O3.2

Finding the latitude

Builds on: Sections O3.1, 1.4 and O1.6
Reading time: 35 min
Exercises: 3
Olympiad problems: 1

For centuries sailors found their latitude at night by measuring how high Polaris stood above the sea. The same idea answers the first question asked about almost any sky in the planetarium round. In 2019 in Keszthely the second sky on the dome came from a place that was not named, and the first question was its latitude. In this section we find the latitude of a sky in several ways, so that one of them works whichever part of the sky is easiest to read.

O3.2.1The height of the pole

Before you readMake a guess first

On the dome, Polaris stands 21° above the north point of the horizon. Where on Earth was this sky seen?

Section 1.4 showed that the altitude of the celestial pole above the horizon equals the latitude of the observer, by (1.11). On the dome the north celestial pole stands above the north point in the northern hemisphere, and the south celestial pole above the south point in the southern hemisphere. Finding the pole is therefore the quickest way to the latitude.

In the north, Polaris lies only 0.63° from the pole, and it is found from the two Pointers of the Big Dipper. The line from Merak through Dubhe, carried on about five times the distance between them, reaches Polaris. In the south there is no bright star near the pole. The long axis of the Southern Cross, from Gacrux through Acrux, points towards it, and the pole lies about four and a half times the length of the Cross beyond Acrux, in an empty part of the sky. Section O1.6 gives other ways to the south pole.

Figure O3.2The celestial pole on the domeinteractive

LatitudeSidereal time

View

Draw

Star	Tap a star on the dome.
Altitude and azimuth
Declination

The figure shows the whole sky on the dome, with the zenith Z at the centre and the horizon as the outer circle, and the visible celestial pole marked P. The pole stands above the north point N or the south point S at an altitude equal to the latitude. The sliders set the latitude and the sidereal time. As the sidereal time changes, the sky turns about P while P itself stays still. Above the Pointers of the Big Dipper lead down towards the horizon to Polaris, 40° above N. From the Southern Cross stands high, and its long axis points down to the south pole, 34° above S. At both poles lie on the horizon, at N and S, and the celestial equator passes through Z, and at P is at Z and the celestial equator is the horizon.

Quick checkThe pole above the south point

On a dome, the long axis of the Southern Cross points down to the south celestial pole, which stands 34° above the south point. What is the latitude?

degrees south

O3.2.2The zenith and the meridian

Sometimes the pole is low and hard to judge, as near the equator, or lies behind the edge of a worksheet chart. The zenith then gives the latitude in another way. The zenith lies on the celestial sphere at a declination equal to the latitude, since it is as far from the celestial equator as the observer is from the Earth's equator. A star that passes through the zenith has a declination equal to the latitude.

A star crossing the meridian, the line from N through Z to S, gives the latitude from its altitude. At that moment it is $90 ° - h$ from the zenith along the meridian, by (1.12), and the zenith's declination is the latitude.

ResultLatitude from a star on the meridian(O3.1)

φ = δ \pm (90 ° - h)

$φ$	the latitude, positive to the north
$δ$	the declination of the star
$h$	its altitude as it crosses the meridian above the pole, at upper culmination

The sign is plus when the star crosses south of the zenith and minus when it crosses north of it. A star on the meridian below the pole, at lower culmination, gives

φ = 90 ° - δ + h

for a northern observer instead.

Example O3.2Sirius on the meridian

On the dome, Sirius, at declination −16.7°, is on the meridian due south, at an altitude of 33.4°. What is the latitude?

Solution

Sirius is $90 ° - 33.4 ° = 56.6 °$ south of the zenith, so the zenith has declination

φ = - 16.7 ° + 56.6 ° = 39.9 ° .

AnswerThe latitude is 39.9° N, that of Ankara or Beijing.

O3.2.3The edge of the sky

The stars that never set and the stars that never rise also give the latitude. By (1.14), for an observer at latitude $φ$ north, stars with declinations more than $90 ° - φ$ never set, and stars south of declination $φ - 90 °$ never rise. A star of known declination that just grazes the north point at its lowest, or the south point at its highest, therefore marks the latitude exactly. A bright star missing from the dome tells us something too. Canopus is below the horizon at every latitude north of 37.3° N, ignoring refraction, so a dome sky with Canopus anywhere on it was seen south of 37.3° N.

Example O3.3Dubhe on the northern horizonfinish it

On a dome, Dubhe, at declination +61.75°, sits exactly on the north point of the horizon, at the lowest point of its daily circle. What is the latitude?

The start

At its lowest, a star that never sets is on the meridian below the pole, and its altitude is $φ - (90 ° - δ)$ ((1.13)).

Your stepNow set that altitude to zero and find

φ

degrees north

Common errorsSection O3.2

The latitude equals the altitude of the pole, which is measured up from the horizon below it. The angle from the pole up to the zenith is $90 °$ minus the latitude.
Polaris is 0.63° from the pole, so it gives the latitude to within about a degree, which is enough for the dome but not for a precise measurement.
A sky whose south pole is visible was seen from the southern hemisphere, and its latitude has to be written as south. The two hemispheres look alike at a glance, so the visible pole should always be checked first.

ExercisesSection O3.23 questions

O3.2.1
On the dome a star of declination +52° passes exactly through the zenith. What is the latitude?
degrees north
O3.2.2
A star of declination −30° crosses the meridian due south at an altitude of 50°. What is the latitude?
degrees north
O3.2.3
On a dome, the Southern Cross stands high above the south point and no bright star marks the pole. Canopus and Achernar are both above the horizon. What can we say about the place?

Olympiad problemsPast IOAA problems, restated1 problem

2019A sky from somewhere on Earth26 points

Adapted from IOAA 2019 (Keszthely), planetarium problem 2. We have restated it in our own words, left out one part about comets, and drawn the sky again from a star catalogue at the latitude and time of the original. The solution was last checked on 20 October 2026.

The dome below shows the sky from somewhere on Earth, with no Solar System objects.

Star	Tap a star on the dome.

(a) What is the latitude of the place, and in which hemisphere is it? (3 points)

(b) Name the three brightest stars on the dome, in order, and estimate the azimuth of each, measured from north through east. The dome draws α Centauri, a pair too close to split by eye, at the magnitude of its brighter star, −0.01, and the eye sees the pair as one star of magnitude −0.28. (12 points)

(d) Mintaka, δ Orionis, is setting at this moment. Roughly how many hours ago did it rise? (2 points)

I solved it without the solution

AnswerThe place is at latitude 25° S. The brightest stars are Sirius at azimuth 262°, Canopus at 222° and α Centauri at 151°. Any nine of the seventeen constellations in the solution answer part (c), and Mintaka rose about 12 hours ago. The official solution gives the azimuths as 82°, 42° and 331°, which are 180° away from these, as if they had been measured from south. From 25° S, α Centauri never strays more than 33° in azimuth from due south, so 331° cannot be measured from north, and measured from north, as the question asks, the azimuths are the values given here.