**Exercise:**

You are lost on a desert island

with a sextant, a chronometer, a carrier pigeon,

and your
copy of Smart's *Spherical Astronomy.*

Explain how you will
save yourself.

(Assume that the chronometer is keeping GMT,

and that you know the date.)

**Step 1: determine your
latitude.**

There are (at least) two possible techniques.

**1**. Measure the altitude
of Polaris above the northern horizon, using the sextant.

This is
approximately equal to your latitude.

(Polaris, the "North
Star", lies very close to the North Celestial Pole.)

There are various problems with
this.

Firstly, if you are in the
southern hemisphere, Polaris will be below the horizon!

Secondly, you need to carry out the
measurement in nautical twilight,

while it is still light enough
to see the horizon,

and Polaris is only a second-magnitude star,

so it may not appear bright enough to measure accurately.

Thirdly, Polaris does not lie
exactly at the North Celestial Pole,

so your result could be
nearly 1 degree in error.

**2**. So, as an
alternative,

measure the altitude of the Sun at midday, using the
sextant.

Knowing the date, calculate the
declination of the Sun

(it varies sinusoidally,

with a period
of 1 year starting at the spring equinox,

and an amplitude of
23.4 degrees.)

The midday altitude, when the
Sun is on the local meridian,

is composed of:

the height of
the celestial equator above the southern horizon (equal to the
co-latitude)

plus the height of the Sun above the celestial
equator (its declination).

(If you are in the southern
hemisphere,

the celestial equator will be closer to the *northern
*horizon;

in this case its distance from the *southern
*horizon, the co-latitude,

will be greater than 90°.)

Knowing the altitude and the solar declination,

calculate the
co-latitude and hence the latitude.

If the sextant can be read to
an accuracy of a few arc-minutes,

you should correct your reading
for refraction.

The apparent zenith angle of an object *z' *is
greater than its true zenith angle *z*

by the value *k
tan(z')*, where *k* is approximately 1 arc-minute.

**Step 2: Determine your
longitude**.

Again there are (at least) two possible
techniques.

**1**. During nautical
twilight,

if you can locate a star whose celestial coordinates
you know,

measure its altitude above the horizon using the
sextant,

and note the time (GMT) using the chronometer.

Knowing the star's altitude,
its declination, and your latitude (previously determined),

calculate its Hour Angle

by applying the cosine rule to "the"
Astronomical Triangle.

Knowing the star's Right
Ascension,

calculate the local sidereal time of the observation

(Local Hour Angle = Local Sidereal Time - Right Ascension).

Knowing the date,

calculate
the Greenwich Sidereal Time

corresponding to the Greenwich Mean
Time of the observation.

GST is equal to GMT at the autumn
equinox,

and GST runs faster than GMT by one day in 365.25 days.

The difference between the
Local Sidereal Time (from your observation)

and Greenwich
Sidereal Time (from the chronometer)

is your longitude east or
west of Greenwich.

**2**. Failing a star with
known coordinates, use the Sun.

Note the time (GMT) when it
reaches its greatest altitude:

this is midday, Local Apparent
Time.

Use the formulae given in
Smart's *Spherical Astronomy*

to calculate the Equation of
Time on that date.

(Or derive it from first principles:

allow
firstly for the non-uniform motion of the Sun around the ecliptic
(Kepler's Second Law);

then allow for the fact that the ecliptic
is tilted to the equator.)

Add or subtract the Equation of
Time to your Local Apparent Time,

to obtain Local Mean Time.

The
difference between Local Mean Time and GMT

is your longitude east
or west of Greenwich.

**Step 3**:

Tear a strip
of paper from the title-page of Smart's *Spherical Astronomy*

to write a message giving your latitude and longitude.

Launch
it by carrier-pigeon and wait to be rescued!

*This question formed part of
the final exam at UCLA in 1961. (Trimble, V., "The
Observatory" 118, 32, 1998).*

Back to "Final exercise".