<p>I don't have a complete answer, but I do know where the $7.6$ factor comes from: It equals
$$(2×0.0167)×(24×60)/(2 \pi).$$</p>

<p>$0.0167$ comes from the eccentricity of the Earth, and the factor of $2$ comes from the above equation. $24×60$ converts from days to minutes, and $2 \pi$ is the number of radians in a complete revolution.</p>

<p>What I still don't completely understand is why these coordinate conversions need to be done, specifically the division by $2 \pi$.</p>

<p>Thanks everyone for your help. </p>

<p><strong>Edit:</strong> I now have a somewhat better explanation for the computation in this answer. <code>2ε sin(M)</code> is the number of radians that the mean Sun is behind the apparent Sun on Noon on a day with mean anomaly <code>M</code>. Dividing the <code>2ε</code> factor by <code>2π</code>gives the fraction of the rotation <em>of the Earth</em> (i.e. one day). When the result is multiplied by the number of minutes in a <em>sidereal</em> day <code>(23×60+54)</code> yields the desired coefficient <code>7.63345</code>.</p>

CC BY-SA 4.0
2016-03-29T19:58:45.667Z
2018-06-29T13:48:05.550Z
2018-06-29T13:48:05.550Z
Spencer Rugaber
<p>This questions concerns the longitudinal aspect of the Equation of Time, also called the Equation of the Center. In some sources the equation looks like the following:</p>

<p>$\nu - M = 2\varepsilon \sin M$       (1)</p>

<p>where $\nu$ is the True Anomaly of the Sun's position from the Earth, $M$ is the Mean Anomaly, and $\varepsilon$ is the eccentricity of the Earth's orbit (0.0167). $\nu - M$ is the difference between the Sun's actual angle and the and the angle that would exist if the Earth's orbit were circular.</p>

<p>Other sources have a first-order approximation that looks like the following:</p>

<p>Time deviation (minutes) = $-7.655 \sin d$       (2)</p>

<p>where d is the day of the year.</p>

<p>My difficulty is reconciling these two equations. None of the sources actually make this connection explicit. I assume that the difference has to do with converting angles and times, and I have tried various approaches to making the numbers work out, without avail. I would appreciate it if someone could tell me how the value of -7.655 minutes is derivable from $2 \varepsilon$ and or point me at a resource that demonstrates the connection between them.</p>

<p>(Note I realize that both (1) and (2) are approximations. At this stage, I am trying to understand the situation in its simplest form before adding refinements.</p>

CC BY-SA 3.0
2016-03-14T23:27:52.063Z
The conversion to LaTex looks good. Thanks for making it. BTW, is there an easy way to include Mathematica equations?
Spencer Rugaber
CC BY-SA 3.0
2016-03-15T01:18:11.403Z
Update: The conversion to LaTex looks good, if d in (2) is taken to mean ((d/365.24)*(2Pi)). Thanks for making it. BTW, is there an easy way to include Mathematica equations?
Spencer Rugaber
<fundamental-astronomy>
Equation of the Center constant factor
Spencer Rugaber
# Equation of the Center constant factor

(Asked by: username_0 on 2016-03-14)

This questions concerns the longitudinal aspect of the Equation of Time, also called the Equation of the Center. In some sources the equation looks like the following:

$\nu - M = 2\varepsilon \sin M$ (1)

where $\nu$ is the True Anomaly of the Sun's position from the Earth, $M$ is the Mean Anomaly, and $\varepsilon$ is the eccentricity of the Earth's orbit (0.0167). $\nu - M$ is the difference between the Sun's actual angle and the and the angle that would exist if the Earth's orbit were circular.

Other sources have a first-order approximation that looks like the following:

Time deviation (minutes) = $-7.655 \sin d$ (2)

where d is the day of the year.

My difficulty is reconciling these two equations. None of the sources actually make this connection explicit. I assume that the difference has to do with converting angles and times, and I have tried various approaches to making the numbers work out, without avail. I would appreciate it if someone could tell me how the value of -7.655 minutes is derivable from $2 \varepsilon$ and or point me at a resource that demonstrates the connection between them.

(Note I realize that both (1) and (2) are approximations. At this stage, I am trying to understand the situation in its simplest form before adding refinements.

**Tags:** fundamental-astronomy

> **Comments:**
> **username_0**: The conversion to LaTex looks good. Thanks for making it. BTW, is there an easy way to include Mathematica equations?  
> **username_0**: Update: The conversion to LaTex looks good, if d in (2) is taken to mean ((d/365.24)*(2Pi)). Thanks for making it. BTW, is there an easy way to include Mathematica equations?  

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## 1 answers

### Answer by username_0 on 2016-03-29. Score: 2

I don't have a complete answer, but I do know where the $7.6$ factor comes from: It equals
$$(2×0.0167)×(24×60)/(2 \pi).$$

$0.0167$ comes from the eccentricity of the Earth, and the factor of $2$ comes from the above equation. $24×60$ converts from days to minutes, and $2 \pi$ is the number of radians in a complete revolution.

What I still don't completely understand is why these coordinate conversions need to be done, specifically the division by $2 \pi$.

Thanks everyone for your help.

**Edit:** I now have a somewhat better explanation for the computation in this answer. `2ε sin(M)` is the number of radians that the mean Sun is behind the apparent Sun on Noon on a day with mean anomaly `M`. Dividing the `2ε` factor by `2π`gives the fraction of the rotation *of the Earth* (i.e. one day). When the result is multiplied by the number of minutes in a *sidereal* day `(23×60+54)` yields the desired coefficient `7.63345`.


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