Tisserand's Parameter in 2 Dimensions

Here is another interactive graph about Tisserand's parameter. My first post about it has the explanation what it is, and also a line graph that adapted to the set orbital parameters semimajor axis a, eccentricity e and inclination i of a perturbed body and the chosen perturber, any giant planet. First, here is the definition again, where T_p is the Tisserand parameter:

$$T_p=\frac{a_p}{a}+2\sqrt{\frac{a}{a_p}(1-e^2)}cosi$$

As stated in the above post, the nice thing about it is that it is a quasi-conserved quantity that stays more or less the same before and after a close encounter of an asteroid, a comet or a space probe with a large body. Even nicer would be to see a whole lot of parameter-combinations at once, as a T_p-heat map combining any two parameters in a plane. Like this:

As you can see, the graph lets you select the parameter combination a vs. i, a vs. e, or e vs. i, and chose a value for the third one separtely, for each of the giant outer planets, which are represented by a red circle in the plane. The currently selected parameters are indicated in the resulting formula at the lower right. The Tisserand value below the crosshair is indicated close to it, along with the peri- and aphelion (closest and farthest distance to the Sun) for the given combination of parameters.

So that means paths along the same line of Tisserand values indicate changes of the plotted parameters achievable by a flyby of the respective giant planet, be it for a spacecraft, asteroid or comet. At least that's what I think it means.

The graph uses D3.js for display and interactions, and MathJax for math-formatting. Check out the source code on GitHub

Tisserand's Parameter

One topic that comes up frequently in orbital dynamics is the Tisserand parameter. So as an interested layman, I'm curious about it, and have several questions about it, like "what is the Tisserand parameter?" or "how is it useful?". Well, first, here is it's definition:

$$T_p=\frac{a_p}{a}+2\sqrt{\frac{a}{a_p}(1-e^2)}cosi$$

As you can see, it relates the orbital parameters semimajor axis a, eccentricity e and inclination i of a perturbed body (i.e. a small body encountering a larger body) to each other and to the semimajor axis of the perturber a_p, like, say, Jupiter, or any larger body in the Solar System. The result is the dimensionless value called the Tisserand parameter T_p, or Tisserand's criterion.

The nice thing about it is that it is a quasi-conserved quantity that stays more or less the same before and after a close encounter of an asteroid, a comet or a space probe with a large body. That makes it a useful tool to identify prior visits of periodic comets, whose orbital parameters may have changed in the mean time, but since the Tisserand parameter stays about the same, old observations can be searched for a similar value.

Another use is the shaping of spacecraft trajectories, since each of the orbital parameters changes through a flyby in a predictable way. It is in fact this property of the Tisserand parameter that made achievements like the Voyager's Grand Tour or Cassini's epic tour of the Saturn system, possible. The former through flybys of all the giant planets, the latter case through regular flybys of Saturn's largest moon Titan.

OK, that's what it does, but I had a hard time visualizing how it behaves for each of the parameters. So I made the formula interactive:

As you can see, the graph lets you select each parameter a, e, and i separately, and chose a value for the other two. Select any kind of trajectory and see how it's Tisserand parameter changes relative to each of the outer planets, which are represented by a red circle. The currently selected parameters are indicated in the resulting formula at the lower right.

For asteroids the value relative to Jupiter is normally Tj < 3 and for comets within 2 < Tj < 3. If you select semimajor axis, you'll see that Tj only gets close to these values if the bodies' orbit goes anywhere close to Jupiter, and if you play around with e or i, the higher you chose them /(i.e. more comet-like) the lower Tj gets.

Maybe this helps somebody else as well.

The graph uses D3.js for display and interactions, and MathJax for consistent math-formatting. Check out the source code on GitHub