Math 2.0

Mechanics of particles and systems Definition

Tisserand's parameter

Builds on Orbital elementsCircular restricted three-body problemJacobi integral

Definition

Let a planet move on a circular orbit of radius aP>0a_P > 0 about the Sun. Let a small body have osculating orbital elements about the Sun with semi-major axis a>0a > 0 and eccentricity 0≤e<10 \le e < 1. Let ii be its inclination measured from the plane of the planet's orbit. Tisserand's parameter of the small body relative to the planet is TP=aPa+2cos⁡iaaP(1−e2).T_P = \frac{a_P}{a} + 2 \cos i \sqrt{\frac{a}{a_P} (1 - e^2)}.

Remarks

In the circular restricted three-body problem the Jacobi integral is exactly conserved. Write it in heliocentric elements and drop the terms of the order of the planet-to-Sun mass ratio. It then says that TPT_P is nearly constant, away from the immediate vicinity of the planet.

Example: the planet itself, with a=aPa = a_P, e=0e = 0 and i=0i = 0, has TP=1+2=3T_P = 1 + 2 = 3. Bodies with TPT_P close to 33 have orbits similar to that of the planet and can meet it at low relative speed.

Non-example: the semi-major axis aa alone is not nearly conserved, since a close encounter with the planet can change it a lot while TPT_P stays nearly fixed.

Relative to Jupiter, Jupiter-family comets typically have 2<TJ<32 < T_J < 3.

Reading path

To understand this page from the ground up, read these 76 pages in order. The foundations come first.

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  6. Ellipse concept
  7. Euclidean space object
  8. Euclidean vector concept
  9. First integral concept
  10. Focus of a conic concept
  11. Function concept
  12. Hamiltonian system concept
  13. Identity matrix object
  14. Interval concept
  15. Matrix definition
  16. Orbital elements concept
  17. Ordinary differential equation concept
  18. Vector space definition
  19. Velocity concept
  20. Kepler problem example
  21. Fréchet derivative definition
  22. Cross product definition
  23. Ordinary differential equation definition
  24. Function of several variables concept
  25. Vector-valued function concept
  26. Lipschitz condition definition
  27. Classical mechanics concept
  28. n-body problem problem
  29. Square matrix definition
  30. Transpose definition
  31. Linear map definition
  32. Momentum concept
  33. Double integral concept
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  35. Limit of a function of several variables concept
  36. Inertial frame of reference concept
  37. Solution of an ordinary differential equation definition
  38. Orthogonal matrix definition
  39. Multilinear map definition
  40. Three-body problem problem
  41. Triple integral concept
  42. Newton's laws of motion statement
  43. Continuity in R^n definition
  44. Differentiable function of several variables definition
  45. Initial value problem definition
  46. Partial derivative definition
  47. Orthogonal group object
  48. Alternating multilinear map definition
  49. Centre of mass concept
  50. Differentiable on an open set definition
  51. Gradient definition
  52. Higher partial derivatives definition
  53. Picard–Lindelöf hypotheses definition
  54. Determinant definition
  55. Reduced mass concept
  56. Iterated derivative definition
  57. Picard operator definition
  58. Special orthogonal group object
  59. Angular velocity concept
  60. C^k function characterised by continuous derivatives statement
  61. Integral equation form of an initial value problem statement
  62. An iterate of the Picard operator is a contraction statement
  63. Rotating reference frame concept
  64. Vector field definition
  65. Picard–Lindelöf theorem statement
  66. Conservative vector field definition
  67. Classical mechanics definition
  68. Momentum definition
  69. Potential energy definition
  70. Angular momentum definition
  71. Central force definition
  72. Newton's law of universal gravitation statement
  73. Two-body problem problem
  74. Circular restricted three-body problem problem
  75. Jacobi integral concept
  76. Tisserand's parameter definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Tisserand%27s_parameter
About this page
TagENHQD5N9D8
URIm2://ENHQD5N9D8
Kinddefinition
WritingW0 stub
MSC70F07, 70F15
DefinesTisserand's parameter
UsesOrbital elements, Circular restricted three-body problem, Jacobi integral
Changed3 November 2026

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