Editing Extraterrestrial sky (section)

==Luminosity and angular diameter of the Sun==
{{main article|Solar luminosity}}
{{Further|Angular diameter}}

The [[Sun]]'s [[apparent magnitude]] changes according to the [[inverse square law]], therefore, the difference in magnitude as a result of greater or lesser distances from different [[celestial bodies]] can be predicted by the following [[formula]]:

:<math> \text{intensity} \ \propto \ \frac{1}{\text{distance}^2} \, </math>

Where "distance" can be in [[km]], [[Astronomical unit|AU]], or any other appropriate unit.

To illustrate, since [[Pluto]] is 40 AU away from the Sun on average, it follows that the parent star would appear to be <math>\frac{1}{1600}</math> times as bright as it is on [[Earth]].

Though a [[Terrestrial planet|terrestrial]] observer would find a dramatic decrease in available [[sunlight]] in these environments, the Sun would still be bright enough to cast shadows even as far as the hypothetical [[Planet Nine]], possibly located 1,200 AU away, and by analogy would still outshine the [[full Moon]] as seen from Earth.

The change in angular diameter of the Sun with distance is illustrated in the diagram below:

[[File:Angular dia formula.JPG|thumb|200px|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 centre of said circle can be calculated using the formula<ref group="nb">This can be derived using the formula for the length of a cord found at http://mathworld.wolfram.com/CircularSegment.html</ref>
:<math>\delta =  2\arctan \left(\frac{d}{2D}\right),</math>
in which <math>\delta</math> is the angular diameter, and <math>d</math> and <math>D</math> are the actual diameter of and the distance to the object. When <math>D \gg d</math>, we have <math>\delta \approx d / D</math>, and the result obtained is in [[radians]].

For a spherical object whose ''actual'' diameter equals <math>d_\mathrm{act},</math> and where <math>D</math> is the distance to the ''centre'' of the sphere, the angular diameter can be found by the formula
:<math>\delta =  2\arcsin \left(\frac{d_\mathrm{act}}{2D}\right)</math>

The difference is due to the fact that the apparent edges of a sphere are its tangent points, which are closer to the observer than the centre of the sphere. For practical use, the distinction is significant only for spherical objects that are relatively close, since the [[small-angle approximation]] holds for <math> x \ll 1</math>:<ref>{{Cite web |url=http://www.mathstat.concordia.ca/faculty/rhall/mc/arctan.pdf |title=Archived copy |access-date=August 5, 2017 |archive-url=https://web.archive.org/web/20150218190328/http://www.mathstat.concordia.ca/faculty/rhall/mc/arctan.pdf |archive-date=February 18, 2015 |url-status=dead }}</ref>
:<math>\arcsin x \approx \arctan x \approx x</math> .