conservation of energy and angular momentum for a comet orbiting the sun

The goal of this page is to guide the reader through a spreadsheet exercise that shows that the total energy (kinetic and gravitational) and the orbital angular momentum of a comet orbiting the sun are the same at each position in the comet's orbit.  Kepler's three laws of planetary motion are seen to apply to comet orbits as well.

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Netscape 4.x is a disaster; don't use it!)

1) obtaining comet data

The Ephemeris Generator at NASA-JPL is a flexible method of generating position-vs-time data for any comet (or any other solar system body, for that matter).  The ephemeris consists of  (a) the comet-sun distance, (b) the rate at which the comet-sun distance changes with time, and (c) the comet's orbital speed with respect to the sun.  A short guide to use the Ephemeris Generator follows:

Step 1: Modify current settings as desired:

Target Body

    for a comet, use the "Select Small Body" portion;
    you must know the comet name and input it using one of the following example formats:
        2P/Encke  or  2P  or  Encke or 1990 XXI;
    limit the search to "Comets Only" in the box, and then click on the "Search" button;
    the generator then returns a list of possible comets; select the appropriate one;
    then click on the "Use Selected Asteroid/Comet" button

Observer Location

    for generating position-vs-time data for a comet orbit around the sun, the observer
        location is irrelevant,
although if you are interested in observing the comet from your
        geographic location, it can't hurt to input
your latitude and longitude

Time Span

    the start time will be today's date; you will want to know the orbital period of the comet if you
        want to insure
your ephemeris covers a complete revolution of the comet about the sun;
    for Output Interval you will have to experiment; if you select a time interval that is too small,
        your ephemeris
will be huge; if you select a time interval that is too large, you may miss
        key positions in the comet orbit
perihelion, especially if the orbit is highly eccentric;
    click on "Use Specified Settings" when finished

    IMPORTANT NOTE: the accuracy of the ephemeris is NOT affected by the time step you
       insert; the model produces the same results no matter whether the time step is 1 minute
       or 10 days; the only difference is the number of values returned

Output Quantities and Format

    The following are important for generating a comet orbit; click in their boxes

       19. Heliocentric range and range-rate  (this is distance from sun and radial velocity relative to sun)
       22. Speed wrt sun and observer (this is the object's speed relative to the sun and and to the observer)

    Uncheck the other boxes (unless you are interested in observing the comet from your
        geographic location);
    when finished, click on "Use Selected Settings"

Step 2. Select desired options:

    if you are interested in importing the results into an Excel spreadsheet, click in the
        "Use CSV (spreadsheet) format " box;
    also check the  "Include Body Information Page"

Step 3. Request the ephemeris:

    click on the "Generate Ephermis" box

The ephemeris will be returned in html table format.

To save the file in spreadsheet-ready format,
    1) select Save As... (or Save Page As...)
    2) change the file extension to .csv in the File Name box
    3) make sure that All Files is selected in Save as type  box.

The three crucial pieces of data that will allow you to generate position-vs-time data for the comet are

    a) the column labeled  r  (the distance of the comet from the sun; the units returned are au)
    b) the column labeled rdot  (the radial velocity of the comet relative to the sun  =  dr/dt ;
        i.e., the component of the comet's orbital velocity along the sun-comet line);
        this will be labeled vr below; the units returned are km/s
    c) the column labeled VmagSn (the orbital speed of the comet relative to the sun);
        this quantity will be labeled v below;  the units returned are km/s

A sample ephemeris for comet Encke (for a time span of 4 years at 30-day intervals) can be found here

2) elliptical orbits: semimajor axis and eccentricity

Kepler's first law is commonly stated as "A planet's orbit about the sun is an ellipse with the sun located at one of the foci."  A comet's orbit can be either an ellipse or a hyperbola.

An elliptical orbit can be characterized by two parameters: the semimajor axis a and the eccentricity e.  The semimajor axis is one-half of the longest axis ( = major axis) of the ellipse. 

The eccentricity parameter characterizes the shape of the ellipse.  An ellipse with an eccentricity of 0 is a circle; in this case, the two foci of the ellipse are coincident at the center of the circle.  (The two foci are always located on the major axis of the ellipse.  One definition of an ellipse is that for any point P on the ellipse, the sum of the distance from P to one focus and the distance from P to the other focus is the same and equal to the length of the major axis: PF1 + PF2 = 2a.)
The eccentricity is defined as the ratio of the distance between the two foci to the length of the major axis: e  =  F1F2 /(2a) .  As the eccentricity approaches 1, the ellipse becomes more elongated.

The perihelion distance of the comet (distance from the sun at closest approach)  =  a (1 - e) ; the aphelion distance of the comet (furthest distance from the sun)  =  a (1 + e).   Note that the semimajor axis is the arithmetic mean of the perihelion and aphelion distances.

3) kinetic and gravitational energy in orbits

a) the comet's kinetic energy (KE)   =  1/2 mv2      where m is the mass of the comet
    the sun's kinetic energy is negligible in comparison  

b) the gravitational energy (GE) of the comet-sun system is   =   - G m Msun/r
    where M is the mass of the sun, G is Newton's Gravitational Constant

c) the total energy (TE) is therefore  =  1/2 mv2  - G m Msun/r

    because the mass m of the comet is generally unknown or known inaccurately,
    it is best to reformat these
three energies as energies per (comet) mass:

    KE/mass   =  1/2 v2  ;  in order to have the units in standard SI (J/kg), convert the orbit speed from km/s to m/s
    GE/mass   =  - GM
sun/r  ;  in order to have units of J/kg, convert the comet-sun distance from au to m
                                           it is important to use the most accurate values of quantities available
                                           G  =  6.6742 x 10-11 m3/kg/s2
                                           Msun  =  1.9891 x 1030 kg
                                           1 au  =  1.495 978 707  x 1011 m

    TE/mass  =  1/2 v2  - G Msun/r

4) Is the total energy per mass constant?

Open the  .csv  file created by the Ephemeris Generator in Part 1 above.  The next task is to create columns in the spreadsheet that calculate the KE/mass,. the GE/mass, and the TE/mass according to the above rules.  (Familiarity with Excel or similar spreadsheet s is assumed.)  You should find that the total energy is constant throughout the comet orbit, if you did the calculations correctly.  It may not be precisely constant, as there will be some uncertainty in the data returned and in the values of the constants used; the first 4 or 5 significant digits should be the same for each of the total energy numbers.

The total energy will be negative if the comet is bound to the sun, i.e., if it has an elliptical orbit and not a hyperbolic one; the total energy will be positive if the comet is unbound and has a hyperbolic orbit about the sun. 

The total energy (per mass) should also be equal to  - G
Msun/(2a)  where a is the semimajor axis of the comet orbit.  A calculation of the quantity TE/mass  =  - G Msun/(2a)   should agree well with the values calculated in the previous section.

5) angular momentum in orbits: how to determine the tangential speed and the angular speed

The angular momentum of the comet is  L  =  m r vt    where  vt  is the tangential component of the orbit velocity (vt is perpendicular to the radial compent of the velocity  vr ).  Because these two components are perpendicular, they are related to the orbit speed by the Pythagorean theorem:

             v2  =  vr +  vt2

Therefore, the tangential speed  vt  can be calculated using this equation and the values of  v  and  vr  that were produced by the Ephemeris Generator. 

Note that the tangential speed can also be written as   vt  =  r dθ/dt, where θ is the position angle of the comet with respect to some defined direction.  The quantity  dθ/dt  is called angular speed and can be calculated by using the equation in the previous sentence along with the values of  vt  and   r  .  The units of angular speed will be radians/s if tangential speed and the comet-sun distance incorporate the same distance units (e.g., kilometers).

Angular momentum is another quantity that remains constant in an orbit as Kepler knew.  His second law states that the line joining a planet to the sun sweeps out equal amounts of area in equal times.  This is true for any orbiting body including comets.  Kepler's second law is equivalent to saying that the area swept out by the comet-sun line per unit time is constant.  Because the area swept out by the line joining the comet to the sun is that of a triangle, with "height" equal to the comet-sun distance and with "base" equal to the distance the comet moves tangentially during the time interval,

       the area swept out per time interval is  =  1/2 (base) (height)/(time interval) 
                                                                         =  1/2 (r) (r Δθ) /Δt  =  1/2 r v

Note that the area swept out per time by the comet-sun line is just one-half the angular momentum per mass.

6) Is the angular momentum per mass constant?

Generate columns in the spreadsheet to calculate vt , dθ/dt, and the angular momentum per mass (or, alternatively, the area swept out by the comet-sun line per time).  Once again, the first 4 significant digits of the angular momentum should be constant with time if you have done the calculation correctly.

A sample spreadsheet with energy and angular momentum calculations for comet Encke is here.

7) Generating an orbit shape

To be able to plot the orbit of the comet around the sun it is best to use polar coordinates: r and  θ .  Polar coordinate graph paper can be obtained here.

The ephemeris has already returned values of r (comet-sun distance) as a function of time.  A value for  Δθ , the angular change in position during a time interval, can be calculated by multiplying the angular speed (dθ/dt) by the time interval over which that angular speed applies.  (For this calculation, the ephemeris was recalculated so that the time intervals were 10 days; it is important that the time interval be small enough that the perihelion angular speed is small.)  The position angle  θ  has been defined to be zero at the time of perihelion in both figures 1 and 2 above; this is purely arbitrary, and the θ  = 0 position can be set by the user..  Once the angular position changes during each time interval have been calculated, then can be summed, starting at the position defined to be  θ  = 0  in order to find the angular position at any particular time.

Generate columns in the spreadsheet for  Δθ  and   θ  as a function of time.

The orbit of comet Encke generated in the manner described is shown at the right.  The red dots show the position of the comet at monthly intervals. The sun is located at the black dot at the center of the polar plot.  Note the labels of 2 au and 4 au radially. 

Also note that, as expected, the comet moves much more slowly when farthest from the sun (i.e., the red dots are closer together) than when nearest the sun.

8) Is Kepler's 3rd law obeyed?

The final calculation is a test of Kepler's third law:   P2  =   a3   where P is the period of the comet's revolution about the sun (in years) and   a  is the semimajor axis of the comet orbit (in au).  If you prefer to test the law in standard SI units, the equivalent form (due to Newton) of Kepler's third law is

                                        G Msun P2  =  4 π2 a3