Objects in orbit generally do so in an elliptical orbit. A circular orbit is actually a special case of ellipse where the eccentricity e = 0. Ellipses have eccentricity between 0 (circle) and less than 1.
$$ 0 \leq e < 1 $$
At e = 1 , the curve is no longer closed. It becomes a parabola. And for e > 1, it is then a hyperbola. We will look at parabolas and hyperbolas later, they pertain to when something reaches escape velocity.
First we need to learn some of the properties of ellipses. It will also be useful to describe them using polar coordinates.
We call the long axis the major axis. Can you guess what the short axis is called? Yes, the minor axis.
What we want to use are half of those values. The
semi-major axis = \( a \)
and the
semi-minor axis = b.
When we looked at circular orbits before, we used r which is the radius of the circular orbit. Here, the important parameter is the semi-major axis \(a\).
The eccentricity of an ellipse is a measure of how "elongated" the ellipse is, and is related to the semi major and semi minor axes by
$$ e = \sqrt{1- \frac{b^2}{a^2}} $$For a circle a = b. And the eccentricity is thus zero.
You can draw an ellipse using a loop of string and two pins. Stick two pins, not too far apart, on a sheet of paper, then put the loop of string around them. Put a pencil inside the loop and try to draw a circle. You will get an ellipse instead.
It happens that the positions of the two pins are the foci of the ellipse.
Why is this important? When something is in an elliptical orbit around a massive body such as a planet, that massive body will be at one of the foci of the ellipse. We will do most calculations as if that focus is the centre.
The closest point is called the periapsis, while its farthest point is called its apoapsis.
Those points can be referred to by other names too, depending on what the massive body is. For example, if studying objects orbiting the Sun, the periapsis is called the perihelion while the apoapsis is called the aphelion. (Helios is an ancient Greek god of the Sun, also called Apollo). For things orbiting Earth, we use the terms perigee and apogee, because geo refers to Earth in most sciences.
$$ r = \frac{a(1-e^2)}{1 + e \cos \theta } $$
Where r is the distance between the satellite and the centre of the massive object, a is the semi-major axis, e is the eccentricity, and \(\theta\) is the angle between the semimajor axis at the periapsis and line of r, and is called the true anomaly.
Just for the sake of consistency, let us make the orbit move anticlockwise, starting from the peripasis.
Equation of ellipse using polar coordinates
$$ r = \frac{a(1-e^2)}{1 + e \cos \theta } $$Comets have orbits that are highly eccentric ellipses. Planets have orbits that are almost circular.
The Second Law is that the orbital period T is related to the semimajor axis a, $$ T^2 \propto a^3 $$ or $$ T \propto a^{3/2} $$
For circular orbit, we already derived $$ T = \frac{2 \pi}{\sqrt{GM}} r^{\frac{3}{2}} $$ It turns out we can substitute r with a so that $$ T = \frac{2 \pi}{\sqrt{GM}} a^{\frac{3}{2}} $$ The Third Law describes the speed of something orbiting the massive body and how it depends on the distance to the massive body. It moves faster when it is closer to the massive body, amd more slowly when it is far away.If we take a time period of t, the object will cover a greater distance if it is near periapsis than when near apoapsis. If we drew two triangles with areas A1 and A2 like below, they will have the same area.
The variation in velocity can be described using the Vis-Viva Equation:
$$ v^2 = GM \left( \frac{2}{r} - \frac{1}{a} \right) $$We can use the ellipse equation and set the semi major axis and eccentricity, calculate the distance \(r\) when it is at the angle \(\theta\). Then we can use the Vis Viva Equation to calculate its speed.
The next section will show some applications of these equations.