Editing Angular diameter (section)

==Formulation==
[[File:Angular diameter formula.svg|thumb|400px|right|Diagram for the formula of the angular diameter]]

The angular diameter of a [[circle]] whose plane is perpendicular to the displacement vector between the point of view and the center of said circle can be calculated using the formula<ref>This can be derived using the formula for the length of a chord found at {{cite web |url=http://mathworld.wolfram.com/CircularSegment.html |title=Circular Segment|access-date=2015-01-23 |url-status=live |archive-url=https://web.archive.org/web/20141221042937/http://mathworld.wolfram.com/CircularSegment.html |archive-date=2014-12-21 }}</ref><ref>{{Cite web |title=Angular Diameter {{!}} Wolfram Formula Repository |url=https://resources.wolframcloud.com/FormulaRepository/resources/Angular-Diameter |access-date=2024-04-10 |website=resources.wolframcloud.com}}</ref>
:<math>\delta =  2\arctan \left(\frac{d}{2D}\right),</math>
in which <math>\delta</math> is the angular diameter (in units of angle, normally radians, sometimes in degrees, depending on the [[arctangent]] implementation), <math>d</math> is the linear diameter of the object (in units of length), and <math>D</math> is the distance to the object (also in units of length). When <math>D \gg d</math>, we have:<ref>{{Cite web |title=7A Notes: Angular Size/Distance and Areas |url=https://w.astro.berkeley.edu/~casey_lam/7A_Angular_Distance_and_Area.pdf}}</ref>
:<math>\delta \approx d / D</math>,
and the result obtained is necessarily in [[radians]].

===For a sphere===
For a spherical object whose linear diameter equals <math>d</math> and where <math>D</math> is the distance to the {{em|center}} of the sphere, the angular diameter can be found by the following modified formula{{Citation needed|date=April 2024}}
:<math>\delta =  2\arcsin \left(\frac{d}{2D}\right)</math>

Such a different formulation is because the apparent edges of a sphere are its tangent points, which are closer to the observer than the center of the sphere, and have a distance between them which is smaller than the actual diameter. The above formula can be found by understanding that in the case of a spherical object, a right triangle can be constructed such that its three vertices are the observer, the center of the sphere, and one of the sphere's tangent points, with <math>D</math> as the hypotenuse and <math>\frac{d_\mathrm{act}}{2D}</math> as the sine.{{Citation needed|date=April 2024}}

The formula is related to the [[Horizon#Zenith angle|zenith angle to the horizon]], 
:<math>\delta = \pi - 2\arccos\left(\frac{R}{R+h}\right)</math>
where ''R'' is the radius of the sphere and ''h'' is the distance to the near {{em|surface}} of the sphere.

The difference with the case of a perpendicular circle is significant only for spherical objects of large angular diameter, since the following [[small-angle approximation]]s hold for small values of <math>x</math>:<ref>{{cite web |url=http://www.mathstat.concordia.ca/faculty/rhall/mc/arctan.pdf |title=A Taylor series for the functionarctan |access-date=2015-01-23 |url-status=dead |archive-url=https://web.archive.org/web/20150218190328/http://www.mathstat.concordia.ca/faculty/rhall/mc/arctan.pdf |archive-date=2015-02-18 }}</ref>
:<math>\arcsin x \approx \arctan x \approx x.</math>