental Keplerian parameters mass, latus rectum and eccentricity and such important derived parameters as period and total energy. When a quantity is changed using variables drop down all related quantities and graphics change consistently everywhere else within the calculator and on the screen.

For any orbit, the start and stop times of a path and the interval within this interval over which the Lagrangian is integrated can be set with the variables drop down Here also the location, form, and amplitude of the variation q to be imposed on the path is set.

Besides allowing for incremental change, the variables drop down presents a data field for input from the operator, which can be quantities developed in the bottom row of the main screen, which may in turn originate in the table of 6.

As the orbital phenomena presented by the Kepler calculator cover many orders of magnitude in length, time, and action, and presentation of concepts from calculus and variational calculus itself requires presentation of extremes of linear dimension, all graphics and charts presented by the calculator are adjustable exponentially in amplitude using the respective items listed in the variables drop down. To the right of the variables drop down are the buttons for change of these quantities.

Inversion of the variation imposed on a path and inversion of the differentials is done by buttons to the right of the variables drop down. It is helpful to make the amplitude of a variation q imposed on a path sufficiently large so that it, and its opposite -q, are visible in graphic Newton. This will typically lead conveniently to a visible display of the corresponding change in the Lagrangian. Analytically, extremely small invisible variations are acceptable, even preferred. Graphic Lagrangian_n presents the Lagrangian function auto scaled so that it will be visible for small q.

The escape velocity at the moon surface is estimated with an orbit of eccentricity .95 and perihelion equal to the moon radius. The interval and resolution are set using differential control to DT=200 sec.

Parameters

Below the variables drop down the parameters drop down has numerous items for organization of data.

Latus Eccentricity Period presents basic data including the mass,primary mass, period, and DT.

Lagrange presents the action integrals computed for path and varied path, and the parameters of the imposed variation as set by the operator. This includes the duration of the imposed variation window, the magnitude qt of the transverse variation, and the magnitude qa of the axial variation within this window. Here the quadratic dependence of the change in action on variation amplitudes q can be demonstrated directly.

px vx ax presents the current position, velocity, and acceleration of the body, x and y coordinates.

F=ma presents the force on the mass and its acceleration in x and y coordinates. It is used in conjunction with other tables to show convergence of quotients of differentials (velocity and acceleration) with their nominal values in the limit DT small.

Nest From To presents the start time and stop time together with the limits of integration of the action integral nested within.

Period Start Stop presents the time, DT, and dt, along with the period and start and stop times.

The row of buttons on the lower right may be used in conjunction with Nest From To for modification of paths on the screen. The times specifying path and action integral and disturbance can be translated collectively around the ellipse by the differential of time dt. All intervals within the path can be stretched using the parts per unit differential. This can be done to modify the time resolution DT or interval of the action integral to a more convenient value. Of course the resolution DT with which a path is represented can be more directly altered by changing the number of points in the interval, or the start and stop times.

After the physical parameters of an orbit are established, any section of that orbit may be selected for examination. A button on the lower right may used to specify optimum parameters to present the periapsis of the orbit, which always occurs at time = 0. Another button specifies parameters for overlapping display of the entire ellipse to provide two crossings of periapsis

Quantities

The quantities drop down in the lower right has these items among others:

px vx ax is used to display the x positions, velocities, and accelerations of the body in orbit, four points sequentially.

F=ma presents the increment of a velocity v by a.dt to yield the new velocity in the next parallelogram is shown numerically

YT presents numerically the yellow triangle of Principia subtended by the body during one DT, to show Kepler's second law in its differential form. The vertices, the azimuths, the area, and the sectorial velocity of the triangle are shown. Also shown are the period and DT and the product of the sectorial velocity and the period, which is always the area of the ellipse.

WT presents numerically the white triangle of Principia subtended by the body during DT, to show Kepler's second law in its differential form. The vertices, the azimuths, the area, and the sectorial velocity of the triangle are shown. Also shown are the period and DT and the product of the sectorial velocity and the period, which is always the area of the ellipse.

The bottom row on the main screen of the calculator can do several things. Quantities appearing in

the table of 6 just above can be moved to the bottom row for presentation in a different system of units. Also the usual arithmetic combinations of the quantities presented in the table above can be configured for computation in the bottom row. Here astronomical reference data, such as planetary masses, semi major axes, and periods can be called for insertion in to the variables drop down on the right.

Another important operation of the bottom row is differentiation. The derivative of quantity A with respect to B in the table of 6 can be computed as a quotient of differentials. An example is the derivative of the planet kinetic energy with respect to time, which is the work done by gravity in accelerating the planet during its orbit. The table of 6 can be configured to show that for all orbits and times this is equal to the dot product of the planet velocity and force of gravity.

Area

The following graphics show the presentation of Kepler's second law by the calculator. First the differential area dA swept by a body during DT is computed, then the sectorial velocity dA/DT. As dA/DT is constant, the area swept in one period is the ellipse area. This is shown by placing the sectorial velocity in the table of 6 (entry A) for multiplication by the period (entry B). The product A*B appears in the bottom row.

YT in the quantities drop down presents the area dA swept by Osiris during DT = 1.02e4 sec. The sectorial velocity of the body is dA/DT. dA/DT times the period of the orbit is the area of the ellipse, which is pi times the product of the semi major and minor axes.

Sectorial Velocity

Inverse Square Law F = (g m1 m2)/r^2

(F/m2) r^2 = g m1

(a r^2 )/g=m1

The table of 6 and bottom row can be configured to show the above relation for any orbit whatever, every where along the path.

Inverse square law applied to Mercury

Inverse square law applied to moon.

Inverse square law applied to Osiris.

Escape Velocity

Calculus of Variations

The Kepler calculator follows the presentation of the principle of least action and Kepler's problem found in

Landau and Lifshitz.

Mechanics Third Edition

L D Landau and E M Lifshitz

Volume 1 of Course of Theoretical Physics

ISBN 0-08-029141-4

The paths presented by the calculator are stationary. This means that the time integral of the Lagrangian KE-PE of the path under consideration, called the action, is a minimum for the path: any small deviation whatever from the path yields a small increase in the action. By the calculus of variations this strong statement is equivalent to F=ma along the path.

The calculator is configured so that the operator can impose an arbitrary small variation q, or its opposite -q, upon a path and observe the computed change in action which results to show that this is always the case. For any given form or location of small variation imposed, the computed change in action will vary linearly and quadratically with the amplitude q of the variation. As the action is a minimum for the given path, the linear terms in the action expansion in q will be negligible, and quadratic terms will remain. In the absence of third order and higher terms, the change in action will be proportional to q^2 = ( -q )^2, the same, for both the imposed variation q and its opposite -q. This vivid condition appears on the screen as a demonstration of the stationary property. It is of course essential that, as q is a differential quantity, this symmetry will obtain for all small q, -q, independent of the magnitude of q, as long as the variation vanishes at the end points of the action integral, which requirements are set out in the calculus of variations.

The quadratic dependance of the change in action on the variation amplitude q can also be shown by doubling this amplitude on the screen, which should quadruple the change in action as long as q is small enough that third order and higher terms in q may be neglected. The calculator illustrates with rigorous validity this central idea: it is an interface for graphical exposition of this branch of higher mathematics.

The principle of least action requires that the action be stationary for any variation of path whatever. To effectively demonstrate this the operator can impose pointwise along the path small deviations qa in the coordinates in the local direction of the path, and independently deviations qt transverse to this. These independent variations are added pointwise along the path, localized together somewhere along the path by a window function of duration set by the operator. This generality of variation can be considered as almost complete.

If the change in action after variation is expressed in powers of qa and qt, linear terms must be absent because the action has a minimum for qa=qt=0. For sufficiently small qa, qt, symmetric quadratic terms will predominate.

The operator can impose the opposite variation on the path by negating together qa and qt. This will not change the varied action if the quadratic terms are predominate and the path is stationary

A small variation q of duration .09 sec is imposed upon the path of a body of mass 1 kg. The path is of duration 1.1 sec. The variation is transverse to the velocity of the body. Lagrangian shows the kinetic energy (yellow) and potential energy (red) , and the Lagrangian autoscaled in green

The opposite variation is imposed. The kinetic and potential energies are changed, and the Lagrangian is changed,

The position, velocity and acceleration of the body during the interval. Small changes in x position and x velocity during the variation imposed in blue are shown.

The magnitude q of the transverse variation is halved, yielding one fourth the change in action, a quadratic dependance.

Examples

The graphics Newton and x_Variation display the basic geometric properties of the orbit. Here the start time (green asterisk) is negative, and the stop time (red asterisk) exceeds one period, so two clockwise passages through periapsis are displayed. The action interval, indicated in white in both graphics, also encompasses both periapses. The form and time of the disturbance imposed upon the path is shown in both graphics in blue. The current time is shown in magenta.

Lagrangian shows the kinetic energy (yellow) and potential energy (red) during the path. The Lagrangian and the change in Lagrangian is shown in green, autoscaled. The change in Lagrangian course coincides with the form, width, and location of the imposed variation in blue.

Impulses and Principia are used to present Newton's decomposition of the motion into continuous and impulsive components. The green and blue vectors show the magnitude and direction of the gravitational impulse. Principia

shows the force parallelograms superimposed on a path, and triangles swept out over time.

Impulses and Principia show the gravitational impulse after time step DT = .1078 sec.

Parallelogram shows the rectilinear motion of the planet between the impulses of gravity, both the motion due to inertia and that caused by the gravitational impulse, which are added using the parallelogram rule.

Galileo shows the planet kinetic energy proportional to the area of the circle, and the planet velocity as a radius of that circle. At bottom kinetic energy in yellow and potential energy in red sum to the total energy at bottom in red, which is always negative.

The angular momentum of Osiris if found by differentiation of the mass azimuth phi with respect to time.

The angular momentum is given by mass * r^2 * dphi/dt.

The angular momentum of Mercury if found by differentiation of the planet azimuth phi with respect to time.

The angular momentum is given by mass * r^2 * dphi/dt.

The time rate of change of the kinetic energy, the work done by gravity on Mercury, is found by differentiation. This equals the dot product of the planet velocity with the gravitational force on the planet during the orbit.

The time rate of change of the kinetic energy of Osiris is found by differentiation. This equals the dot product of the satellite velocity with the gravitational force on the body.

Stationary Path of Asteroid Eros

The path of asteroid eros after perihelion from 4.198e6 sec to 11.21e6 sec. The variation in path is imposed at 6.702e6 sec.

The Lagrangian KE-PE of the asteroid will be integrated from 4.45e6 sec to 11.00e6 sec, for the actual path and the changed path. This interval is 6.6e6 sec

The time resolution of the path is 5e4 sec . Eros period is 5.579e7 sec

The calculated action for the asteroid over the interval is 4.65e31 J-sec

The calculated action for the asteroid over the interval for the inverse variation. The change is the same, indicating a stationary path.

The kinetic energy, the potential energy, and the Lagrangian in green for the asteroid over the time interval.

Planet Mercury

The planet Mercury at aphelion, 6.997 e10 m from sun. Velocity is 3.883 e4 m/s.

The planet Mercury at perihelion, 4.61 e10 m from sun. Velocity is 5.891 e4 m/s.

The acceleration of planet Mercury at perihelion is .0624 m/s^2.

The component velocities of planet Mercury at time 2.902 e6 sec. Vx = -2.468 e4 m/s Vy =-3.213 e4 m/s.

Time derivative of kinetic energy is shown.

Moon Surface

Perihelion for a 1 kg mass in orbit near moon surface.

The moon mass is 7.342 e22 kg. Velocity at perihelion is 2304 m/s.

Velocity at perihelion 2304 m/s. Escape velocity is 2375 m/s Time rate of change of kinetic energy is 246 W.

Kinetic energy at time 1410 sec is 1857 mJ

Earth Orbit

Geostationary satellite of mass 1000 kg. Earth mass is 5.972 e24 kg. Period at 4.224e7 m altitude is 8.64e4 sec, 1 day

Global positioning satellite at altitude 2.661 e7 m. Period is 12 hr.

1000 kg mass at altitude 1.279 e7 m Period is 4 hr

1000 kg mass at altitude 6.399e6 Period is 1.415 hr

Escape velocity at earth surface is 11.16 km/sec

Gravitational acceleration at earth surface is 9.22 m/s^2

Osiris

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.

Eros

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