Newtonian restricted three-body gravitational problem with positive and negative point masses

The Newtonian restricted three-body problem involving a positive primary point mass, <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msub><a:mi>m</a:mi><a:mo>+</a:mo></a:msub></a:math>, and a negative secondary point mass, <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:msub><c:mi>m</c:mi><c:mo>−</c:mo></c:msub></c:math>, in a circular orbit, and a positive or negative tertiary point mass, <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:msub><e:mi>m</e:mi><e:mn>3</e:mn></e:msub></e:math>, with <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:msub><g:mi>m</g:mi><g:mo>+</g:mo></g:msub><g:mo>&gt;</g:mo><g:mo stretchy="false">|</g:mo><g:msub><g:mi>m</g:mi><g:mo>−</g:mo></g:msub><g:mo stretchy="false">|</g:mo><g:mo>≫</g:mo><g:mo stretchy="false">|</g:mo><g:msub><g:mi>m</g:mi><g:mn>3</g:mn></g:msub><g:mo stretchy="false">|</g:mo></g:math>, is solved. Five Lagrange points are found for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:msub><m:mi>m</m:mi><m:mn>3</m:mn></m:msub></m:math>, three of which are coplanar with <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:msub><o:mi>m</o:mi><o:mo>+</o:mo></o:msub></o:math> and <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:msub><q:mi>m</q:mi><q:mo>−</q:mo></q:msub></q:math>, and two of which are not, a subtle consequence of the gravitational repulsion from <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:msub><s:mi>m</s:mi><s:mo>−</s:mo></s:msub></s:math>. All Lagrange points are linearly unstable, except for one point in the regime <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:msub><u:mi>m</u:mi><u:mo>+</u:mo></u:msub><u:mo>≳</u:mo><u:mn>8.4</u:mn><u:mo stretchy="false">|</u:mo><u:msub><u:mi>m</u:mi><u:mo>−</u:mo></u:msub><u:mo stretchy="false">|</u:mo></u:math>, which is linearly stable and collinear with <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"><y:msub><y:mi>m</y:mi><y:mo>+</y:mo></y:msub></y:math> and <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"><ab:msub><ab:mi>m</ab:mi><ab:mo>−</ab:mo></ab:msub></ab:math>. Published by the American Physical Society 2024