Turns out that when we try to get into how two bodies, only affected by the gravitational force, move around each other, we find that the solution is the general equation for a conic section. We'll explain how in more detail in the next post, but for now, let's brush up on ellipses and other conic sections.
First, when we describe these systems, it's often easiest to use polar coordinates. You can think of them as round coordinates- they work very well for round systems! Simply put, the points on the plane are determined by a distance from a reference point and an angle from a reference direction. Thus, \(x = rcos{\theta}\) and \(y = r\sin{\theta}\) (see Figure 1). When using vectors, we can use unit vectors to determine their direction. For polar coordinates we use \(\vec{e}_r\) and \(\vec{v}_{\theta}\).
Okay!
An ellipse is a conic section - a curve defined by the intersection of a cone with a plane.
?What sort of conic section we have depends on the incline of the plane, as you can see from Figure 2. All conic section can be described using polar coordinates by
\(r = \frac{p}{1+e\cos{f}}\)
where \(p=a(1-e^2)\) .
Ellipses can be close to a circular shape, or far from it, depending on what we call its eccentricity! We define this as:
\(e = \sqrt{1+\frac{b^2}{a^2}}<1\)
Recall that the equation for an ellipse centered at the origin is
\(\frac{x^2}{a^2}+\frac{y^2}{b^2} = 1\)
where a is the width and b is the height.
The eccentricity of an ellipse will always be less than 1 because its width, a, will always be larger than its height, b. If we know something about how 'stretched' our ellipse is, we can find the focal points by multiplying the eccentricity with the width a.
When talking about planetary orbits we refer to a as the semi major axis, and b as the semi menor axis. We also place our sun in one of the two focal points!
If the ellipse is a description of a body in orbit, what of hyperbolas and parabolas? These represent bodies with a high enough velocity to not get caught by, for example, the stars gravitational field.
We place a star in the middle of a system, and let an object pass by it.
A body with a very high angular velocity to the gravitational force will have its orbit affected a little, from the gravitational force, but ultimately won't be caught in orbit. This leads to a hyperbola sort of orbit. Recall from our previous post that the kinetic energy of the object is here larger than the potential energy of the gravitational field of the star.
If the kinetic energy is exactly the same as the potential energy from the gravity field of the star, the object will pass in a slightly steeper orbit than otherwise, a parabola orbit.
Finally, if the kinetic energy is lower than the potential energy, we will get the familiar elliptical orbit- and this is what we want to look at for our planets.