Trigonometrische substitutie

Molniya satellites, known for their highly elliptical orbits, trace distinct regions of space during each revolution around Earth.

To estimate the area enclosed, the orbit is modeled as an ellipse in its standard form, which is then rearranged to an expression involving a square root term.

Because the ellipse is symmetric about both axes, the area in the first quadrant is computed using integration and then multiplied by four to get the total area.

Here, direct integration is difficult due to the square root. To resolve this, trigonometric substitution is used by substituting x in terms of the sine function. This substitution converts the square root term into a simple cosine function, and the differential changes accordingly.

The integration limits are then updated to correspond to the angular domain, ranging from zero to pi over two. The integrand is rewritten using a trigonometric identity to simplify the integration process. This definite integral is evaluated over the new limits.

The final expression yields the total area swept by the satellite during one complete orbit.