**Applying the Equation of Time when calculating longitude at the Sun’s meridian Passage.**

Although the imaginary Mean Time gives us an accurate measurement of time, it presents the navigator with a problem. When fixing his position by an observation of the Sun, he measures the altitude of the True Sun which keeps apparent solar time. However, he notes the time of the observation from a deck watch that keeps mean solar time. To enable us to connect mean solar time with apparent solar time, we have the Equation of Time which is defined as follows:

**Equation of Time = Mean solar time – Apparent Solar Time**

In other words, the equation of time is the difference between apparent solar time and mean solar time taken at the same instant at one place.

The equation of time can be either positive or negative depending on the time of the year.

- The values range from approximately +15 to -15 mins. but can be as much as
__+__64 mins. - The values are positive from 15
^{th}April to 14^{th}June and from 1^{st}September to 24^{th} - The values are negative from 15
^{th}June to 31^{st}August and from 25^{th}December to 14^{th}

**Nautical Almanac.**

The Equation of Time for 00^{h} (lower meridian) and 12^{h} (upper meridian) for each day is printed at the foot of the Nautical Almanac daily page as shown in this extract. The Local Mean Time of the Sun’s Meridian Passage is shown in the column to the right of the EOT. (This is the apparent time of Mer Pas adjusted for EOT to give the LMT and rounded up to the nearest minute).

If the mean time of Mer. Pas is shown to be greater than 1200 then the EOT must be negative, indicating that apparent time is slow compared to mean time. Conversely, if the mean time of Mer. pas. is less than 1200 then EOT is positive, indicating that apparent time is ahead of mean time.

To calculate longitude we simply find the difference between LMT of Mer. Pas. and the GMT of our observation of Mer. Pas. then, by converting the time difference to arc we are able to find the difference in degrees of longitude.

Let’s try an example:

Date: 22 June. Zone Time: 1140 (+4). DR Pos: 32^{0} 30’N. 61^{0} 55’W. At meridian passage, the deck watch time was 16^{h} 08^{m}^{ }25.1^{s} and the Deck Watch Error was -05.0^{s}. The daily page for that date shows that the Eqn. of Time is 02^{m} 02^{s} and that Mer. Pas. is 1202 indicating that EOT is negative.

**Procedure:**

**Calculate time difference.**

Deck Watch Time: 16^{h} 08^{m}^{ }25.1^{s}

DWE: -05.0^{s}

GMT: 16^{h} 08^{m}^{ }20.1^{s}

LMT Mer.Pas: 12^{h}^{ } 02^{m} 00^{s}^{ }

Time Diff: 04^{h} 06^{m}^{ }20.1^{s}

(Longitude West, GMT Best)

**Convert Time to Arc**

4^{h}^{ }^{ }= 4 x 15 = 60** ^{o }**00’ 00”

06^{m}^{ }= 6 ÷ 4 = 1** ^{o}** 30’ 00”

20.1^{s} = 20.1 ÷ 4 = 0** ^{o}** 05’ 01″.5

Therefore, Long = 61^{o }^{ }35’ 01″.5 W

The navigator seldom requires the time of meridian passage to accuracies greater than one minute. Therefore, use the time listed under the “Mer. Pass.” column unless extreme accuracy is required.

**Alternative Method. **Notwithstanding the slight inaccuracy caused by the rounding up of the listed time of Mer. pas., practicing navigators may prefer the above method but for students and tutors of astro navigation, it provides very little understanding of the equation of time. To overcome this problem, it may be more beneficial for students to calculate the time of Mer. Pas themselves by applying the equation of time. There is also the benefit of a greater degree of accuracy.

So if, in the above example, we wished to recalculate the time difference by applying the equation of time to the local apparent noon (LAN) instead of simply using the published time of Mer. Pas., we would proceed as follows:

On 22 June, the EOT is 01^{m} 55^{s} at 00^{h} and it is 02^{m} 02^{s }at 12^{h} so the hourly rate of change is (02^{m} 02^{s} – 01^{m} 55^{s}) ÷ 12 = 0.58^{s}. Therefore, at 16^{h} 00^{m}, the EOT will be 02^{m }02^{s} + (4 x 0.58^{s}) = 02^{m} 04.32^{s} which we can approximate to 02^{m} 04^{s}. The table shows that Mer. Pas is greater than 1200 indicating that apparent time is slow compared to mean time. This means that EOT is negative and must be added to the apparent time to give mean time.

**Calculate time difference.**

Deck Watch Time: 16^{h} 08^{m}^{ }25.1^{s}

DWE: -05.0^{s}

GMT Mer. Pas: 16^{h} 08^{m}^{ }20.1^{s}

LAN: 12^{h}^{ } 00^{m} 00^{s}^{ }

EOT: 02^{m} 04^{s}^{ }

LMT Mer. Pas.: 12^{h}^{ } 02^{m} 04^{s}^{ }

Time Diff (GMT – LMT) = 04^{h} 06^{m} 16.1^{s}

*(Longitude West, GMT Best)** *

**Convert Time to Arc**

4^{h}^{ }^{ }= 4 x 15 = 60** ^{o }**00’ 00”

06^{m}^{ }= 06 ÷ 4 = 1** ^{o}** 30’ 00”

16.1^{s} = 16.1 ÷ 4 = 0** ^{o}** 04’ 01″.5

**Therefore, Long = 61 ^{o }**

^{ }**34’ 01″.5 W**

**Links:** Understanding Meridian Passage, Meridian Passage Short Method, Meridian Passage Long Method What is the point of meridian passage? Zone Time, Local Hour Angle and Greenwich Hour Angle, Converting GMT to GHA , Altitude Corrections

**Where to buy books of the Astro Navigation Demystified series:**

Celestial Navigation at Amazon.com

Celestial Navigation at Amazon.uk

Astro Navigation Demystified at Amazon.com

Astro Navigation Demystified at Amazon.uk

Applying Mathematics to Astro Navigation at Amazon .com

Applying Mathematics to Astro Navigation at Amazon .uk

Astronomy for Astro Navigation at Amazon.com

Astronomy for Astro Navigation at Amazon.uk

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