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Coulomb Scattering The trajectory from an inverse square force forms a conic section. e < 1 ellipse e =1 parabola e >1 hyperbola. The force center is at a focus. Hyperbolic Orbits q attractive focus a r a ae repulsive focus Orient the incident axis to horizontal.

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Presentation Transcript
hyperbolic orbits
The trajectory from an inverse square force forms a conic section.

e < 1 ellipse

e =1 parabola

e >1 hyperbola.

The force center is at a focus.

Hyperbolic Orbits

q

attractive focus

a

r

a

ae

repulsive focus

reoriented view
Orient the incident axis to horizontal.

Scattering mass forms a hyperbolic trajectory.

Reoriented View

q

attractive focus

a

r

a

ae

repulsive focus

h

v0

q

m

b

x

impact parameter
The potential is defined at infinity.

The impact parameter would be closest approach for no force.

Compare to angular momentum

Impact Parameter

h

v0

q

m

x

b

rutherford cross section
Scattering cross section is based on the potential.

Impact parameter

Differential impact parameter

The result is the Rutherford scattering cross section

Rutherford Cross Section

q

b

y

lunar miss
Problem

A spaceship of mass m moving with velocity v0 approaches the Moon.

The impact parameter is b.

The velocity v0 is perpendicular to the orbital velocity V of the moon.

Show that if the spaceship passes behind the Moon it gains kinetic energy as it leaves the Moon.

Lunar Miss

q

M

b

x

m

lunar frame
The moon is much more massive.

Lunar frame is CM

Incident and final energies same in CM

Lunar Frame

b

m1

v0

q

x

b

M

energy boost
In the Moon’s frame the velocity components come from the scattering angle.

In the observer’s frame the velocity is boosted by the moon.

Energy Boost

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