Effective cross-section of catching a comet with the Sun

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SUMMARY

The effective cross-section (σ) for catching a comet with the Sun is defined by the formula σ=πb², where b represents the semi-minor axis of the comet's elliptical trajectory. To determine b, one must consider the mass of the Sun (M), its radius (R), and the initial velocity (v) of the comet. The challenge lies in calculating the semi-major axis and linear eccentricity, which are essential for accurately determining the trajectory and effective cross-section. Utilizing principles of energy conservation and angular momentum is crucial in this analysis.

PREREQUISITES
  • Understanding of orbital mechanics and elliptical trajectories
  • Familiarity with the concepts of semi-major and semi-minor axes
  • Knowledge of energy conservation in gravitational fields
  • Basic principles of angular momentum
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  • Study the conservation of energy in celestial mechanics
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Hello.

1. Homework Statement
I have to find formula for effective cross-section (σ) of catching a comet with the Sun. I know mass of the Sun (M), radious of the Sun (R) and initial velocity (v) of the comet (at a big distance from the Sun).

Homework Equations


σ=πb^2, where b is on the picture (b>R)It is the biggest distance when the Sun still catch the comet.
obrázek.jpg


The Attempt at a Solution


I really don't know how to work with the b. It is semi-minor axis in in an elliptical trajectory of the comet. However, I don't know semi-major axis or linear eccentricity...
I just know the initial velocity v of the comet. I think I sould work with energy or angular momentum at the beginning and near the catching. However, I still have there some distances which I don't know.

Please, could you give me an advice?
Thank you.
 
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