Tisserand Parameter & Encounter velocities Tisserand parameter Tisserand paramet

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Tisserand Parameter & Encounter velocities

Tisserand parameter Tisserand parameter with respect to Jupiter is given by T_J = (a_J/a) + 2cos(i) [(a/a_J)(l - e^2)]^1/2, where a_J is Jupiter's semi-major axis. An active comet is one that gets close enough to the Sun for its icy surface to sublimate; active comets usually have perihelia q lessthanorequalto 2.5 AU. Such bodies are usually crossing the orbit of Jupiter, so their dynamics tend to be dominated by that planet. The long period comets, also called Oort Cloud comets, have parabolic orbits with semimajor axes greaterthanorequalto 10^4 AU. Show that an active comet from the Oort Cloud will have T_J lessthanorequalto 2. In 1977, astronomer Charles Kowal discovered a minor planet in the outer solar system; it was subsequently named 95P/Chiron, and was the first member of the class of small solar system bodies called Centaurs. Given its orbital parameters of a = 13.68 AU, e = 0.379, i = 6.9 degree, calculate T_J for Chiron and comment on what dynamical classification would be appropriate for this object. Encounter velocities About two years ago, Comet C/2013 A1 (Siding Spring), an Oort Cloud comet, had a close encounter with Mars. Given that its orbit plane is inclined by about 130 degree to Mars' orbit plane, estimate the speed of encounter. (Neglect Mars' gravity.) Comet Encke is the shortest period comet known, and is unusual in many aspects. Given its orbital parameters of a = 2.214 AU, e = 0.848, i = 12 degree, calculate the encounter speed between Earth and Comet Encke (assuming Encke could have a node at 1 AU, and neglecting Earth's gravity).

Explanation / Answer

4a. Oort clouds are the long period comets with charcterisitic lenghty scale i.e Roc 100,000 AU and orbital period is 106-107 years. Total number of oort cloud which are larger than 1 km radius is Noc=1012. These are thought to be originated protoplanetary disk between neptune and jupiter. These comets have T<2 and passes extremely closer to the sun.

b.Since semi major axis of Jupiter aj=5.2 Au

Given a=13.68

e=0.379

i=6.9

Now Tj=(aj/a)+2cosi(a/aj)(1-e^2)1/2)

=(5.2/13.68)+2cos6.9((13.68/5)(1-0.379^2)1/2)

=0.38+2*0.99((2.6)(1-0.143)1/2)

=0.38+0.75(2.6*0.857)1/2

=0.38+0.75(2.22)1/2

=0.38+0.75*1.48

=0.38+1.11

=1.49

Since the T<2 these comets passes extremely closer to the sun and are called sungrazer comets.

5.a=2.214AU

e=0.848

i=1.2

semi major axis of earth=1 AU

Tj=(1/2.214)+2cos12(2.214/1)(1-0.848^2)1/2

=0.45+2*0.97(2.214)(0.719)1/2

=0.45+1.94(1.59)1/2

=0.45+1.94*1.2

=0.45+2.3

=2.75

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