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.
(browser
issues: Mozilla 1.5x & IExplorer 6.x
& Netscape 7.x
display this page properly;
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 = - GMsun/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 = vr2 + vt2 Therefore, the tangential
speed vtcan 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 vt
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