Kepler Orbit - Mathematical Solution of The Differential Equation (<cite Id="equation_1" Style="font-style: Normal; | Mathematical Solution Differential Equation (<cite Id=equation

Mathematical Solution of The Differential Equation ( and polar coordinates

we have,

(2)

And the differential equation (1) takes the form (see "Polar coordinates#Vector calculus")

(3)

Taking the time derivative of (2) one gets

(4)

Using the chain rule for differentiation one gets

(5)

(6)

Using the expressions for of equations (2), (4), (5) and (6) all time derivatives in (3) can be replaced by derivatives of r as function of . After some simplification one gets

$\ddot{r} - r \cdot {\dot{\theta}}^2 = \frac {H^2} {r^4} \cdot \left ( \frac{d^2 r} {d\theta ^2} - 2 \cdot \frac{\left (\frac {dr} {d\theta} \right ) ^2} {r} - r\right )= - \frac {\mu} {r^2}$

(7)

The differential equation (7) can be solved analytically by the variable substitution

(8)

Using the chain rule for differentiation one gets:

(9)

$\frac {d^2r} {d\theta^2} = \frac {2} {s^3} \cdot \left (\frac {ds} {d\theta}\right )^2 - \frac {1} {s^2} \cdot \frac {d^2s} {d\theta^2}$

(10)

Using the expressions (10) and (9) for and one gets

(11)

with the general solution

(12)

where e and are constants of integration depending on the initial values for s and .

Instead of using the constant of integration explicitly one introduces the convention that the unit vectors defining the coordinate system in the orbital plane are selected such that takes the value zero and e is positive. This then means that is zero at the point where is maximal and therefore is minimal. Defining the parameter p as one has that

(13)

This is the equation in polar coordinates for a conic section with origin in a focal point. The argument is called "true anomaly".

For this is a circle with radius p.

For this is an ellipse with

(14)

(15)

For this is a parabola with focal length

For this is a hyperbola with

(16)

(17)

The following image illustrates an ellipse (red), a parabola (green) and a hyperbola (blue)

The point on the horizontal line going out to the right from the focal point is the point with for which the distance to the focus takes the minimal value, the pericentre. For the ellipse there is also an apocentre for which the distance to the focus takes the maximal value . For the hyperbola the range for is

and for a parobola the range is

Using the chain rule for differentiation (5), the equation (2) and the definition of p as one gets that the radial velocity component is

(18)

and that the tangential component (velocity component perpendicular to ) is

(19)

The connection between the polar argument and time t is slightly different for elliptic and hyperbolic orbits.

For an elliptic orbit one switches to the "eccentric anomaly" E for which

(20)

(21)

and consequently

(22)

(23)

and the angular momentum H is

(24)

Integrating with respect to time t one gets

(25)

under the assumption that time is selected such that the integration constant is zero.

As by definition of p one has

(26)

this can be written

(27)

For a hyperbolic orbit one uses the hyperbolic functions for the parameterisation

(28)

(29)

for which one has

(30)

(31)

and the angular momentum H is

(32)

Integrating with respect to time t one gets

(33)

i.e.

(34)

To find what time t that corresponds to a certain true anomaly one computes corresponding parameter E connected to time with relation (27) for an elliptic and with relation (34) for a hyperbolic orbit.

Note that the relations (27) and (34) define a mapping between the ranges

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