1. $PZX$ Spherical Triangle Setup

In the celestial sphere, consider the $PZX$ triangle for Polaris:

• $P$: Elevated Pole
• $Z$: Observer’s Zenith
• $X$: Position of Polaris
• $PZ = c = 90^\circ - L$ (co-latitude, where $L$ is Latitude)
• $ZX = z = 90^\circ - a$ (zenith distance, where $a$ is True Altitude)
• $PX = p$ (polar distance of Polaris, where $p \approx 1^\circ$)
• $\angle ZPX = h$ (Local Hour Angle of Polaris, $\text{LHA}_\star$)

Let the zenith distance be $z = c + k$, where $k$ represents the correction to be applied to the altitude. Substituting $z = 90^\circ - a$ and $c = 90^\circ - L$:

$$90^\circ - a = (90^\circ - L) + k \implies L = a + k$$

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2. Fundamental Spherical Trigonometry Relation

Applying the Cosine Formula to triangle $PZX$:

$$\cos z = \cos p \cos c + \sin p \sin c \cos h$$

Substituting $z = c + k$ into the left-hand side:

$$\cos(c + k) = \cos c \cos k - \sin c \sin k$$

Equating the two expressions for $\cos z$:

$$\cos c \cos k - \sin c \sin k = \cos p \cos c + \sin p \sin c \cos h$$

Dividing both sides by $\sin c$:

$$\cot c \cos k - \sin k = \cos p \cot c + \sin p \cos h$$

Rearranging terms:

$$\sin k = \cot c (\cos k - \cos p) - \sin p \cos h$$

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3. Small Angle Expansion

Since $p$ and $k$ are small angles, express $\sin$ and $\cos$ as series expansions in radians:

$$\sin k \approx k, \quad \cos k \approx 1 - \frac{k^2}{2}$$

$$\sin p \approx p, \quad \cos p \approx 1 - \frac{p^2}{2}$$

First Approximation

Neglecting higher-order terms ($\cos k \approx 1$ and $\cos p \approx 1$):

$$-k \approx p \cos h \implies k \approx -p \cos h$$

Second Approximation

Substituting the second-order terms back into the rearranged equation:

$$\cot c \left(1 - \frac{k^2}{2}\right) - k = \cot c \left(1 - \frac{p^2}{2}\right) + p \cos h$$$$\cot c - \frac{1}{2} k^2 \cot c - k = \cot c - \frac{1}{2} p^2 \cot c + p \cos h$$$$-k = \frac{1}{2} k^2 \cot c - \frac{1}{2} p^2 \cot c + p \cos h$$$$k = -p \cos h + \frac{1}{2} \cot c (p^2 - k^2)$$

Substitute $k \approx -p \cos h \implies k^2 \approx p^2 \cos^2 h$:

$$p^2 - k^2 = p^2 - p^2 \cos^2 h = p^2 (1 - \cos^2 h) = p^2 \sin^2 h$$

Since $c = 90^\circ - L \implies \cot c = \tan L$:

$$k = -p \cos h + \frac{1}{2} p^2 \sin^2 h \tan L \quad \text{(in radians)}$$

Expressing $p$ and $k$ in minutes of arc ($p', k'$) using $p' \sin 1' = \sin p$:

$$k = -p' \cos h + \frac{1}{2} p' \sin p \sin^2 h \tan L$$

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4. Derivation of Table Correction Factors $a_0, a_1, a_2$

To avoid negative quantities in Nautical Almanac lookups, the correction $k$ is decomposed into three positive values ($a_0, a_1, a_2$):

1. $a_0$ (Function of $\text{LHA}_\gamma$): Computed using the main term $-p' \cos h$ and the second term evaluated at a reference baseline latitude of $50^\circ\text{N}$, using adopted mean values of SHA ($327^\circ 39'$) and declination ($89^\circ 09.5'$), plus an added constant of $+58.8'$ to ensure $a_0 > 0$:

   $$a_0 = -p' \cos h + \frac{1}{2} p' \sin p \sin^2 h \tan 50^\circ + 58.8'$$

2. $a_1$ (Function of $\text{LHA}_\gamma$ and Latitude $L$): Corrects for the difference between the observer’s actual latitude $L$ and the $50^\circ\text{N}$ reference latitude, plus an added constant of $+0.6'$:

   $$a_1 = \frac{1}{2} p' \sin p \sin^2 h (\tan L - \tan 50^\circ) + 0.6'$$

3. $a_2$ (Function of $\text{LHA}_\gamma$ and Date / Declination): Adjusts for small seasonal/annual variations of Polaris’s actual SHA and declination from the mean values, plus an added constant of $+0.6'$:

   $$a_2 = (\text{variation correction}) + 0.6'$$

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5. Final Latitude Formula

The total offset constant added across all three factors is:

$$\text{Sum of constants} = 58.8' + 0.6' + 0.6' = 60.0' = 1^\circ$$

Subtracting $1^\circ$ to compensate for the added constants gives the final formula:

$$\text{Latitude } (L) = \text{True Altitude} - 1^\circ + a_0 + a_1 + a_2$$

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