1. 3-4. The Tisserand Relation
Let’s consider a comet with initial semi-major axis a, eccentricity e, and inclination I. Following a close approach to Jupiter the comet’s orbital elements become a’, e’, and I’.
We can use the Jacobi integral and some simple approximations to relate these two sets of elements. The Jacobi integral, given by CJ=2U-v2, remains constant throughout the encounter. In three-dimensional inertial space, the comet has a position vector r=(,,) and velocity vector In this system we can use Eq. (3.36) to write the Jacobi constant
(3.40)
where r1 and r2 are the distance of the comet from the Sun and Jupiter respectively. We will also choose units such that the semi-major axis and mean motion of Jupiter’s orbit are unity and, since the mass of Jupiter and the comet are much smaller than the Sun’s mass,
(3.41)
where mSUN, mJUPITER, and mCOMET are the masses of the Sun, Jupiter and the comet respectively. From the energy integral of the Sun-comet two-body problem (Eq. (2.34)) we have
(3.42)
where we have taken =1 in accordance with our system of units and where we are assuming r1≈r since the mass of the comet and Jupiter are effectively negligible compared with the mass of the Sun.

2. The angular momentum per unit mass of the comet’s orbit is given by
(3.43) r2
comet
The component of the angular momentum vector is given by
(3.44) r1
Therefore Eq. (3.40) can be written r
(3.45) O
If we assume that the comet is not close to Jupiter so that 1/r2 is always a small quantity and neglect the 2 term, we have 
(3.46)
Therefore the approximate relationship between the orbital elements of the comet before and after the encounter with Jupiter is given by
(3.47)
This is known as the Tisserand relation (Tisserand 1896).

3. This figure shows an example of such an encounter with Jupiter
This figure shows an example of such an encounter with Jupiter. The close approach to Jupiter alters the orbital elements of the comet with the semi-major axis increasing by almost 8 AU. The initial orbital elements of the comet are:
a=4.8 AU e= I=7.5° (before the encounter)
a’=10.8 AU e’= I=21.4 ° (after the encounter).
Although Tisserand relation is only an approximation to the Jacobi constant, which are derived by assuming that Jupiter is in a circular orbit, it is still a good approximation of the motion in the case where the eccentricity of Jupiter is nonzero (actually eJ=0.048).

4. In this figure, the variation in the quantity (1/2a)+sqrt[a(1-e2)]cosI as a function of time for two separate numerical integrations. In the lower case, Jupiter is taken to move in a circular orbit, while in the upper case the eccentricity of Jupiter is taken to have its current value of
Apart from the closest approach where the approximation is invalid (because 1/r2 is not small in this case, see the assumption of Eq. (3.46)), the value of Tisserand constant changes by <1% in the former case and <2% in the latter.

5. Dynamical Evolution of Comets
Comets have traditionally been divided into two major dynamical groups: long-period (LP) comets with orbital periods >200 yr, and short-period (SP) comets with periods <200 yr. Long-period comets typically have random orbital inclinations while short-period comets typically have inclinations relatively close (within ~35°) to the ecliptic plane.
In addition it has become common to divide the SP comets into two subgroups: Jupiter-family comets (JFC), with orbital periods <20 yr and a median inclination of ~11°, and Halley-type comets (HTC), with periods of 20–200 yr and a median inclination of ~45°.
A more formal dynamical definition of the difference between the JFC and HTC comets was proposed based on the Tisserand parameter. As we studied, Tisserand parameter is an approximation to the Jacobi constant, which is an integral of the motion in the circular restricted three-body problem. It was originally devised to recognize returning periodic comets that may have been perturbed by Jupiter.

6. An example of such an encounter for Jupiter family comet 81P/Wild 2 is shown below. A close approach to Jupiter alters the orbital elements of the comet with the perihelion distance from 4.95 AU to 1.49 AU (the semi-major axis from 12.3AU to 3.36AU).
81P/Wild 2 in early 1970’s 
Jupiter encounter
Jupiter 5.2
Saturn 9.5
Uranus 19.2
Neptune 30.1

7. Cometary Dynamics: Evolution to Asteroidal Orbits (2)
Jupiter-family comets have T > 2 while long-period and Halley-type comets have T < 2. Levison (1996) proposes that comets with T > 2 be known as ecliptic comets, while comets with T < 2 be known as nearly isotropic comets.
JFC
HTC
LPC

8. Comets on planet-crossing orbits are transient members of the solar system. Close and/or distant encounters with the giant planets, in particular Jupiter, limit their mean dynamical lifetimes to ~0.4–0.6 m.y. (Weissman, 1979; Levison and Duncan, 1997). Thus, they must be continually re-supplied from long-lived dynamical reservoirs.
The different inclination distributions of the ecliptic and nearly isotropic comets reflect their different source reservoirs. Nearly isotropic comets (Long-Period and Halley-Type Comets) are believed to originate from the nearly spherical Oort cloud. Ecliptic comets (Jupiter-family comets) are fed into the planetary system from the highly flattened Kuiper belt.

9. Fernández et al. (2001) compared measured albedos for 14 cometary nuclei and 10 NEOs with Tisserand parameters T<3, and 34 NEOs with T > 3. They showed that all of the comets and 9 out of 10 of the NEOs with T < 3 had low albedos, ≤0.07, while most of the NEOs with T >3 had albedos >0.15. This result suggests that the T <3 objects have a cometary origin.
© IAU: Minor Planet Center