Editing Gaussian gravitational constant (section)

===In modern terms===
Gauss is notorious for leaving out details, and this derivation is no exception. It is here repeated in modern terms, filling out some of the details.

Define without proof

:<math>h=2\frac{dA}{dt},</math>

where<ref name="Smart77-100">{{cite book | last = Smart | first = W. M. | title = Textbook on Spherical Astronomy | url = https://archive.org/details/textbookonspheri00smar | url-access = limited | publisher = Cambridge University Press |location=Cambridge | year = 1977|edition=6th |isbn=0-521-29180-1|page=[https://archive.org/details/textbookonspheri00smar/page/n99 100]}}</ref>

*{{math|{{sfrac|''dA''|''dt''}}}} is the time rate of sweep of [[area]] by a body in its [[orbit]], a constant according to [[Johannes Kepler|Kepler]]'s [[Kepler's laws of planetary motion#Second law|second law]], and
*{{mvar|h}} is the [[Specific relative angular momentum|specific angular momentum]], one of the constants of [[Two-body problem|two-body motion]].

Next define

:<math>h^2=\mu p,</math>

where<ref name="Smart77-101">Smart, W. M. (1977). p. 101.</ref>
*{{math|''μ'' {{=}} ''G''(''M'' + ''m'')}}, a [[Standard gravitational parameter|gravitational parameter]],<ref group=note>Do not confuse {{mvar|μ}} the gravitational parameter with Gauss's notation for the mass of the body.</ref> where
**{{mvar|G}} is [[Isaac Newton|Newton's]] [[gravitational constant]],
**{{mvar|M}} is the [[mass]] of the primary body (i.e., the [[Sun]]),
**{{mvar|m}} is the mass of the secondary body (i.e., a [[planet]]), and
*{{mvar|p}} is the semi-parameter (the [[Ellipse#Latus rectum|semi-latus rectum]]) of the body's orbit.
Note that every variable in the above equations is a constant for two-body motion. Combining these two definitions,

:<math>\left(2\frac{dA}{dt}\right)^2=G(M+m)p,</math>

which is what Gauss was describing with the last of his laws. Taking the [[square root]],

:<math>2\frac{dA}{dt}=\sqrt{G}\sqrt{M+m}\sqrt{p},</math>

and solving for {{math|{{sqrt|''G''}}}},

:<math>\sqrt{G}=\frac{2dA}{dt\sqrt{M+m}\sqrt{p}}.</math>

At this point, define {{math|''k'' ≡ {{sqrt|''G''}}}}.<ref name="Smart53"/> Let {{mvar|dA}} be the entire area swept out by the body as it orbits, hence {{math|''dA'' {{=}} π''ab''}}, the area of an [[ellipse]], where {{mvar|a}} is the [[semi-major axis]] and {{mvar|b}} is the [[semi-minor axis]]. Let {{math|''dt'' {{=}} ''P''}}, the time for the body to complete one orbit. Thus,

:<math>k=\frac{2\pi ab}{P\sqrt{M+m}\sqrt{p}}.</math>

Here, Gauss decides to use Earth to solve for {{mvar|k}}. From the geometry of an [[ellipse]], {{math|''p'' {{=}} {{sfrac|''b''<sup>2</sup>|''a''}}}}.<ref name="Smart77-99">Smart, W. M. (1977). p. 99.</ref> &nbsp; By setting Earth's semi-major axis, {{math|''a'' {{=}} 1}}, {{mvar|p}} reduces to {{math|''b''<sup>2</sup>}} and {{math|{{sqrt|''p''}} {{=}} ''b''}}. Substituting, the area of the ellipse "is evidently" {{math|π{{sqrt|''p''}}}}, rather than {{math|π''ab''}}. Putting this into the [[numerator]] of the equation for {{mvar|k}} and reducing,

:<math>k=\frac{2\pi}{P\sqrt{M+m}}.</math>

Note that Gauss, by normalizing the size of the orbit, has eliminated it completely from the equation. Normalizing further, set the mass of the Sun to 1,

:<math>k=\frac{2\pi}{P\sqrt{1+m}},</math>

where now {{mvar|m}} is in [[solar mass]]es. What is left are two quantities: {{mvar|P}}, the [[orbital period|period]] of Earth's orbit or the [[sidereal year]], a quantity known precisely by measurement over centuries, and {{mvar|m}}, the mass of the Earth–Moon system. 
Again plugging in the measured values as they were known in Gauss's time, {{mvar|P}} = {{val|365.2563835}} days, {{mvar|m}} = {{sfrac|{{val|354710}}}} solar masses,{{clarify|date=June 2018}}<!--surprisingly far from the modern value of 1:328900, needs an explanation or at least a citation, how can k still be accurate if P was correct but M was off by >7%?--> yielding the result {{mvar|k}} = {{val|0.01720209895}}.