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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"

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"

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 v

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

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: PF

The eccentricity is defined as the ratio of the distance between the two foci to the length of the major axis: e = F

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.

the sun's kinetic energy is negligible in comparison

b) the gravitational energy (GE) of the comet-sun system is = - G m M

where M is the mass of the sun, G is Newton's Gravitational Constant

c) the total energy (TE) is therefore = 1/2 mv

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 v

GE/mass = - GM

it is important to use the most accurate values of quantities available

G = 6.6742 x 10

M

1 au = 1.495 978 707 x 10

TE/mass = 1/2 v

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 M

v

Therefore, the tangential speed v

Note that the tangential speed can also be written as v

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

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

G M