Elliptical orbit
This activity belongs to the GeoGebra book The Domain of the Time.
Not all orbits of artificial satellites are circular. For example, the Molniya orbit is a highly eccentric ellipse (0.74) designed to cover high-latitude areas where geostationary satellites cannot reach.
Furthermore, no planetary or lunar orbit is circular. Artificial satellites often need to correct the magnitude and direction of their acceleration to maintain a circular orbit, something that celestial bodies cannot do.
Until now, we have assumed that the gravitational acceleration g remains constant. However, a planet M orbiting the Sun (S) does not always maintain the same distance from it, meaning it follows an elliptical orbit (although the ellipse is never perfect and varies slightly each year).
We will now refer to g as the gravitational acceleration caused by the mass of the Sun. Remember that this acceleration varies with the square of the distance d:
where G is the universal gravitational constant and mS is the mass of the Sun.
Now, instead of using this formula to conveniently scale the distances, and subsequently scale the time, we will create a much more open construction that facilitates observing the essence of the motion for different initial data. That is, instead of scaling the screen to represent reality, we will imagine that reality adjusts to the dimensions of the screen. In other words, in order to facilitate visualization, we will represent only the idea of that motion here. We represent the Earth, initially, at 9 units from the Sun, with an initial acceleration |g| = 1 and initial velocity |v| = v0. And we will let the slider anima do its work.
The result is an elliptical trajectory (although in many cases, such as with Earth, with very little eccentricity, it appears almost identical to a circular path).
Note that now the magnitude of g does not remain constant (except for v0 = 3), nor does the magnitude of v. Also observe that at the point in the orbit closest to the Sun (perihelion), as g increases, the maximum velocity v is reached, and the minimum at the farthest point (aphelion).
SCRIPT FOR SLIDER anima
# Calculate the elapsed seconds dt; add one second if t1(1) < tt
SetValue(tt, t1(1))
SetValue(t1, First(GetTime(), 3))
SetValue(dt, (t1(1) < tt) + (t1(1) − tt)/1000)
# Record the orbit time and the number of completed orbits
SetValue(reg, If(x(v) > 0 ∧ x(v + dt g) ≤ 0, Append(t, reg), reg))
SetValue(orbits, If(x(v) > 0 ∧ x(v + dt g) ≤ 0, orbits + 1, orbits))
# Move M
SetValue(v, v + dt g)
SetValue(M, M + dt v)
# Adds the position M to the record for the polyline trace
SetValue(regM, Append(regM, M))
Author of the activity and GeoGebra construction: Rafael Losada.