Guide
Positional Astronomy
Coordinate Systems

Coordinate Systems

In order to specify the position of objects in the sky, we need a way to describe their exact location. This can be done by setting up a coordinate system, such that each point in the sky can be described by a pair of numbers. There are several coordinate systems used in astronomy, each with its own advantages and disadvantages. In this chapter, we will cover the most commonly used ones: horizontal (alt-az), equatorial, ecliptic, and galactic.

Horizontal (Alt-Az) Coordinate System

Any point in the sky can be described by its altitude aa and azimuth AA. The horizontal coordinate system is based on the observer’s local horizon, with the zenith (the point directly above the observer) at 90∘90^\circ altitude and the nadir (the point directly below the observer) at −90∘-90^\circ altitude.

An arc passing through the zenith and perpendicular to the horizon is called a vertical circle. The altitude aa of a point XX is the angle between the horizon and the point, as measured along the vertical circle passing through that point. The zenith angle zz is the angle between the zenith and the point, equal to the complement of the altitude:

z=90∘−a(1.2.1)\tag{1.2.1} z = 90^\circ - a

The altitude is positive above the horizon and negative below it, ranging from −90∘-90^\circ at the nadir to 90∘90^\circ at the zenith.

The azimuth AA of a point XX is the angle between the north cardinal point and the projection of the point XX onto the horizontal plane. The azimuth starts from 0∘0^\circ at the north point and increases eastwards: 90∘90^\circ at the east point, 180∘180^\circ at the south point, and 270∘270^\circ at the west point. In the southern hemisphere, azimuth is measured from the south point eastwards.

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The altitude-azimuth coordinate system is based on the observer's local horizon. (Source: Wikipedia)

The coordinates of a point in the horizontal coordinate system are given by the pair (a,A)(a, A), where aa is the altitude and AA is the azimuth. These change with the observer’s location and the time of observation.

Equatorial Coordinate System

The equatorial coordinate system is based on the celestial equator, which is the projection of the Earth’s equator onto the celestial sphere. The coordinates in this system are right ascension (RA/α\alpha), hour angle (HA/hh), and declination (Dec/δ\delta).

An arc passing through the north celestial pole and perpendicular to the celestial equator is called an hour circle. The declination δ\delta of a point XX is the angle between the celestial equator and the point, as measured along the hour circle passing through that point. It is positive above the celestial equator and negative below it, ranging from −90∘-90^\circ at the south celestial pole to 90∘90^\circ at the north celestial pole.

The hour angle hh of a point XX is the angle between the observer’s meridian (the hour circle passing through the observer’s zenith) and the hour circle passing through the point XX. The hour angle is positive when the point is west of the observer’s meridian and negative when it is east of it, ranging from −12h-12^h to 12h12^h, with 0h0^h at the observer’s meridian.

The hour angle can also be interpreted as the time since the point XX last crossed the observer’s meridian, or the time since its upper culmination.

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The Earth rotates 360∘360^\circ in 2424 hours, so the hour angle increases by 15∘15^\circ per hour. To convert between degrees and hours, we use the conversion factor 1h=15∘1^h = 15^\circ.

The hour angle of a point changes with the observer’s location and the time of observation. This is because the hour angle is defined with respect to the observer’s meridian. To remove this dependence, the right ascension (RA) is introduced.

We first define the equinoxes: the equinoxes are the two points where the ecliptic intersects the celestial equator. The vernal equinox ♈︎ (also known as the first point of Aries) is the point where the Sun (moving along the ecliptic) crosses the celestial equator from south to north. The autumnal equinox ♎︎ (also known as the first point of Libra) is the point where the Sun crosses the celestial equator from north to south. These two points are fixed on the celestial sphere and do not depend on the observer’s location or the time of observation.

The right ascension α\alpha of a point XX is the angle between the vernal equinox and the projection of the point XX onto the celestial equator. The right ascension starts from the vernal equinox at 0h0^h and increases eastwards (opposite to the direction of increasing hour angle): 6h6^h at the summer solstice, 12h12^h at the autumnal equinox, and 18h18^h at the winter solstice.

The conversion from hour angle to right ascension will be discussed in chapter 1.3.

The RA/Dec coordinates of a point are given by the pair (α,δ)(\alpha, \delta), where α\alpha is the right ascension and δ\delta is the declination. These coordinates do not depend on the location of the observer or the time of observation, as they are defined with respect to the celestial equator and the vernal equinox, which are fixed points on the celestial sphere.

Culmination

A star is said to culminate when it is at the observer’s meridian.

The upper culmination occurs when the star is at its highest point in the sky, which corresponds to the moment when the star’s hour angle is 0h0^h. At this point, the altitude of the star is at its maximum, and it is directly south of the observer in the northern hemisphere (or directly north in the southern hemisphere).

The lower culmination occurs when the star is at its lowest point in the sky, which corresponds to the moment when the star’s hour angle is 12h12^h. At this point, the altitude of the star is at its minimum, and it is directly north of the observer in the northern hemisphere (or directly south in the southern hemisphere).

The maximum altitude or minimum zenith distance of a star (during upper culmination) can be calculated using the formula:

amax=90∘−∣ϕ−δ∣(1.2.2)\tag{1.2.2} a_{\text{max}} = 90^\circ - |\phi - \delta|

zmin=∣ϕ−δ∣(1.2.3)\tag{1.2.3} z_{\text{min}} = |\phi - \delta|

where ϕ\phi is the observer’s latitude, and δ\delta is the star’s declination. If amax<0a_{\text{max}} < 0, the star will never rise above the horizon for the observer. If ϕ>δ\phi > \delta, the star will culminate south of zenith, and if ϕ<δ\phi < \delta, it will culminate north of zenith.

The minimum altitude or maximum zenith distance of a star (during lower culmination) can be calculated using the formula:

amin=ϕ+δ−90∘(1.2.4)\tag{1.2.4} a_{\text{min}} = \phi + \delta - 90^\circ

zmax=180∘−(ϕ+δ)(1.2.5)\tag{1.2.5} z_{\text{max}} = 180^\circ - (\phi + \delta)

If amin>0a_{\text{min}} > 0, the star is circumpolar and will never set, as discussed in the following section.

Circumpolar Stars

A star is said to be circumpolar if it never sets below the horizon. For a star to be circumpolar, we must have

ϕ+δ≥90∘(1.2.6)\tag{1.2.6} \phi + \delta \geq 90^\circ

where ϕ\phi is the observer’s latitude and δ\delta is the star’s declination. This can be derived from the amin>0a_\text{min} > 0 condition from the previous section.

Some stars never rise above the horizon. For a star to never rise above the horizon, we must have

ϕ−δ≤−90∘(1.2.7)\tag{1.2.7} \phi - \delta \leq -90^\circ

where ϕ\phi is the observer’s latitude and δ\delta is the star’s declination. This can be derived from the amax<0a_\text{max} < 0 condition from the previous section.

Ecliptic Coordinate System

The ecliptic coordinate system is based on the ecliptic plane, which is the plane of the Earth’s orbit around the Sun. The coordinates in this system are the ecliptic longitude λ\lambda and the ecliptic latitude β\beta.

The ecliptic longitude λ\lambda of a point XX is the angle between the vernal equinox and the projection of the point onto the ecliptic plane. The ecliptic longitude starts from the vernal equinox at 0∘0^\circ and increases eastwards: 90∘90^\circ at the summer solstice, 180∘180^\circ at the autumnal equinox, and 270∘270^\circ at the winter solstice.

The ecliptic latitude β\beta of a point XX is the angle between the point and the ecliptic plane. The ecliptic latitude is positive northwards from the ecliptic plane and ranges from −90∘-90^\circ at the south ecliptic pole to 90∘90^\circ at the north ecliptic pole.

The ecliptic coordinates of a point are given by the pair (λ,β)(\lambda, \beta), where λ\lambda is the ecliptic longitude and β\beta is the ecliptic latitude. These coordinates do not depend on the location of the observer or the time of observation, as they are defined with respect to the ecliptic plane and the vernal equinox, which are fixed points on the celestial sphere. This coordinate system is particularly useful for describing the positions of objects in the solar system, as their orbits are generally close to the ecliptic plane.

The geocentric ecliptic coordinates are the ecliptic coordinates of a point as seen from the center of the Earth, while the heliocentric ecliptic coordinates are the ecliptic coordinates of a point as seen from the center of the Sun. The two coordinate systems are equal only if the object is very far away. The geocentric coordinates, moreover, depend on Earth’s position in its orbit around the Sun, while the heliocentric coordinates do not.

Galactic Coordinate System

The galactic coordinate system is based on the plane of the Milky Way galaxy. The coordinates in this system are galactic longitude ll and galactic latitude bb.

The galactic longitude ll of a point XX is the angle between the galactic center and the projection of the point onto the galactic plane. The galactic longitude is measured eastwards from the Sun and starts from 0∘0^\circ at the galactic center, 90∘90^\circ at the first galactic quadrant, 180∘180^\circ at the second galactic quadrant, and 270∘270^\circ at the third galactic quadrant.

The galactic latitude bb of a point XX is the angle between the point and the galactic plane, measured northwards from the galactic plane. It ranges from −90∘-90^\circ at the south galactic pole to 90∘90^\circ at the north galactic pole.

The north galactic pole (NGP) is the point in the sky that is perpendicular to the plane of the Milky Way galaxy. Its equatorial coordinates are (αG,δG)=(12h51.4m,27∘8′)(\alpha_G, \delta_G) = (12^h 51.4^m, 27^\circ 8').

The zero point of galactic longitude ll is defined as the direction towards the galactic center, located in the constellation Sagittarius. The zero point of galactic latitude is defined as the plane of the Milky Way galaxy. It is such that the angle θ=∠PGL≈123.0∘\theta = \angle PGL \approx 123.0^\circ, where PGLPGL is the position of the galactic north pole in the equatorial coordinate system.

Transformation Between Coordinate Systems

The easiest way to transform between coordinate systems is to draw a spherical diagram and apply the identities of spherical trigonometry.

≃ 23°26' P P' Q Q' Z Z' O C α δ τ φ A h τ 360°-a 90°-φ z=90°-h p=90°-δ N S W E (I) (H) (M) zeru-ekuatorea tokiko zeru-horizontea ekliptika azimut-zirkulua ordu-zirkulua tokiko zeru-meridianoa Lurraren ardatza Zeruko Ipar Poloa Zeruko Hego Poloa Zenita Nadirra ♎ - Libra puntua ♈ - Aries puntua Martxoko ekinozioan eguzkia bertan dagoela dirudien zeru-esferaren tokia. φ - tokiko latitudea δ - deklinazioa a - azimuta α - igoera zuzena h - altitude edo garaiera τ - ordu-angelua z - zenit-distantzia (angeluarra) p - distantzia polarra (angeluarra) (O) equador celeste horitzó celeste local eclíptica cercle azimutal cercle horària meridià celeste local eix de la Terra Pol Nord Celeste Pol Sud Celeste Zenit Nadir ♎ - Punt de Balança ♈ - Punt d'Àries Lloc de l'esfera celeste on sembla estar el sol a l'equinocci de març. φ - latitud local δ - declinació a - azimut α - ascensió recta h - altura τ - angle horari z - distància zenital (angular) p - distància polar (angular) (O) equatore celeste orizzonte celeste locale eclittica cerchio azimutale cerchio orario meridiano celeste locale asse terrestre Polo Nord Celeste Polo Sud Celeste Zenit Nadir ♎ - Punto della Bilancia ♈ - Punto vernale Il luogo sulla sfera celeste in cui il sole sembra essere all'equinozio di marzo. φ - latitudine locale δ - declinazione a - azimut α - ascensione retta h - altezza τ - angolo orario z - distanza zenitale (angolare) p - distanza polare (angolare) (O) Himmelsäquator Horizont Ekliptik Azimutkreis Stundenkreis Meridian Erdachse Himmels- nordpol Himmels- südpol Zenit Nadir ♎ - Herbstpunkt ♈ - Frühlingspunkt Die Stelle auf der Himmelskugel, an der sich die Sonne zur Tagundnachtgleiche im März zu befinden scheint. φ - lokale Breite δ - Deklination a - Azimut α - Rektaszension h - Vertikalwinkel τ - Stundenwinkel z - Zenitwinkel p - celestial equator local celestial horizon ecliptic azimuth circle hour circle local celestial meridian Earth axis North Celestial Pole South Celestial Pole Zenith Nadir ♎ - First point of Libra ♈ - First point of Aries The place on the celestial sphere where the Sun seems to be at the March equinox. φ - local latitude δ - declination A - azimuth α - right ascension h - altitude or elevation τ - hour angle z - zenith distance (angular) p - polar distance (angular) Felix Oleaga Larrazabal, Busturia 1904-06-21, Algorta 1977-02-04, Busturia. Miren Olabarria Torrontegi Algorta 1921-03-14, Algorta 2011-05-06, Busturia
Illustration of the different celestial coordinate systems. (Source: Wikipedia)

Horizontal and Equatorial

To interconvert between horizontal coordinates (a,A)(a, A) and equatorial coordinates (h,δ)(h, \delta), given the observer’s latitude ϕ\phi, we use the following identities:

sin⁡Acos⁡a=sin⁡hcos⁡δcos⁡Acos⁡a=−cos⁡hcos⁡δsin⁡ϕ+sin⁡δcos⁡ϕsin⁡a=cos⁡hcos⁡δcos⁡ϕ+sin⁡δsin⁡ϕsin⁡hcos⁡δ=sin⁡Acos⁡acos⁡hcos⁡δ=−cos⁡Acos⁡a+sin⁡acos⁡ϕsin⁡δ=cos⁡acos⁡Acos⁡ϕ+sin⁡asin⁡ϕ \tag{1.2.8} \begin{align*} \sin A \cos a &= \sin h \cos \delta \\ \cos A \cos a &= -\cos h \cos \delta \sin \phi + \sin \delta \cos \phi \\ \sin a &= \cos h \cos \delta \cos \phi + \sin \delta \sin \phi \\ \\ \tag{1.2.9} \sin h \cos \delta &= \sin A \cos a \\ \cos h \cos \delta &= -\cos A \cos a + \sin a \cos \phi \\ \sin \delta &= \cos a \cos A \cos \phi + \sin a \sin \phi \end{align*}

Equatorial and Ecliptic

To interconvert between equatorial coordinates (α,δ)(\alpha, \delta) and ecliptic coordinates (λ,β)(\lambda, \beta), we use the following identities:

sin⁡αcos⁡δ=−sin⁡βsin⁡ε+cos⁡βcos⁡εsin⁡λcos⁡αcos⁡δ=cos⁡βcos⁡λsin⁡δ=sin⁡βcos⁡ε+cos⁡βsin⁡εsin⁡λsin⁡λcos⁡β=sin⁡δsin⁡ε+cos⁡δcos⁡εsin⁡αcos⁡λcos⁡β=cos⁡δcos⁡αsin⁡β=sin⁡δcos⁡ε−cos⁡δsin⁡εsin⁡α \tag{1.2.10} \begin{align*} \sin \alpha \cos \delta &= - \sin \beta \sin \varepsilon + \cos \beta \cos \varepsilon \sin \lambda \\ \cos \alpha \cos \delta &= \cos \beta \cos \lambda \\ \sin \delta &= \sin \beta \cos \varepsilon + \cos \beta \sin \varepsilon \sin \lambda \\ \\ \tag{1.2.11} \sin \lambda \cos \beta &= \sin \delta \sin \varepsilon + \cos \delta \cos \varepsilon \sin \alpha \\ \cos \lambda \cos \beta &= \cos \delta \cos \alpha \\ \sin \beta &= \sin \delta \cos \varepsilon - \cos \delta \sin \varepsilon \sin \alpha \end{align*}

where ε≈23.44∘\varepsilon \approx 23.44^\circ is the axial tilt of the Earth.

Equatorial to Galactic

To convert from equatorial coordinates (α,δ)(\alpha, \delta) to galactic coordinates (l,b)(l, b), we use the following identities:

sin⁡(θ−l)cos⁡b=cos⁡δsin⁡(α−αG)cos⁡(θ−l)cos⁡b=−cos⁡δcos⁡δGcos⁡(α−αG)+sin⁡δsin⁡δGsin⁡b=cos⁡δcos⁡δGcos⁡(α−αG)+sin⁡δsin⁡δG \tag{1.2.12} \begin{align*} \sin (\theta - l) \cos b &= \cos \delta \sin (\alpha - \alpha_G) \\ \cos (\theta - l) \cos b &= - \cos \delta \cos \delta_G \cos (\alpha - \alpha_G) + \sin \delta \sin \delta_G \\ \sin b &= \cos \delta \cos \delta_G \cos (\alpha - \alpha_G) + \sin \delta \sin \delta_G \end{align*}

where θ≈123.0∘\theta \approx 123.0^\circ is the angle between the galactic north pole and the vernal equinox, and (αG,δG)=(12h51.4m,27∘8′)(\alpha_G, \delta_G) = (12^h 51.4^m, 27^\circ 8') are the coordinates of the galactic north pole in the equatorial coordinate system.

Problems

Derive all the coordinate system transformation identities stated in the section above using spherical trigonometry.
Find the hour angle of an object XX with declination δ\delta, when it rises and sets for an observer at latitude ϕ\phi. From this, find the time the object is above the horizon.

At the time of rising or setting, the altitude of the object is a=0∘a = 0^\circ. Using equation (1.2.8.3):

sin⁡a=cos⁡hcos⁡δcos⁡ϕ+sin⁡δsin⁡ϕ\sin a = \cos h \cos \delta \cos \phi + \sin \delta \sin \phi

  ⟹  cos⁡h=−tan⁡δtan⁡ϕ(1.2.13)\tag{1.2.13} \implies \boxed{\cos h = - \tan \delta \tan \phi}

This is a very important result and is used often in positional astronomy. For rising, hh is negative, and for setting, hh is positive. The time the object is above the horizon is given by

Δt=hset−hrise=2cos⁡−1(−tan⁡δtan⁡ϕ)\Delta t = h_\text{set} - h_\text{rise} = 2 \cos^{-1} (- \tan \delta \tan \phi)

where Δt\Delta t is in hours.

Find the azimuth of an object XX with declination δ\delta, when it rises and sets for an observer at latitude ϕ\phi.

At the time of rising or setting, the altitude of the object is a=0∘a = 0^\circ. Using equations (1.2.8.2) and (1.2.13):

cos⁡Acos⁡a=−cos⁡hcos⁡δsin⁡ϕ+sin⁡δcos⁡ϕ\cos A \cos a = - \cos h \cos \delta \sin \phi + \sin \delta \cos \phi

  ⟹  cos⁡A=tan⁡ϕtan⁡δcos⁡δsin⁡ϕ+sin⁡δcos⁡ϕ\implies \cos A = \tan \phi \tan \delta \cos \delta \sin \phi + \sin \delta \cos \phi

  ⟹  cos⁡A=sin⁡δcos⁡ϕ(1.2.14)\tag{1.2.14} \implies \boxed{\cos A = \frac{\sin \delta}{\cos \phi}}

This result too is often used in solving problems. The azimuth at rising is between 0∘0^\circ and 180∘180^\circ, and at setting is between 180∘180^\circ and 360∘360^\circ.

Calculate the azimuths of the star Procyon (δ=5∘\delta = 5^\circ N) when its zenith distance is 80∘80^\circ as seen by an observer at latitude 56∘56^\circ N.

A zenith distance of 80∘80^\circ corresponds to an altitude of 10∘10^\circ. Using equation (1.2.9.3):

sin⁡δ=cos⁡acos⁡Acos⁡ϕ+sin⁡asin⁡ϕ\sin \delta = \cos a \cos A \cos \phi + \sin a \sin \phi

  ⟹  cos⁡A=sin⁡δ−sin⁡asin⁡ϕcos⁡acos⁡ϕ\implies \cos A = \frac{\sin \delta - \sin a \sin \phi}{\cos a \cos \phi}

=sin⁡10∘sin⁡56∘−sin⁡5∘cos⁡10∘cos⁡56∘= \frac{\sin 10^\circ \sin 56^\circ - \sin 5^\circ}{\cos 10^\circ \cos 56^\circ}

≈−0.103\approx -0.103

Thus, the possible azimuths are A=96∘A = 96^\circ or A=264∘A = 264^\circ.

Given that h1h_1, h2h_2 are the hour angles of a star of declination δ\delta on the prime vertical west and at setting respectively for an observer in north latitude, show that

cos⁡h1cos⁡h2+tan⁡2δ=0\cos h_1 \cos h_2 + \tan^2 \delta = 0

From equation (1.2.13), at the time of setting the hour angle is given by

cos⁡h2=−tan⁡δtan⁡ϕ(1)\tag{1} \cos h_2 = - \tan \delta \tan \phi

When the star is on the prime vertical west, its azimuth is A=270∘A = 270^\circ. Using equation (1.2.8.2):

cos⁡Acos⁡a=−cos⁡hcos⁡δsin⁡ϕ+sin⁡δcos⁡ϕ\cos A \cos a = - \cos h \cos \delta \sin \phi + \sin \delta \cos \phi

  ⟹  0=−cos⁡h1cos⁡δsin⁡ϕ+sin⁡δcos⁡ϕ\implies 0 = - \cos h_1 \cos \delta \sin \phi + \sin \delta \cos \phi

  ⟹  cos⁡h1=tan⁡δcot⁡ϕ(2)\tag{2} \implies \cos h_1 = \tan \delta \cot \phi

Thus, from (1) and (2) we get

cos⁡h1cos⁡h2+tan⁡2δ=0\boxed{\cos h_1 \cos h_2 + \tan^2 \delta = 0}
A star has declination δ\delta and is seen by an observer located in the northern hemisphere at latitude ϕ<δ\phi < \delta. Find the maximum azimuth of the star as seen by the observer.

Since ϕ<δ\phi < \delta, the star has both its upper culmination and lower culmination north of zenith, and thus can never achieve all values of the azimuth. To find the maximum azimuth, we draw a spherical triangle with vertices at the north celestial pole PP, the zenith ZZ, and the star XX. We know two sides and two angles of the triangle:

  • PZ=90∘−ϕPZ = 90^\circ - \phi
  • PX=90∘−δPX = 90^\circ - \delta
  • ∠PZX=A\angle PZX = A
  • ∠ZXP=90∘\angle ZXP = 90^\circ

Using the sine rule, we have

sin⁡Asin⁡(90∘−δ)=sin⁡90∘sin⁡(90∘−ϕ)\frac{\sin A}{\sin (90^\circ - \delta)} = \frac{\sin 90^\circ}{\sin (90^\circ - \phi)}

  ⟹  sin⁡A=cos⁡δcos⁡ϕ\implies \sin A = \frac{\cos \delta}{\cos \phi}

Thus, the maximum azimuth seen is A=sin⁡−1(cos⁡δsec⁡ϕ)\boxed{A = \sin^{-1} (\cos \delta \sec \phi)}

Regulus (α Leo) is the brightest star in the constellation Leo. It appears to be a single star to the naked eye, but is actually a quadruple star system, composed of four stars that are organized into two pairs. One of the stars, Regulus A, has equatorial coordinates (α,δ)=(10h8m,11∘58′ N)(\alpha, \delta) = (10^h 8^m, 11^\circ 58' \text{ N}). Calculate the ecliptic coordinates of Regulus A.

We are given that δ=11∘58′\delta = 11^\circ 58' and α=10h8m\alpha = 10^h 8^m. Thus, using equations (1.2.11), we have

sin⁡λcos⁡β≈0.5038cos⁡λcos⁡β≈−0.8640sin⁡β≈0.0075 \begin{align*} \sin \lambda \cos \beta &\approx 0.5038 \\ \cos \lambda \cos \beta &\approx -0.8640 \\ \sin \beta &\approx 0.0075 \end{align*}

From these, we get (λ,β)=(9h59m,0∘26′)\boxed{(\lambda, \beta) = (9^h 59^m, 0^\circ 26')}. Since the ecliptic latitude is very small, we conclude that Regulus A lies very close to the ecliptic plane.

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