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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.

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  • $\begingroup$ I edited your post to use TeX. Please confirm I haven't changed the meaning, thanks! $\endgroup$
    – user21
    Mar 14, 2016 at 3:41
  • $\begingroup$ Two thoughts: 1) you must convert minutes to radians, 2) you must convert sin(days) to sin(radians), possibly using the product rule for sines. $\endgroup$
    – user21
    Mar 14, 2016 at 3:44
  • $\begingroup$ The conversion to LaTex looks good. Thanks for making it. BTW, is there an easy way to include Mathematica equations? $\endgroup$ Mar 14, 2016 at 23:27
  • $\begingroup$ 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? $\endgroup$ Mar 15, 2016 at 1:18
  • $\begingroup$ Mathematica has a "TeXForm" you could use. After converting to TeXForm, put the result between two dollar signs. $\endgroup$
    – user21
    Mar 15, 2016 at 14:08

1 Answer 1

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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 factor by 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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