<p>Yes.  Assuming that the moon is distant enough to neglect parallax effects, the semi major axis is just the radius of the moon, <span class="math-container">$r$</span>, and the semi minor axis is  <span class="math-container">$|r\cos\theta|$</span> where <span class="math-container">$\theta$</span> is the Sun-Moon-Earth angle.</p>
<p>So when the sun is behind the Earth, the angle is 0 and the ellipse is a circle (full moon) so the semiminor axis = <span class="math-container">$r$</span>.  When the sun is at right angles to the Earth, the ellipse is a line, semiminor axis=0(half-moon) and when the sun is (nearly) behind the moon, the angle is 180, and again the semi-minor axis =<span class="math-container">$r$</span>.</p>

CC BY-SA 4.0
2021-12-27T15:59:25.607Z
Great answer, just one nitpick. @Wanderer is looking for the angle as a function of the axes, so a full answer would be that $\theta(b) = \text{arccos}(b/r)$ where $b$ is the semi-minor axis.
zephyr
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2021-12-27T18:20:25.690Z
@zephyr May I know how you got that?
Particle king
CC BY-SA 4.0
2021-12-27T21:00:37.013Z
Zepher just rearranged $b=r\cos(\theta)$ to make theta the subject.
James K
CC BY-SA 4.0
2021-12-27T08:33:26.260Z
2021-12-27T08:33:26.260Z
James K
<p>I want to do a theoretical prediction before trying out an experiment. I just noticed that the arc separating the illuminated side of moon and the shadowed side is an ellipse arc. So Is it possible to write down the angle between Sun and moon (as seen from earth) in terms of the semi major and minor axes of that ellipse.</p>

<observational-astronomy><the-moon><mathematics>
Sun Moon earth Angle
Particle king
# Sun Moon earth Angle

(Asked by: username_0 on 2021-12-27)

I want to do a theoretical prediction before trying out an experiment. I just noticed that the arc separating the illuminated side of moon and the shadowed side is an ellipse arc. So Is it possible to write down the angle between Sun and moon (as seen from earth) in terms of the semi major and minor axes of that ellipse.

**Tags:** observational-astronomy, the-moon, mathematics

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## 1 answers

### Answer by username_1 on 2021-12-27. Score: 4 | Accepted answer

Yes. Assuming that the moon is distant enough to neglect parallax effects, the semi major axis is just the radius of the moon, $r$, and the semi minor axis is $|r\cos\theta|$ where $\theta$ is the Sun-Moon-Earth angle.

So when the sun is behind the Earth, the angle is 0 and the ellipse is a circle (full moon) so the semiminor axis = $r$. When the sun is at right angles to the Earth, the ellipse is a line, semiminor axis=0(half-moon) and when the sun is (nearly) behind the moon, the angle is 180, and again the semi-minor axis =$r$.

> **Comments:**
> **username_2**: Great answer, just one nitpick. @Wanderer is looking for the angle as a function of the axes, so a full answer would be that $\theta(b) = \text{arccos}(b/r)$ where $b$ is the semi-minor axis.  
> **username_0**: @zephyr May I know how you got that?  
> **username_1**: Zepher just rearranged $b=r\cos(\theta)$ to make theta the subject.  

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