Effective cross-section of catching a comet with the Sun

In summary, the task is to find the formula for the effective cross-section (σ) of catching a comet with the Sun, with given values for the mass (M) and radius (R) of the Sun, and the initial velocity (v) of the comet at a large distance from the Sun. The equation for σ is σ=πb^2, where b is the semi-minor axis in an elliptical trajectory of the comet. However, the values for the semi-major axis and linear eccentricity are not known. The approach could involve using energy or angular momentum at the beginning and near the catching, but there are still unknown distances. Advice is requested for solving this problem.
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Oomph!
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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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1. What is the effective cross-section of catching a comet with the Sun?

The effective cross-section of catching a comet with the Sun refers to the area of the comet that would be directly impacted by the Sun's gravitational pull, causing the comet to change its orbit and potentially collide with the Sun.

2. How is the effective cross-section of catching a comet with the Sun calculated?

The effective cross-section is calculated by taking into account the size and mass of the comet, as well as its distance from the Sun and the strength of the Sun's gravitational force. It can be calculated using mathematical formulas and computer simulations.

3. Why is the effective cross-section important in understanding comet behavior?

The effective cross-section is important because it helps us understand the potential risks of a comet colliding with the Sun. It also gives us insight into the comet's trajectory and how it may be affected by other celestial bodies or forces.

4. Can the effective cross-section change over time?

Yes, the effective cross-section can change over time as the comet's orbit and distance from the Sun changes. It can also be affected by other factors such as the comet's composition or interactions with other objects in space.

5. How does the effective cross-section of a comet compare to other celestial bodies?

The effective cross-section of a comet is typically much smaller than that of larger celestial bodies such as planets or stars. This is due to the relatively small size and mass of comets compared to these other objects. However, the effective cross-section can still have a significant impact on the behavior of a comet as it interacts with the Sun and other bodies in its path.

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