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The celestial equator is a projection of the terrestrial equator on the celestial sphere. So doesn't that mean the geographical latitude and the declination will be "exactly" equal, as they are measured from the same reference point (Celestial Equator = Geographical Equator)?

$$\lambda = \delta \pm (90° - \rm{Altitude})$$

latitude = the star's declination ± its zenith distance = the star's declination ± (90° - the star's altitude) Source : https://cseligman.com/laboratory/navcalc.htm

So what is the meaning of this formula as Latitude = Declination? Also I'm unable to understand that plus-or-minus sign (north of zenith and south of zenith).

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    $\begingroup$ One more question In that equation we take the altitude of lower culmination or upper culmination. For example If a question is like this - " The altitudes of a circumpolar star at culminations are 70° and 10°, both culminations being north of zenith". And we have to find declination there will be two choices for altitude to put in that equation. Also I'm confused in south/north of zenith ( the plus minus sign) as mentioned above. $\endgroup$ Commented Dec 26, 2021 at 11:08
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    $\begingroup$ You should clarify what the equation is supposed to represent. I do not recognize the "random" formula. Once we know what that formula is about, we can answer the other questions. $\endgroup$ Commented Dec 27, 2021 at 1:43
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    $\begingroup$ @JohnHoltz cseligman.com/laboratory/navcalc.htm "latitude = the star's declination ± its zenith distance = the star's declination ± (90° - the star's altitude)" $\endgroup$ Commented Dec 27, 2021 at 4:06
  • $\begingroup$ Yes, Studied them for Indian Astronomy Olympiad. $\endgroup$ Commented Dec 27, 2021 at 7:58
  • $\begingroup$ That plus - minus sign. $\endgroup$ Commented Dec 27, 2021 at 8:00

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It may be easier to visualize the formula by calculating the altitude based on the observer's latitude (lat) and the object's declination ($\delta$), and then re-arranging the equation to find the observer's latitude. First, the equation is only true when the object is on the meridian. Here is a diagram through the meridian (for an observer at a northern latitude): cross section through zenith

Since the declination at the zenith equals the latitude, the altitude of the celestial equator = 90-lat. Then the altitude of an object south of the zenith is $$alt_s = (90-lat)+\delta_s$$ $$lat=\delta_s+(90-alt_s)$$ where $\delta_s$ is the declination of the object south of the zenith and $alt_s$ is the altitude measured from the southern horizon. Do not confuse $\delta_s$ to mean an object south of the equator!

For an object between the zenith and the celestial pole, the altitude is 180 minus the above formula, or $$alt_n = 180-[(90-lat)+\delta_n]$$ $$lat=\delta_n-90+alt_n$$ $$lat=\delta_n-(90-alt_n)$$ where $\delta_n$ is the declination of the object north of the zenith and $alt_n$ is the altitude measured from the northern horizon.

For an object between the celestial pole and the north horizon (that is, at lower culmination), another formula is required!

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