Question

The general Kepler orbit is given in polar coordinates by (8.49). Rewrite this in Cartesian coordinates for the cases that € = 1 and € > 1. Show that if e = 1, you get the parabola (8.60), and if € > 1, the hyperbola (8.61). For the latter, identify the constants a, b, and & in terms of c and €. Ueff (min) = E.

Answer 1

For e = 1, the general Kepler orbit yields a parabola. For
`\(\epsilon > 1\)`, it forms a hyperbola.
`\(U_{\text{eff}}(r_{\text{min}}) = E\)` signifies the effective potential energy equals total energy at
`\(r_{\text{min}}\)`.

The general Kepler orbit in polar coordinates is given by
`\(r(\theta) = (c)/(1 + e \cos(\theta))\)`, where c is a constant and
`\(\epsilon\)` is the eccentricity. To rewrite this in Cartesian coordinates:

1. For e = 1:

When
`\(\epsilon = 1\)`, the orbit becomes
`\(r(\theta) = (c)/(1 + \cos(\theta))\)`. By using the identity

`\(\cos(\theta) = (x)/(r)\)`, this simplifies to
`\(r = (c)/(1 + (x)/(r))\)`, which leads to the equation of a parabola.

2. For
`\(\epsilon > 1\)`:

When
`\(\epsilon > 1\)`, the orbit becomes
`\(r(\theta) = (c)/(1 + \epsilon \cos(\theta))\)`. Using

`\(\cos(\theta) = (x)/(r)\)`, it transforms to
`\(r = (c)/(1 + \epsilon (x)/(r))\)`. Identifying constants a, b, and c, this corresponds to the equation of a hyperbola.

Finally,
`\(U_{\text{eff}}(r_{\text{min}}) = E\)` implies that at the minimum radial distance

`\(r_{\text{min}}\)`, the effective potential energy is equal to the total energy E.