errors of position line

In celestial navigation, position lines (PLs) derived from astronomical observations rely on precise measurements of altitude and time. Errors in reading the sextant (Index Error and Dip) or recording time (Chronometer Error) directly affect the True Altitude ($T_{\text{alt}}$), True Zenith Distance ($\text{TZD}$), or the Geographical Position ($\text{GP}$) of the observed body, causing the calculated Position Line to shift.

1. Index Error (IE) — On the Arc vs. Off the Arc

Index Error occurs when the index mirror and horizon glass of a sextant are not parallel when the index arm is set exactly to $0^\circ$.

On the Arc vs. Off the Arc

• On the Arc: The sextant reads above $0^\circ$ when zeroed. Because all altitude readings taken with this instrument will be too high, Index Error on the arc must be subtracted from the sextant altitude.
• Off the Arc: The sextant reads below $0^\circ$ (on the arc of excess/micrometer drum) when zeroed. Because all altitude readings taken will be too low, Index Error off the arc must be added to the sextant altitude.

Mnemonic Rule: “If it’s on the arc, it’s off (subtract); if it’s off the arc, it’s on (add).”

Application in Numericals

• Minus ($-$) for IE on the arc.
• Plus ($+$) for IE off the arc.

Effect of Misapplication / Omission on Position Lines

• Wrong Application (e.g., applying $2'$ on the arc as $2'$ off the arc):
  • Instead of subtracting $2'$, you add $2'$, causing the calculated True Altitude ($T_{\text{alt}}$) to be too high by $4'$ ($2 \times \text{IE}$).
  • Since $\text{TZD} = 90^\circ - T_{\text{alt}}$, the calculated True Zenith Distance becomes too small by $4'$.
  • Because TZD is the angular distance from the body’s GP to the Position Line, a smaller TZD places the incorrect PL too close (towards) to the GP by $4'$.
• Rectification in Numericals: To correct the position line, shift the PL away from the GP by $2 \times \text{IE}$ (or subtract $2 \times \text{IE}$ from a “towards” intercept).

2. Error of Chronometer (Time Error)

Chronometer error is the difference between the deck watch/chronometer time and Greenwich Mean Time (GMT/UTC). Since GHA increases as the Earth rotates, an error in GMT translates directly into a positional error in longitude.

Rate of Rotation & Longitude Conversion

The Earth rotates $360^\circ$ in 24 hours ($15^\circ$ per hour or $15'$ of arc per minute of time):

(Note: For star observations, the rate of travel is approximately $15.04'$ per minute of time).

Effect on Numericals

• Chronometer Fast: Reading a fast chronometer means using a later GMT. The GHA extracted from the almanac will be too large (shifted West), which shifts the calculated Position Line Westwards. To rectify, apply a $\text{d'long}$ correction Eastwards.
• Chronometer Slow: Reading a slow chronometer means using an earlier GMT. The GHA extracted will be too small (shifted East), shifting the calculated Position Line Eastwards. To rectify, apply a $\text{d'long}$ correction Westwards.

Relationship Between Longitude Error and Intercept Error

The relationship between an error in longitude ($\delta\,\text{d'long}$) and the corresponding shift in intercept ($\delta\,\text{Intercept}$) is given by:

3. Error of Dip (Height of Eye — HE)

Dip is the angular depression of the visible sea horizon below the true sensible horizon caused by the observer’s height of eye above sea level.

Finding Dip Given Height of Eye (HE)

Dip is calculated using the standard formula:

Application in Numericals

Because elevating the eye depresses the visible horizon, sextant altitudes measured relative to the sea horizon are always too large. Therefore, Dip is ALWAYS SUBTRACTED:

Effect of Omitting Dip in Numericals

• If Dip correction is omitted:
  • The True Altitude ($T_{\text{alt}}$) remains too high by the value of Dip.
  • The True Zenith Distance ($\text{TZD}$) becomes too small by the value of Dip.
  • The resulting Position Line is drawn too close (towards) to the GP by an amount equal to Dip.
• Rectification in Numericals: To rectify the omission of Dip, shift the Position Line away from the GP by the Dip value (or reduce a “towards” intercept by Dip).

Errors of Cocked Hat

    flowchart TD
    A[Error on three position Lines] --> B[Similar Error- Index / Dip error]
    A --> C[Random Error- chronometer/refraction error]
    B --> D[Angular Spread greater than 180 - position inside cocked hat]
    B --> E[Angular spread lesser than 180 - position outside cocked hat]
    C --> F[Cocked hat may form but the intersection point is not the correct position]
    C --> G[Cocked-hat if forms cannot be resolved geometrically]