Skip to main content Accessibility help
×
Home

A Note on a Separation of Equations of Variation of the Elliptic Restricted Three-Body Problem Into Hill's Equations

  • V. Matas (a1)

Abstract

The equations of variation of the three-dimensional elliptic restricted three-body problem corresponding to the equilibrium solutions (the libration points) have been separated into three Hill's equations. As regards the equation ‘corresponding’ to the motion of the infinitesemal body in the z-axis (perpendicular to the plane of motion of the primaries), the matter is trivial one since the initial equation - as known - reads d2 z/dv 2 + (Ai + e cosv)/(l + e cosv) = 0 (e, 0<e< 1, and v are the eccentricity and the true anomaly of the relative motion of the primaries) with Ai > 1 for the straight-line libration points Li (i= 1, 2, 3) and Ai =l for the triangular libration points Li, i=4, 5. As concerns the remaining two components, x and y, of the motion of the infinitesimal body (x, y and z are the Nechvíle's variables), in the case of the straight-line libration points, L 1, L 2 and L 3, the corresponding equations of variation have been transformed and separated into two further - mutually independent - Hill's equations without any limitation. In the case of the equilateral triangle libration points, L 4 and L 5, the separation has been found only when the eccentricity e and the dimensionless mass μ, 0<μ≦1/2 of the ‘minor’ primary satisfy the additional conditions: Let us write the latter two Hill's equations obtained in the form where Ik, k= 1, 2, are 2π-periodic even functions of the true anomaly v. The functions Ik, k = 1, 2, are real functions in the case of the straight-line libration points, L 1, L 2 and L 3, without a limitation but in the case of the triangular libration points, L 4 and L 5, they are real only if Provided the functions Ik, k= 1, 2, are complex-valued functions of the real variable v.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      A Note on a Separation of Equations of Variation of the Elliptic Restricted Three-Body Problem Into Hill's Equations
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      A Note on a Separation of Equations of Variation of the Elliptic Restricted Three-Body Problem Into Hill's Equations
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      A Note on a Separation of Equations of Variation of the Elliptic Restricted Three-Body Problem Into Hill's Equations
      Available formats
      ×

Copyright

A Note on a Separation of Equations of Variation of the Elliptic Restricted Three-Body Problem Into Hill's Equations

  • V. Matas (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed