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    <p>You're not going to find an analytic solution to these equations because they're transcendental, containing <code>a</code> both inside and outside of a trigonometric function.</p> <p>I think the trouble you're having with numerical solutions is that the range of acceptable values for <code>a</code> is constrained by the <code>arcsin</code>. Since <code>arcsin</code> is only defined for arguments between -1 and 1 (assuming you want <code>a</code> to be real), your formulas for <code>alpha</code> and <code>beta</code> require that <code>a &gt; s/2</code> and <code>a &gt; (s-c)/2</code>.</p> <p>In Python, you can find a zero of your third equation (rewritten in the form <code>f(a) = 0</code>) using the <code>brentq</code> function:</p> <pre><code>import numpy as np from scipy.optimize import brentq s = 10014.6 c = 6339.06 mu = 398600.0 dt = 780.0 def f(a): alpha = 2*np.arcsin(np.sqrt(s/(2*a))) beta = 2*np.arcsin(np.sqrt((s-c)/(2*a))) return alpha - beta - (np.sin(alpha)-np.sin(beta)) - np.sqrt(mu/a**3)*dt a0 = max(s/2, (s-c)/2) a = brentq(f, a0, 10*a0) </code></pre> <p><strong>Edit:</strong></p> <p>To clarify the way <code>brentq(f,a,b)</code> works is that it searches for a zero of <code>f</code> on an interval <code>[a,b]</code>. Here, we know that <code>a</code> is at least <code>max(s/2, (s-c)/2)</code>. I just guessed that 10 times that was a plausible upper bound, and that worked for the given parameters. More generally, you need to make sure that <code>f</code> changes sign between <code>a</code> and <code>b</code>. You can read more about how the function works in the <a href="http://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.brentq.html" rel="nofollow">SciPy docs</a>.</p>
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    1. POusing python to solve a nonlinear equation
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    1. COWhy did you need to define a0 and then say brentq(f,a0,10*a0)? I am asking because I have never used Python, and I would like to know why so I could try on my own if something like this is encountered again.
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    2. COThe way brentq works is that it starts at a (here a0) and looks for a zero of the given f function between a (here a0) and b (10*a0). Defining a0 is necessary because the brentq function needs a place to start looking for zeros. As Ray mentioned, this is explained in the [documentation for the brentq function](http://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.brentq.html)
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      1. USJames Porter
 

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