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At this time of the year in lat S41 it is not possible to take noon sights with an artificial horizon. However the sun crosses the prime vertical morning and evening so a long by chron is possible  - the ideal time for such an observation is when the azimuth of the sun is 90 deg.

Error in long  = Error in lat  *   C correction

From Norie's ABC tables A+-B = 0 when the azimuth is 90 deg. i.e. an accurate DR lat is not needed when the sun is on the prime vertical.

This morning I made a spur of the moment decision to do a long by chron. I cheated and used a phone app to determine that the sun's azimuth would be N90E at about 0854 NZDST (-13). I took three sights. The second I had to discard because two cats distracted me and I forgot to read the sextant - see the attached photo.

My procedure for working the sights was as follows:

First I calculated an intercept (using my gps position as my AP) with the cos formula to determine the accuracy of the sight.

Next I calculated the azimuth with Norie's ABC tables. This confirmed that the sun's azimuth was near 90 deg and also gave me the C correction.

Then  I determined t using the formula cos t = (sin h - sin lat  sin dec)/(cos lat cos dec)

After some experimentation I determined that in an east long and with a morning sight the long can be found from

360  -  t  - GHA

All calculations were done with a casio fx-82AU calculator which I highly recommend.


SIGHT A

Hs 68° 10.2'
IE 0
Correction (from NA)  -17.5'
Ho 33° 47.6'

Zone date 29/11/2016
ZT 08 54 43
Zone -13
GMT 19 54 43
GD  28/11/2016

GHA 121° 37.8'

DR lat  S 41° 06.5'
DR long  E 175° 05.2'

First work as an intercept

Hc  3°  46.5
Intercept 1.1' T

C = .01
Azimuth  N  89.6°  E

Finally work as a time sight

long = E 175° 06.6'  (gps 175° 05.2')
A 10' error in lat  would give a long error of  10 *.01  = 0.1'


SIGHT B

Hs 71°  24.2'
IE 0
Correction (from NA)  -17.4'
Ho 35° 24.7'

Zone date 29/11/2016
ZT 09 03 23
Zone -13
GMT 20 03 23
GD  28/11/2016

GHA 122° 56.8'

DR lat  S 41° 06.5'
DR long  E 175° 05.2'

First work as an intercept

Hc 35° 24.5'
Intercept 0.2' T

C = .04
Azimuth  N  88.3°  E

Finally work as a time sight

long = E 175° 05.4'  (gps 175° 05.2')
A 10' error in lat  would give a long error of  10 *.04  = 0.4'


My next step is to run the long up to noon, do a meridian altitude and so determine my noon position. The first part is easy because my location is fixed. The second part is impossible (with an artificial horizon) at this time of year (-;



SUMMARY

GPS long E 175° 05.2'

I have not measured the GPS long since the recent magnitude 7.8 earthquake. My gps is accurate to 5m and AFAIK parts of NZ moved horizontally by only a few cm.

Sight A

intercept (gps DR)  1.1' T
azimuth N 89.6° E
C  .01
long E 175° 06.6' (by time sight)


Sight B 

Intercept (gps DR) 0.2' T
azimuth N  88.3° E
C = 0.04
long 175°  05.4'  (by time sight)

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