Two Stars That Point
Two stars in the Big Dipper's bowl point to Polaris, which marks true north; altitude and azimuth then give everything in your sky a two-number address, and your fist at arm's length is the ruler. · 11 min
This whole course begins with a single skill: step outside on a clear night and find true north. The northern sky offers one star that holds still while everything else drifts — Polaris, the North Star. Find it once and you have found north, tonight and every clear night after, without a compass or a phone. Two stars in the Big Dipper will take you there, and a trick with your own fist will let you measure the sky along the way.
Guess before you learn
Before we go find it — what do you already believe about Polaris, the North Star?
Polaris ranks about 48th in brightness — dozens of stars outshine it. Its fame is position: it stands within one degree of the point the whole sky turns around, so it alone appears to hold still. If you guessed brightest, you are in the majority — it is probably the most common belief about the night sky, and correcting it is this folio's first job. Overhead-everywhere and winter-only fail too, for reasons the next two folios make plain.
9–12
3–5
The night sky turns slowly all night long, and every star moves with it — except one. Polaris sits at the still center of the turning, so it alone stays in place, and it always marks north. To find it, find the Big Dipper's bowl, take the two stars on its outer edge, and follow the line they make. The first bright star you reach is Polaris. Face it, and you face north.
You can also measure the sky with your own hand. Hold your fist out at arm's length: from edge to edge it covers about ten degrees of sky, and that works for almost everyone, because bigger fists come with longer arms. Five fists, stacked one on the next, reach halfway up the sky.
6–8
The two stars at the outer edge of the Dipper's bowl are Merak and Dubhe — the pointers. The line from Merak through Dubhe, extended about five times their separation, lands on Polaris. Polaris sits within one degree of the north celestial pole, the fixed point the whole sky wheels around, so it marks true north to better than the width of your little finger held at arm's length.
Once you have north, every object in the sky gets a two-number address. Azimuth is the compass direction, measured in degrees clockwise from north: 0 is north, 90 east, 180 south, 270 west. Altitude is height above the horizon: 0 on the horizon, 90 straight overhead. Your fist at arm's length spans about 10 degrees, so you can measure both numbers with nothing but your arm.
9–12
Altitude and azimuth make up the horizontal coordinate system: positions measured from your local horizon and your north point. They are wonderfully direct — you can point along them — and completely private. Two observers in different cities assign the same star different altitudes and azimuths, and one observer assigns it different numbers minute by minute, because the sky keeps turning. Alt-az answers where is it now, from here — nothing more.
Polaris earns its job by position, not brightness: it stands about two-thirds of a degree from the north celestial pole, so its daily circle is too small for your eye to notice. One more fact to carry: the altitude of Polaris above your horizon equals your latitude. You will prove that in the next folio and use it for the rest of your observing life.
K–2
The sky has one special star. Its name is Polaris. All the other stars slowly slide through the night. Polaris stays put. It always hangs over the same spot on the ground: north.
Seven bright stars make the Big Dipper. It looks like a soup pot with a long handle. The two stars on the pot's front side point at Polaris. Follow them, and you have found north.
Undergrad
The horizontal system is a spherical coordinate frame whose fundamental circle is the observer's horizon and whose pole is the zenith. Azimuth runs 0–360° from north through east; altitude is 90° minus the zenith distance. Because the frame is fixed to the rotating Earth, a star's alt-az coordinates are functions of both time and location — the price paid for being able to point along them. Converting to a star-fixed frame requires your latitude and the local sidereal time, machinery folio 3 begins to assemble.
Polaris holds a temporary appointment. Earth's axis precesses with a period of about 26,000 years, carrying the celestial pole in a slow circle through the constellations: Thuban held the post when the pyramids were built, and Vega takes a turn in about 12,000 years. Polaris's pole distance is in fact still shrinking — it passes closest, under half a degree, around the year 2100.
Postgrad
Formally the alt-az frame is topocentric and rotating; the transformation to equatorial coordinates is the spherical-triangle solution sin a = sin φ sin δ + cos φ cos δ cos H, with φ the latitude, δ the declination, and H the hour angle. Atmospheric refraction lifts every apparent altitude — about 34 arcminutes at the horizon, which is why the Sun you watch setting has, geometrically, already set.
The pole itself is defined dynamically. Lunisolar torque on Earth's equatorial bulge drives precession at about 50.3 arcseconds per year along the ecliptic, with nutation superposed; the modern reference pole is realized through VLBI to microarcsecond precision. Polaris, at declination near +89.3°, is a convenience for observers rather than a definition — celestial navigation reduces Polaris sights with an almanac correction for its small offset circle.
azimuth
The compass direction to an object, in degrees clockwise from north: 0 north, 90 east, 180 south, 270 west.
Now calibrate your ruler. Make a fist, hold it at arm's length, and sight past it: from edge to edge it covers about ten degrees of sky. This works for nearly everyone — larger fists ride on longer arms. The pointer gap, Merak to Dubhe, is about five and a half degrees: half a fist. Polaris sits about 28 degrees from Dubhe — five pointer gaps, or nearly three fists. And since no other bright star lives in that patch of sky, when your eye lands on a lone, modest star at the end of the line, you have it.
Finding one star is good; telling a friend where anything is, precisely enough to find it, is better. Astronomers do that with two numbers. First: face the object and ask what compass direction you are facing. That angle, in degrees clockwise from north, is the azimuth. Second: how high above the horizon is it? That angle is the altitude — 0 on the horizon, 90 at the point straight overhead. Together they name one spot in your sky. "Azimuth 225, altitude 45" means: face southwest, look four and a half fists up.
altitude
The angle of an object above the horizon: 0° on the horizon, 90° at the point straight overhead (the zenith).
Here is the fact that makes Polaris more than a compass. Measure its altitude — stack fists from the horizon up to the star — and the number you get is your latitude. From latitude 40° north, Polaris stands 40° up: four fists. From Quito, almost on the equator, it grazes the horizon. From the north pole it hangs straight overhead. The next folio proves why this works; tonight you can simply use it, the way navigators did for a thousand years before anyone could prove anything.
Find true north from a backyard at latitude 40°N — the steps fade as you master them
Merak and Dubhe — the pointers.
About five times the pointer gap — nearly three fists at arm's length.
Four fists — about 40°, matching the latitude.
True north.
You now own the night's first tools: a star that marks north, a two-star line that finds it, a fist that measures degrees, and a two-number address for anything you can see. What you do not know yet is why everything except Polaris refuses to hold still — why the whole sky drifts westward while one point stays fixed. That is the next folio's work: the turning of the sky itself.
Practice — new ink and old, interleaved
1.A friend reports a bright object at azimuth 270°, altitude 10°. Where is it?
2.You measure Polaris at altitude 51°. What is your latitude, in degrees north?
3.Put the steps of the Dipper-to-Polaris hop in order.
- Find the Big Dipper's bowl
- Take the line from Merak to Dubhe
- Extend it about five pointer gaps
- Land on the lone modest star — Polaris
4.Without looking back: what are the two numbers of a sky address, and what does each one measure?
Azimuth is the compass direction in degrees clockwise from north; altitude is the angle above the horizon, from 0 at the horizon to 90 straight overhead.
How close were you? Grade yourself honestly — it sets your review date.