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After an interesting morning's work I have deduced that a suitable AP location for me would be   > 22°S  lat  <52°S  and long - 174°E. I do not have an orange but an old globe proved handy. It has scratches on it from George, a cat my wife had many decades ago. I am on terra firma and the globe suggests Stewart Island,  Norfolk Island, Auckland Island, the Antipodes  or Matiu/Somes island would be good starting APs. Any other large  land mass in the area would also be suitable.

Later today I plan to perform a long by chron when the sun is on the prime vertical to confirm the longitude. It should be an easy sight as I need not worry about index error - I will be working to the nearest degree.

I use an AH so it will be later in the year  before I can use a noon sight to  get a better estimate of the latitude. If I can obtain a better compass then a reverse entry into a table such as Burdwood may give me a better estimate of latitude.

Part A

Firstly what I know;

(a) The sun is rising. My phone comapass is sufficiently accurate to tell me where north is. Hence I am in the southern hemisphere and  lat is S.

(b) The solstice was a month or so ago so the sun's  dec is approximately 22°.

(c) At noon the sun's alt is > 60° so will assume lat and dec are of same names. Therefore dec = 22°S.  Thiis is confirmed by the weather patterns although global warming may soon change this.

(d) I use an AH so know from observation that the sun's alt at noon is >60°. At noon   lat = 90  - Ho +- 22 from which   Lat <52°S.

(e) I have observed the rising sun crossing the prime vertical. From Bowditch (1959) table 25 I learn that the lat must be >22° for the sun to cross the prime vertical.

(f) I am on terra firma.

Conclusion A:     >22°S Lat <52°S

Part B

At this point I planned to take a bearing of the sun and estimate local apparent time by a reverse entry into Burdwood's azimuth tables. Unfortunately the compass was not accurate enough so I abandoned this method.

Part C

From previous  playing with a compass I know I know which point on an adjacent hill is true north. I will observe the shadow in my AH to determine the approx UT of noon from which I can compute the approximate longitude.

Sun true north at 0035  2020/1/26 UT

EOT for 26th Jan =  subtract 12 min. This is an approcimation as I am using a 1932 edition of Inmans tables.

Hence apparant time at Greenwhich is 0023.

GHA = 1223

From a time to arc table in Raper (1906)  1223 min = 186°

GHA = LHA + Long +-360

LHA = 0 so  long = GHA +-360

From which long = 174°

Part D

From previous playing with a compass I also know the a neighbour's chimney is due west. I will perform a  long by chron. to determine the approx longitude.

sin Z  =  sin t cos dec sec h

Z=0 so    sin t = sec dec  cos h

Because of the precision I am working to I will not need to worry about index error.

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