Mean anomaly as a function of the true anomaly elliptical orbits

[Dustinsfl]

The book Orbital mechanics by Curtis says that the function (I labeled M2evals) is monotonically increasing. However, at pi, I have a discontinuity and the graph jumps below the negative axis when it should continue on; that is, the e = 0 should be the line y = x.

import pylab
import numpy as np

e = np.arange(0.0, 1.0, 0.15).reshape(-1, 1)
nu = np.linspace(0, 2 * np.pi, 50000)
M2evals = (2 * np.arctan(((1 - e) / (1 + e)) ** 0.5 * np.tan(nu / 2)) -
           e * (1 - e ** 2) ** 0.5 * np.sin(nu) / (1 + e * np.cos(nu)))

fig2 = pylab.figure()
ax2 = fig2.add_subplot(111)

for Me2, _e in zip(M2evals, e.ravel()):
    ax2.plot(nu.ravel(), Me2, label = str(_e))

pylab.legend()
pylab.xlim((0.0, 7.75))
pylab.ylim((-np.pi, np.pi))
pylab.savefig('eccentrueanomfunc.eps', format = 'eps', dpi = 1000)
pylab.show()

Here is the plot:
where the plot jumps down it is supposed to be shifted up to meet the first half.
What is wrong?
http://img96.imageshack.us/img96/5397/pfquestion.png

 

[TSny]

The problem may lie in the arctan function which gives "principle values" as output.

Thus, arctan(tan(x)) does not yield x if x is an angle in the second or third quadrant. If you plot arctan(tan(x)) from x = 0 to x = Pi, you will find that it has a discontinuous jump at x = Pi/2.

I think you can correct for this by using the function arctan2(x1, x2). See http://docs.scipy.org/doc/numpy/reference/generated/numpy.arctan2.html

For your case, instead of writing arctan(arg), I believe you would write arctan2(1, 1/arg) where arg is the argument of your arctan function. That way, when arg becomes negative, arctan2 will yield an angle in the second quadrant rather than the fourth.

I'm not familiar with python, but it appears arctan2 is similar to a function in Mathematica which I have used in similar circumstances.

At least it would be easy to give it a try.

 

[Dustinsfl]

With some help of stackoverflow, I was able to implement your suggestion of arctan2 and it works as intended now.

 

[TSny]

Good.

 

[paranormal]

There appears to be a mistake in the calculation of the mean anomaly as a function of the true anomaly for elliptical orbits. The function as described in the book by Curtis is supposed to be monotonically increasing, but there is a discontinuity at pi where the graph jumps below the negative axis. This is not consistent with the expected behavior of the function.

It is possible that there is an error in the equation used to calculate the mean anomaly, or there may be a mistake in the implementation of the equation in the code. It would be helpful to double check the equation and the code to identify and correct any errors.

Additionally, it is important to consider the physical significance of this discontinuity. If the mean anomaly is meant to be a continuous function, then this discontinuity may have implications for the accuracy of any calculations or predictions based on this function. Further investigation and analysis may be needed to fully understand the cause and potential impact of this discontinuity.