TISSERAND PARAMETER
-----------------
John Pazmino
NYSkies Astronomy Inc
www.nyskies.org
nyskies@nyskies.org
2020 August 23
Introduction
----------
When orbital elements are given for an asteroid or comet, among
the parameters is the 'Tisserand parameter' (tih-seh-RAHND). It is
listed in clear text or symbolcd as 'T-Jup' or 'Tj". Since this number
isn't directly needed to track the asteroid or comet and it is not
requested in solar system software, home astronomers generally pass it
over.
The Tisserand parameter is rarely described in the casual
astronomy litterature. It does show up in the technical articles,
usually as an inert number, like in a table of asteroids discussed in
the article.
Tisserand parameter is sometimes called Tisserand criterion,
invariant, equality. That the clue for home astronomers to get
familiar with it, specially for asteroids and comets whose orbits are
distorted by close passes of the major planets. In fact, it was
invented precisa mente to help identify such objects.
Lost comets
---------
In the mid 19th century, study of comets exploded partly from the
connecttion of comets with meteor showers and from ongoing advances in
astrodynamics.
With increasing numbers of comets discovered and tracked, one
strange problem came up. Some comets after returning to view for a few
rounds of their orbits, suddenly stopped showing up. Altho their
orbits seemed secure and searches were careful the comet simply didn't
come back.
A collateral problem related to new comets, initially treated as
a genuine discoveries, Some of them had orbits that should have let
them be observed on previous returns. They weren't.
What happened to the lost comets? How did they go missing? What's
with the new comets? Why were they missed on earlier returns?
Jupiter jiving
------------
In the 19th century, into the mid 20th century, we could observe
comets only as far as 2 or 3 AU from the Sun. Farther out than that
the comets were just too faint for the eyes and instruments of the
time. Recall that comet Koklhoutek in 1973 was at first a sensation
because when discovered it was around 5 AU from Sun. For the comet to
be seen that far away, it must be highly luminous. Kohoutek could be a
major feature in the sky when it rounded perihelion.
OK, Kohoutek fizzled out. it never cast shadows at night. It
didn't spit flaming debris over the world. Its tail didn't wrap around
the sky. It did show the prevailing limits on detecting and following
comets beyond the asteroid belt.
We already knew of comets that, by calculation, passed close to
Jupiter and had their orbits changed. Lexell's comet, from the 1700s,
is one famous example. Because orbit computations were done by hand,
it was a delicate and tedious chore to analyze the behavior of a comet
caught in Jupiter's gravity. In particular, the orbit after the close
approach could be that of a 'new' comet that escaped previous notice.
It now is making its initial visit in the new orbit.
An old comet went missing because it no longer runs in its former
orbit. After its last known visit it suffered orbit distortion near
Jupiter and now is in a new orbit, perhaps to show up as a newly found
comet. .
One method to identify such a comet is to plot the comet's old
path until it enters Jupiter's gravity field. Then plot backwards the
path of the new comet until it leaves Jupiter's gravity. If thetwo
paths intersect in space and time near Jupiter, a link is established
between a lost comet and a false new one.
Comets, and asteroids, can be disturbed by any planet, even Earth.
By far the greatest number and severity of comet perturbations come
from Jupiter. It alone has most of the solar system mass outside of
the Sun. Its gravity field alone extends much farther out than for any
other planet, making the volume of space around it a dangerous trap
for nearby comets.
Jupiter is also the best planet for gravity-assist maneuvers of
spacecraft. The path arriving at Jupiter is deliberately chosen so
that the path leaving Jupiter heads the probe to its target.
Restricted orbits
---------------
We have a case of three bodies in mutual gravity freefall: Sun,
comet, Jupiter. The path of the comet is primarily a Kepler orbit
around the Sun.
As long as the comet stays away from Jupiter, the rough rule being
more than 1/2 AU away, it follows a two-body path. Home astronomers,
now with calculettes and computers, delight in plotting comet orbits
with the two-body Kepler model.
When the comet comes too close to Jupiter, within 1/2 AU,
Jupiter's gravity is a significant fraction of the sun's gravity. It
can divert the comet into a new orbit. During this time we are playing
with two large seats of gravity acting on a point. mass. Sun and
Jupiter are the large masses; the comet, the point mass.
There is no overall method for working out the general case of
three bodies in gravity freefall about themselfs. We can work with
special three-body cases, like the Sun-Jupiter-comet problem.
To make life easier, we make the large masses orbit in circular
orbits, zero excentricty and constant orbit radius. With Sun some one
thousand times more massive than Jupiter, the Sun-Jupiter center of
gravity is within the photospheric globe of the Sun. The Sun can be
fixed with all mutual motion assigned to Jupiter.
This model is the restricted circular three-body model.
But Jupiter has an elliptical orbit and it is inclined against the
ecliptic. This deviation from the ideal model must complicate working
out the comet path.
By good fortune, Jupiter's orbit is almost circular, with small
excentricity. and its inclination is small, 1.3 degree. For good
approximation we got a textbook restricted circular three body model.
If is used by astrodynamicists for most work, except for extremely
delicate cases such as planning spacecraft maneuvers near planets.
Conserved properties
------------------
In a two-body system a large mass with a point mass orbiting it,
the large body is the origin of a Cartesian coordinate grid, thru
which the point mass moves. As it moves, its position, velocity, and
acceleration continuously change. For a stable orbital motion the
changes cycle thru each lap of the orbit. For one direction, of the
three in the xYZ grid, we have X for position, del(X)/del(T) for
velocity, del2(X)/(delT)2 for acceleration. All three properties, in
each direction, change -- in value and direction -- along the orbit.
In the mid 19th century Carl Jacobi discovered that suitable
mathematical combinations of these properties, remain constant around
the point mass's orbit. While each property does vary, the set of them
together does not.
This, he found, is true a;sp for a point in an unstable orbit
around the large body, not in a nitid orbit. An example today is a
spaceprobe missing its intended solar orbit due to rocket malfunction.
He expressed this constancy of properties in equations used in
astrodynamics today as the Jacobi integral. It has a value
Jacobi expanded his work to the restricted three-body case, like
Sun-Jupiter-point. The maths are more complex because the gravity
fields of the Sun and Jupiter overlap around the comet. As long as the
comet is mainly in solar orbit, with only disturbance from Jupiter,
its Jacobi integral is constant.
Tiiserand parameter
-----------------
Francois Tisserand flourished in the late 1880s as director, in
turn, of Paris Observatory and Toulouse Observatory. He was among the
first to study the astrodynamics of comets. He examined many instances
of lost and new comets. Maybe the Jacobi integral can screen for
candidate pairs of comet that were jived by Jupiter. The maths for
The Jacobi equations were fiddly to work with. But orbit
computations were far more odious. In Tisserand's day -- into the mid
20th century -- astrodynamics was carried out by hand. There were only
clumsy mechanical calculators that handled arithmetic. Slide rulers
were the main 'computer' on the astronomer's desk. Tables of maths
functions were quickly well-worn.
An other factor was that astronomers work with the classical orbit
elements, not physics XYZ coordinates. The two are commutable, but
Tisserand wanted a way to directly use the orbit elements, which are
to hand for any comet.
Tissernad's massaging of Jacobi's integral yielded a simple
formula with only the orbit elements, which are taken recta mente from
observations of comets. It can be quickly worked out with a sci/tech
calculette as
Tj = (aj / a) + (2 * cos(i) * sqrt(((a / aj) * (1 - e^2)))
Th is the usual symbol for Tisserand parameter. aj is Jupiter's
semimajor axis. a, e, i are the comet's semimajor axis, excentricty,
inclination.
For a pair of candidate comets possibly suffering orbit diversion
from Jupiter, the formula is evaluated for the 'before' and 'after'
comet. If the Tisserand value is the same, or nearly so, chances are
the pair is one comet running in two different orbits.
It helps to write the formula with fill-in boxes for the element
values. Then work out each segment in turn.
For repetitive calculations put the formula into computer code.
The code requests the input items. aj can be hardcoded as '5.20' since
it is a constant for all instances. Check the argument of cosine. Most
calculettes allow degree. Computers typicly need radians. Code the
deg->rad conversion to let degree be keyed in.
Tisserand parameter is a dimensionless number. The AU for the
semimajor axes of Jupiter and comet cancel in the fractions. The
degree of inclination is swallowed in the cosine. EXcentricity is
already dimensionless.
Stricta mente we need the comet's inclination relative to
Jupiter's orbit. This involves some fancy geometry to obtain. Because
Jupiter is only 1.3 degree titled from the ecliptic, we neglect it.
use the comet's ecliptic inclination as is and live with any slight
inequity of Tisserand's parameter across the Jupiter encounter.
Comet Schwassmann-Wackman-2
-------------------------
comet Schwassmann-Wackman-2 is a short-period comet that behaved
itself for several decades. After its perihelion in 1994 the next
expected return would be in early 2001. it didn't show up.
An apparent new comet was found in 2002, also short-period, with
no record of a previous visit.
Orbit tracking showed that in 1997 SW2 suffered orbit distortion
near Jupiter. The 2002 comet was SW2 in its new orbit. The orbits are
-------------------------------------
perihelion |SMA, AU| exc |inc, d | per, y | Tj
------------+-------+-------+-------+--------+----- -
1994 Dec 22 | 3.444 | 0.399 | 3.753 | 6.391 | 2.991
2002 Jan 18 | 4.235 | 0.195 | 4.550 | 8.715 | 2.992
--------------------------------------------------------
The Tsserand parameter for the old adnd new comet are almost
equal.
By the 1990s, astrodynamics advanced to offer confident orbit
simulation of comes when interacting with Jupiter. We figured that SW2
will not return in 2001 and keep watch for it in 2002.
Comet Wolf
-- ---
Comet Wolf ran against Jupiter between its 1918 and 1925
perihelia. The old and new elements are
-------------------------------------
perihelion |SMA, AU | exc |inc, d | per, y | Tj
------------+-------+-------+--------+--------+----- -
1918 Dec 13 | 3.582 | 0.559 | 25.283 | 6.794 2.696
1925 Nov 08 | 4.092 | 0.405 | 27.294 | 8.279 | 2.712
--------------------------------------------------------
In this instance the two Tj are different by a couple percent. The
parameter invariance is not perfect. In general, matching before and
after the Jupiter proximity, at first we would put the two comets
aside as candidates for more detailed study.
Wolf's orbit is steeply tilted on the ecliptic, which could
suggest that it resists distortion by Jupiter. In the 1910s-20s it
probably wasn't feasible to track Wolf as it was perturbed by Jupiter.
Manual maths indicated it did get too close to Jupiter in 1921 and got
its orbit wrinkled.
Modern computer simulation -- with home astronomy software --
shows what happened with Jupiter and Wolf. It turns out that the
ascending node of Wolf's orbit is right at Jupiter's orbit! This
guarantees encounters every few decades.
Comet Oterma
----------
Comet Oterma was a well-behaved comet thru 1958 with a 7.9 year
period. In 1963 it suffered an orbit shift by proximity to Jupiter.
Its new orbit stayed too far from the Sun to ignite its gases and
Oterma became inactive. From 1958 until 2001 it was lost from sight by
its lack of self luminance. The elements are
-------------------------------------
perihelion |SMA, AU| exc |inc, d | per, y | Tj
------------+-------+-------+-------+--------+----- -
1958 Jun 10 | 3.958 | 0.144 | 3.986 | 7.874 | 3.036
2002 Dec 22 | 7.237 | 0.244 | 1.943 | 19.469 | 3.005
----------------------------------------------------
The Tj values differ by only about one percen, within the limit to
send the two Oterma comets to detailed study.
Oterms'a current orbit brings it close to both Jupiter and Saturn
for several interactions in the 24st century. It is difficult to
observe now due to its inert state like a small Centaur asteroid.
Asteroid Apophis
----- ------
This example is asteroid Apophis, supposedly to whack Earth in
2036. It is now in an Aten orbit until 2029. It then does a very
close flyby of Earth that throws Apophis into an Amor orbit. That
orbit, maybe?, aims Apophis into Earth in 2036.
-------------------------------------
perihelion |SMA, AU | exc |inc, d | per, y | Te
------------+-------+-------+-------+--------+----- -
2029 Jul 19 | 0.922 | 0.191 | 3.345 | 0.885 | 2.966
2036 Jun 11 1.007 | 0.180 | 4.298 | 1.011 | 2.967
----------------------------------------------------
Tisserand parameter is for Earth, Te, this time because Earth is
the perturbing object. aj is the Tisserand formula is replaced by ae =
1.00 AU. All else is the same. The parameters before 2029 and after
then are almost the same.
Because Apoplhis starts as an Aten asteroid and Earth throws it
into an Amor orbit, we debated about naming the asteroid. Atens are
traditionally named for Egyptian figures while Amors carry names of
Greek figures. The solution was to apply the Greek name of an Egyptian
deity. In this case the god of destruction was chosen, from the
fears of collision with Earth.
Spaceprobe Ulysses
----------------
We work the Tisserand parameter for the Ulysses spaceprobe. This
probe explored the polar regions of the Sun but sapacefaring arts and
skills could not put it into a polar solar orbit directly from Earth.
We used Jupiter in 1991 to do a back flip of Ulysses, out of the
ecliptic plane, into a polar orbit around the Sun.
This is an extreme case of orbit perturbation and of gravity
assist. Gravity assist is merely a deliberate approach to Jupiter
along a prescribed orbit to change that orbit to an other desired one.
The second orbit typicly aims toward a target else where in the solar
system.
Ulysses did not go to the Sun. It went around the Sun along a
huge ellipse, with perihelion about 1.3 AU and period about 6.1 year.
It passed over the Sun's north, and then south, pole about 2 AU
away. This is a remote standoff for studying the Sun's poles. Ulysses
had instruments to magnify the Sun's image about as well as those on
Earth. The principle advantage was that the poles were observed face-
on, not tangently along the limb of the Sun.
The orbital elements for the segment Earth-Jupiter and Jupiter-Sun
are
-------------------------------------
path segment |SMA, AU | exc | inc, d | per, y | Tj
-------------+-------+-------+-------+--------++------
E-J bef 1991 | 8.992 | 0.889 | 1.991 | 26.650 | 1.782
J-S aft 1991 | 3.373 | 0.603 | 79.128 | 6.195 | 1.784
----------------------------------------------------
Tisserand parameters for the two segment are almost equal.
Achieving the solar orbit was a major f3qt of astronautics.
Ulysses was sent to Jupiter along a long skinny ellipse, noted by its
high excentricity. It swooped over Jupiter's south pole in 1991 and
was pulled around and back toward the Sun. it left Jupiter on a large
mildly elliptical orbit. It still runs in it today, long after the
probe's project was finished. In some 166 years, 27 laps of Ulysses,
Jupiter and Ulysses may meet up again for a new orbit perturbation.
Closer to Sun?
------------
The actual Ulysses mission was deprecated by some space fans
because the probe didn't come close to the Sun. At closest it was 1.3
AU away, farther than Earth. Couldn't the orbit be closer by a Jupiter
gravity assist?
Since we purposely set up the approach path of Ulysses and its
reproach path toward Sun, we can play with other paths around the Sun.
let's put Ulysses perihelion at 1/2 AU, so the polar crossings are
some one AY away. Is this orbit possible?
Tisserand parameter for the two paths are equal, so once the
approach path is chosen, there are constraints on the reproach path.
To keep things simple, we hold to the 79 degree inclination to reduce
the degrees of freedom.
The answer is easiest found by trial-&-error by repeated
application of the Tisserand formula. The loose variable is the orbit
excentricity. This is a trivial exercise for the formula written into
computer code.
First we need the semimajor axis of the proposed orbit. We have an
apohelion near Jupiter and perihelion at 1/2 AU.
a = (apoh + perih) / 2
= (5.20 + 0.50 / 2
= 2.85 AU
This orbit has an excentricity of
e = (apoh - perih) / (apoh + perih)
= (5.20 - 0.50) / (5.20 + 0.50)
= 4.70 / 5.70
= 0.825
Couldn't Ulysses upon reaching Jupiter be aimed into this orbit in the
stead of its actual one? It could if the Tisserand value of the
departure path equals that of the arrival path.
Keeping with the same inclination of 79.128 degree, we find Rj =
1.982. This is too high for a feasible gravity assist. It can be
realized with heavy rocket thrust applied during the assist maneuver,
but not by letting Jupiter do all the work.
As much a bonus to human spacefaring is the gravity assist
process, it can not allow an arbitrary choice of paths to and from
Jupiter. A given approach path yields a family of possible reproach
paths and the proposed one here is not one of them.
We enter the trials with Tj = 1.784, a = 2.85 and i = 79.000
It is the task of the mission designer and planner to select
arriving orbits to yield a feasible departure orbit to get the
spacecraft to its target.
Conclusion
--------Tisserand parameter is a truly fun way to exploit the cloudy
nights. Using solar system simulators working by astrodynamics, not
trolley-track, orbit the path of a comet near Jupiter can be watched
on screen in a couple hours.
The parameter is also a vivid way to illustrate energy and angular
momentum conservation in a gravity field, a principle not so easily
grasped thru most astronomy tuition