the origin of comets: Kuiper belt or Oort cloud?

The goal of this page is to guide the reader through an exercise that will determine whether a comet approaching the sun originated in the Kuiper belt (a region just outside that of Pluto's orbit) or the Oort cloud (a region much further away, a good fraction of the way to the nearest stars).  The reader will determine the distance of the comet from the sun and the orbit speed of the comet at the point where the comet is nearest the sun (perihelion), and then use conservation of energy and angular momentum to determine the distance of the comet from the sun at the point where the comet is farthest from the sun (aphelion).

(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) Kuiper belt and Oort cloud

It is believed that all comets that approach the sun originated in either of two comet reservoirs, one nearby (the Kuiper belt) and the other far (the Oort cloud).  Although neither reservoir can be seen from Earth, their existence is inferred from studying the orbits of the comets that do approach the sun.  A table summarizing the properties of comets from these two reservoirs follows.  The comets in these reservoirs have orbits that keep them at roughly the same distance from the sun, until they are perturbed by a gravitational interaction with another body. 

If the gravitational interaction reduces the energy of the comet, the orbit size of the comet will become smaller, and the comet orbit shape will change from one of low eccentricity (and somewhat circular) to one of high eccentricity (and very elongated).  In general, the comet's aphelion distance will remain in the belt/cloud in which it originated, whereas its perihelion distance will be greatly reduced.  On the other hand, if the gravitational interaction increases the comet's energy, the comet will likely be ejected from the solar system.

The goal of this page is to allow the reader to determine the aphelion distance of a sun-approaching comet in order to determine whether the comet originated (at least most recently) in the Kuiper belt or in the Oort cloud.

comet families in the solar system

Jupiter family
 Kuiper belt
 Oort cloud

semimajor axis
of orbit
4.4 au  <  a  <  10 au
30 au  <  a  <  100 au
3000 au  <  a  <  20,000 au (inner)
20,000 au  <  a  <  100,000 au (outer)
orbit period around the sun
9 yr  <  P  <  30 yr
160 yr  <   P  <  1000 yr
 2 x 105 yr   <  P  <  3 x 106 yr  (inner)
3 x 106 yr  <   P  <   3 x 107 yr  (outer) 
plane of
comet orbits
low inclination
to the ecliptic plane
 slight concentration to the ecliptic plane (inner)
random orientation to the ecliptic plane (outer)

formation site
either in the Kuiper belt or the Oort cloud
in situ:
formed where they now orbit
 formed at distances from the sun comparable to those of Uranus and Neptune, ejected outward
to present location by gravitational interaction with Jupiter
why comets
leave their
belt/cloud to approach sun
 gravitational
interaction with Jovian planets or passing stars		 gravitational interaction with
outer solar system planets causes Kuiper belt comets
to  approach sun
 gravitational interaction with random stars
passing near the sun causes Oort cloud
comets to approach sun


2) the data

The cometary orbit data was generated by using data from the Ephemeris Generator at NASAS/JPL.  The additional step of generating the orbit from this data (distance from the comet to the sun, orbital speed of the comet, radial speed of the comet with respect to the sun, angular position of the comet all as a function of time) is described elsewhere.  Orbit data for the comets is provided below in three formats.  Notes about formats, accuracy, etc:

1) the .csv and .ga3 files will produce the most accurate results.  For Oort cloud comets, 6-figure accuracy is needed; for Kuiper belt comets, 5-figure accuracy is needed; for Jupiter family comets, 4-figure accuracy is needed.

the .jpg files are provided mostly for purposes of web browsing.  The image may provide sufficient accuracy for the analysis of a Jupiter-family comet, but not for Kuiper belt or Oort cloud comets.

2) In order to insure the most accurate results, use only the data points nearest perihelion when calculating the orbital speed of the comet.  The smaller the time interval used, the better the results.  The orbital speed is entirely tangential at the point of perihelion.  However, there will be inaccuracies in the comet positions that are inherent in the generation of the position-vs-time data table; these uncertainties will propagate into the speed calculations.

comet data: of X,Y,Z one is a Jupiter family, one is Kuiper belt; one is Oort cloud

file type

software required

notes
 comet X
=
1997 BA6 Spacewatch

(time step used)
 comet Y
=
2P Encke


(time step used)
 comet Z
=
153P
Ikeya-Zhang

(time step used)


.csv file
downloads into Excel and other spreadsheets
(comma separated variables)

(right click on file link to download & save file; IExplorer will open Excel if configured properly) 		.csv files contain a data table with sun-comet distances & comet orbital speeds as a function of time;
.ga3 files contain same data and are accompanied by
a plot of the comet position as a function of time;
the GA plot can be manipulated (zoom in, zoom out,
change scale, etc); it is suggested that the
scale on the x and y axes be kept the same

To determine the most accurate value for the speed
of the comet at perihelion, only the positions
of the comet very nearest should be used. 

For Oort cloud comets, a minimum of 6-signifcant-digit accuracy in the speed and the position are necessary to insure good results; for Kuiper belt comets, 5 are necessary; for Jupiter family, 4.

The green data points representing the comet in the image (and in the data file) are separated by the time interval listed in the adjacent columns. The sun is located at position (0,0) on the graph.

comet X

(1 day)

comet Y

(10 days)

comet Z

(1 hour)


.ga3 file
downloads into Vernier Graphical Analysis

(right click on file link to download & save file; IExplorer will open Graphical Analysis if configured properly)
comet X

(1 day)
comet Y

(10 days)
comet  Z


.jpg file

none;
can be displayed by web browsers
 contains only a graph of comet's position vs time
relative to the sun; probably appropriate only for
display of orbit by web browsers

The green data points representing the comet in the image are separated by the time interval listed in the adjacent columns.  The sun is located at the position (0,0).
  
comet X

(10 days)
comet Y

(10 days)
comet Z


approximate perihelion date
(for method 2 below)		 November,
1999
March,
2002


3) Kepler's laws of planetary motion

a) Kepler's first law:  elliptical orbits

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.

b) Kepler's second law:

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

The angular momentum of the comet is  L  =  m r vt    where  m is the mass of the comet, r is the distance to the comet, vt  is the tangential component of the orbit velocity (vt is perpendicular to the radial component 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  vt  can be calculated using this equation and the values of  v  and  vr  that were produced by the Ephemeris Generator. 


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




c) Kepler's third law:  P2  =   a3                                                                                                                                                                              

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

where  Msun  is the mass of the sun and G is Newton's gravitational constant.

4) kinetic and gravitational energy for orbiting objects

a) the comet's kinetic energy (KE)   =  1/2 mv2     
    where m is the mass of the comet and v is the speed 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
    

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

5) Calculating the total energy (per comet mass)

method 1

This method uses the angular (θ) and radial (r) positions of the comet as a function of time.  The user calculates the comet speed at perihelion from the given data using the method below.

Open the  .csv  file or the .ga3 file created by the Ephemeris Generator in Part 1 above. 

Determine the following at the point of the comet's closest approach to the sun (perihelion)

a) the comet-sun distance  r

b) the orbital speed  v  of the comet.  The best way to determine the speed of the comet at perihelion is to use the 2 positions separated by the smallest time interval closest to the time of perihelion.  At perihelionk, the orbital speed = (change in position)/(change in time)  =  (r * Δθ)/(Δt), where θ must be in radians.  Note that the velocity of the comet at perihelion is entirely tangential (see the Kepler's second law section above); the radial component of the velocity at perihelion is zero.  In other words, at perihelion    v  =  vt  =  r (Δθ/Δt) .

The orbital position data is presented in astronomical units (au) [for r] and degrees [for θ] and time steps are in units of days.  After measuring the two quantities above, convert the quantities into standard SI units (m, m/s).  Use these quantities to calculate the mass/energy in units of J/kg.

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. 


method 2

This method requires the direct use of the JPL Ephemeris Generator to find the precise speed of the comet at perihelion.  Because accurate results require the initial speed data to be good to cm/s accuracy, the user must find the time of perihelion (and, consequently, the perihelion speed) to the nearest minute.  A rough value for the most recent perihelion date of each comet is found the table above.

A short guide to using the JPL Ephemeris Generator for the purposes of this exercise:

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; names of the comets are in the table at top
    limit the search to "Comets Only" in the box, and then click on the "Search" button;
    the generator will either confirm the comet requested or may return a list of possible
        matches; if the latter, select the appropriate one & then click on the
        "Use Selected Asteroid/Comet" button

Observer Location

    for the purposes of this exercise, 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

    approximate perihelion dates for the suggested comets are in the table near the top
        of the page; start with a 30-day range of time that covers the approximate perihelion date;
    for Output Interval you have some options; you might want to start with 1 day and
        thereby find the nearest date nearest to when perihelion occurred;
        then you will have to re-run the Ephemeris with a narrower time range, but with a
        smaller Output Interval (perhaps 1 hour); once you know the approximate hour of
        perihelion, you will want to run the Ephemeris one last time with an even narrow
        time range, but with the Output Interval set to 1 minute...
        of course you could just run the Ephemeris once with a 30-day time span and a
        1-minute Output Interval, but you will then have to look though tens of thousands of data lines
    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:

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

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 two crucial pieces of data that will allow you to determine the comet's total energy at perihelion are:

    a) the column labeled  r  (the distance of the comet from the sun; the units returned are au)
    b) the column labeled VmagSn (the orbital speed of the comet relative to the sun);
        this is the quantity labeled v above (and =  vt  at the time of perihelion);  the units are km/s

6) determining the aphelion distance of the comet

Now let's assume that the principle of conservation of energy applies.  Therefore the comet energy at perihelion (determined in the previous section) must equal the comet energy at aphelion.  Or

Ep  =  Ea

(where the subscripts p and a stand for perihelion and aphelion, respectively)

or

1/2 vp2  - G M/rp   =   1/2 va2  - G M/ra

Our goal is to determine ra, the aphelion distance of the comet (and to see whether it lies in the Kuiper belt or the Oort cloud).  We have already determined  rp  and  vp .  But how are we to determine  va  ?   Fortunately, we can use the principle of conservation of angular momentum (Kepler's second law) here.  The angular momentum of the comet remains constant throughout the orbit.  Therefore, we can set the angular momentum at perihelion equal to the angular momentum at aphelion  (see section 3c above), or

m  rp  vp  =   m  va  ra

and

vp/va   =   ra/rp

However, we also know that the perihelion distance of the comet is much greater than the aphelion distance.  For Kuiper belt comets (and Jupiter-family comets), the latter is likely to be at least 10x greater than the former.  Therefore the speed at perihelion is going to be 10x greater at perihelion than aphelion.  Furthermore, since speed appears squared in the energy equation, that means the kinetic energy at aphelion is more than 100x greater than the kinetic energy at aphelion.  However, this 1% "mistake" will significantly affect the accuracy of a Kuiper belt calculation; remember that the best possible accuracy (employing all significant figures given by the ephemeris) is required for good agreement with known orbital parameters.  For an Oort cloud comet, the error made in assuming that the speed at aphelion is truly negligible and should not affect the results.

We now have enough information to determine the aphelion distance of the comet and then the period of the comet.

When the comet is furthest from the sun, is it in the Kuiper belt or the Oort cloud?