Positional Astronomy - Kapteyn Astronomical...

Positional AstronomyObservational Astronomy 2019 Part 2 Prof. S.C. Trager

Coordinate systems

We need to know where the astronomical objects we want to study are located in order to study them!

We need a system (well, many systems!) to describe the positions of astronomical objects.

First we need the concept of the celestial sphere.

It would be nice if we knew the distance to every object we’re interested in — but we don’t.

And it’s actually unnecessary in order to observe them!

The Celestial Sphere

The Celestial SphereInstead, we assume that all astronomical sources are infinitely far away and live on the surface of a sphere at infinite distance. This is the celestial sphere.

If we define a coordinate system on this sphere, we know where to point!

Furthermore, stars (and galaxies) move with respect to each other. The motion normal to the line of sight — i.e., on the celestial sphere — is called proper motion (which we’ll return to shortly)

Astronomical coordinate systems

A bit of terminology:

great circle: a circle on the surface of a sphere intercepting a plane that intersects the origin of the sphere

i.e., any circle on the surface of a sphere that divides that sphere into two equal hemispheres

Horizon coordinates

A natural coordinate system for an Earth-bound observer is the “horizon” or “Alt-Az” coordinate system

The great circle of the horizon projected on the celestial sphere is the equator of this system.

Horizon coordinates

Altitude (or elevation) is the angle from the horizon up to our object — the zenith, the point directly above the observer, is at +90º

Horizon coordinates

We need another coordinate: define a great circle perpendicular to the equator (horizon) passing through the zenith and, for convenience, due north

This line of constant longitude is called a meridian

Horizon coordinates

The azimuth is the angle measured along the horizon from north towards east to the great circle that intercepts our object (star) and the zenith.

Horizon coordinates

The origin of these angles (coordinates) is the observer

Note that this is a left-handed coordinate system!

Horizon coordinatesNearly all big telescopes (diameter ≥ 4m, telescopes built after ~1990, most “classical” radio telescopes) are in alt-az mounts

This is the natural coordinate system for these telescopes

But this system is dependent on the location of the observer and time of the observation: makes consistent cataloguing of objects difficult!

The William Herschel Telescope is an alt-az telescope, as are the VLTs.

Equatorial coordinates

Let’s consider a coordinate system that is tied to the astronomical objects themselves — and preferably those that don’t move! ♈

+90º

–90º


In equatorial coordinates, the celestial equator is the great circle that intersects both the celestial sphere and the Earth’s equator: it’s the projection of the equator onto the celestial sphere

♈

+90º

–90º

Equatorial coordinatesThe declination δ is the celestial latitude and is measured in degrees, with 0º at the equator, +90º at the North Celestial Pole (NCP) — the intersection of the Earth’s north (rotational) pole with the celestial sphere — and –90º at the South Celestial Pole

♈

+90º

–90º

Equatorial coordinatesThe right ascension (RA) α is the celestial longitude and is measured in units of time, 0–24 hours, from west to east, with 0h at the Sun’s position when it crosses the equator from south to north, approximately at noon on 21 March in Greenwich, UK.

♈

+90º

–90º

Equatorial coordinatesThe position α=0h, δ=0º is called the vernal equinox ♈

this is the sign of the constellation Aries, where the vernal equinox happened 2500 years ago

The equatorial system is a right-handed system

♈

+90º

–90º

Equatorial coordinatesBecause the Earth precesses around an average direction perpendicular to the ecliptic (the plane of the Earth’s orbit around the Sun) due to the torques exerted on by the Moon, Sun, and Jupiter (more later!), the equatorial system slowly changes with time.

♈

+90º

–90º


This means that the vernal equinox and the celestial equator move with respect to the distant background objects (galaxies, quasars).

There we need to assign an epoch — a date — to any equatorial coordinate. (We’ll return to this shortly!)

♈

+90º

–90º

The local equatorial systemThe local equatorial system is used to point polar-axis (or “equatorial”) mount telescopes

These telescopes rotate around an axis parallel to the Earth’s rotation axis

In the Northern Hemisphere, this means that the primary mount axis always points north

Our Gratama Telescope is a polar-axis telescope, as the Isaac Newton Telescope on La Palma

The local equatorial system

These telescopes track a star by rotation around only one axis

Note that this means that the field of the image does not rotate, like it does for an alt-az telescope

The local equatorial systemIn the local equatorial system, the hour angle HA replaces the right ascension: HA=LST–α

Here LST is the local sidereal time (which we’ll define shortly!)

So knowing the time of day (the LST) and the α,δ of an object, it’s very easy to locate your object with a polar-axis telescope.

HA varies from –6 h at the eastern horizon (rising) to 0 h at the zenith to +6 h at the western horizon (setting)

Note that the minus sign makes this a left-handed coordinate system!


A note about fixed angular sizes in (any) equatorial coordinate system:

Fixed angular sizes get longer in longitude of the coordinate system (e.g., right ascension) as one goes goes towards the pole – i.e., towards higher absolute latitude |δ| – by a factor that goes as 1/cos(δ)

Galactic coordinatesIt is sometimes convenient to use the Milky Way itself to define a coordinate system

For example, if you want to know the positions of globular clusters relative to the bugle and disk or need an estimate of the interstellar dust extinction or the stellar density towards an object

Galactic coordinatesIt is sometimes convenient to use the Milky Way itself to define a coordinate system

For example, if you want to know the positions of globular clusters relative to the bugle and disk or need an estimate of the interstellar dust extinction or the stellar density towards an object

Galactic coordinates

In galactic coordinates, the plane of the Galaxy defines the (celestial) equator, assuming that the Sun sits exactly in the plane (which isn’t quite true)


In this system, the galactic longitude l (often written lII) is measured in degrees, with 0º on a line connecting the Sun with the center of the Galaxy (roughly...) and increasing in a right-handed fashion


The galactic latitude b (bII) is also measured in degrees, with b=0º at the equator.


The system is precisely defined by the direction of the North Galactic Pole (NGP):

and by the Galactic longitude of the North Celestial Pole:

↵NGP(B1950) = 192.25� = 12h 49m

�NGP(B1950) = +27.4� = +27�240

lNCP = 123�

Galactic coordinatesThe first set of coordinates implies that the celestial and galactic equators are tilted by 90º–27.4º=62.6º

These two great circles cross at two nodes, and the line of nodes that connect them is the axis that transforms one plane to the other

l=123ºNCPδ=27.4ºα=12h49m

l=33.0º α=18h49m

line of nodes


The equators cross at

for the ascending node

the extra 90º angles in l and α shift from the NCP and the galactic equator to the nodes

lnode = 123� � 90� = 33�

↵B1950node = 12h 49m + 6h = 18h49m

l=123ºNCPδ=27.4ºα=12h49m

l=33.0º α=18h49m

line of nodes


Using the cosine law of spherical trigonometry, one can show that the transformation from α,δ to l,b is

where the last equation gives the sign of b — i.e., the proper quadrant of the Galaxy

cos b cos(l � 33�) = cos � cos(↵� 282.25�)

cos b sin(l � 33�) = cos � sin(↵� 282.25�) cos 62.6� + sin � sin 62.6�

sin b = sin � cos 62.6� � cos � sin(↵� 282.25�) sin 62.6�


To transform from l,b to α,δ use

Note in both transformations that α,δ must be in B1950 coordinates!

cos � sin(↵� 282.25�) = cos b sin(l � 33�) cos 62.6� � sin b sin 62.6�

sin � = cos b sin(l � 33�) sin 62.6� � sin b cos 62.6�

Other coordinate systemsEcliptic coordinates

used mostly for satellite navigation, where knowledge of the Sun–spacecraft angle is critical; uses the plane of the ecliptic as the celestial equator

Supergalactic coordinates

used for determining the positions of galaxies and clusters of galaxies relative to the Virgo Cluster–Local Group–Coma Cluster plane; rarely used

Two issues...1. Epoch: For the equatorial coordinate system, a date must be

specified to know where the vernal equinox was when the positions where defined. Two epochs are commonly used:

B1950, based on the Besselian year and refers to the Earth’s orientation at 22h 09m UT on 1949 December 31

J2000, based on the Julian year and refers to the Earth’s orientation at ≈noon in Greenwich UK on 2000 January 1.

Nearly all astronomers now use J2000, but older papers use B1950 (and the Galactic coordinate system is specified in B1950)

Two issues...2. Reference Frames: Coordinate systems are difficult

to use without signposts, so special calibrating objects are used to determine coordinates. B1950 coordinates were based on the FK4 (Fundamental Katalog 4). J2000 coordinates were originally based on the FK5 but are now based on the ICRS (International Celestial Reference System). ICRS is based on very accurate VLBI-based positions of >600 extragalactic radio sources. The Hipparcos catalog of stellar positions (etc.) has been tied to ICRS to within 0.5 milliarcseconds.

Gaia’s DR2 catalogue will replace Hipparcos in the next few weeks!

Motions of stars, apparent and real

Let’s use the horizon coordinate system and see how stars move on the sky during a night. Consider a northern hemisphere site like Groningen (θlat≈+53º)

zenith

W looking eastN S

polar axis

celest

ial eq

uator

NCP

A B

set

rise

53º

δ θB


Stars with

rise from the east and set in the west, just like the Sun

Stars with

(like star A) are always above the horizon and just circle around the NCP

zenith

W looking eastN S

polar axis

celest

ial eq

uator

NCP

A B

set

rise

53º

δ θB

✓lat � 90� < � < 90� � ✓lat

� > 90� � ✓lat


Stars with

like star B, rise north of east, moves towards the south until they transit the meridian (directly south of the zenith), then move westwards and set north of west. They cross the meridian at a maximum altitude

zenith

W looking eastN S

polar axis

celest

ial eq

uator

NCP

A B

set

rise

53º

δ θB

0� < � < 90� � ✓lat,

✓B = 90� � ✓lat + �


Stars with

like star B, rise north of east, moves towards the south until they transit the meridian (directly south of the zenith), then move westwards and set north of west. They cross the meridian at a maximum altitude

zenith

W looking eastN S

polar axis

celest

ial eq

uator

NCP

A B

set

rise

53º

δ θB

0� < � < 90� � ✓lat,

✓B = 90� � ✓lat + �

ParallaxConsider the Earth, 1 AU away from the Sun at position E1. Six months later, the Earth is at position E2, but the star has remained in the same place relative to the Sun. Then, as seen from Earth, the star appears to have subtended an angle 2ϖ on the sky.

E1 E2Sun1 AU

d

ϖ



Then if r (=1 AU) is the radius of the Earth’s orbit, we find

because ϖ is clearly small; then converting to seconds of arc,

Defining 1 AU such that

and 1 parsec as the distance at which a star would have a parallax of 1″:

The distance to a star with observed parallax ϖ″ is then

r

d= tan� � � rad

�� = 206265� rad

d =206265

�� AU

d =1

�� pc

1 pc = 206265AU = 3.086� 1013 km = 3.26 light years

1 AU=1.496x10^13 cm

If the star lies at the ecliptic pole, it traces out a circle on its parallactic path

If the star lies in the ecliptic plane, it traces out a line

The velocity of the Earth as it orbits the Sun causes another apparent shift of stellar positions called aberration. It’s actually an effect of special relativity, but it can be determined to within ~1 mas using a classical analogy.

Imagine sitting on a train while it’s raining: if the train is sitting still, the rain goes straight down the windows; but if the train is moving, the rain goes diagonally down the windows, because the train has moved during the time it takes the rain to move — straight down, in its reference frame — from the top to the bottom of the window

Aberration of starlight

Consider...

a stationary telescope

a telescope moving with the Earth, seen in the Earth’s frame

and a telescope moving with the Earth, seen in a stationary reference frame

Aberration

Stationary

Earth frame

vEarth

Stationary frame

vEarth

θaber

The maximum effect — for objects perpendicular to the Earth’s orbit (the ecliptic) — is

So a star at the ecliptic pole will trace out a circle of radius 20.5″ every year; stars at the equator will move to and fro on a line

The effect depends entirely on the time of year and the direction of the object

Aberration

✓aber,max ⇡ vEarth

c=

2.979⇥ 106 cm s�1

2.998⇥ 1010 cm s�1= 0.994⇥ 10�4 rad = 20.500

Aberration

ecliptic

Aberration

ecliptic

Note however that all objects in the same direction suffer the same aberration, so it’s impossible to detect aberration from images of small regions of the sky

Note also that the Earth’s rotation also causes an aberration called diurnal aberration, which is a very small effect (<0.3″)

Aberration

Precession and nutation

Precession of the Earth’s rotation axis with respect to the ecliptic, caused by torques from the Sun, Moon, and other planets (mostly Jupiter) on the Earth, causes the equatorial system to precess with time

The Earth is not spherical: it bulges slightly around the equator, so other bodies can exert a torque on it (the Earth has a quadrupole moment)

Precession

Because the Moon’s orbit is almost in the plane of the ecliptic, and because it is so close, it combines with the Sun (and Jupiter, also in the ecliptic plane) to exert a torque that results in a 25770-year precessional period

Ecliptic pole

Earth

track of Sun (ecliptic plane)

celestial equator today

celestial equator in 12885 years

NCP todayNCP in 12885 years

23.45ºfixed stars

fixed stars

fixed stars

fixed stars

Precession

This causes the equatorial coordinate system to move (precess) with respect to the background stars

Ecliptic pole

Earth





23.45ºfixed stars

fixed stars

fixed stars

fixed stars

PrecessionNote that the Earth’s orbit is fixed in space, and therefore so is the ecliptic

As Earth precesses away from its current position, the celestial equator starts to slip to the west, and so the intersection of the celestial equator and the ecliptic — the vernal equinox — does as well

Ecliptic pole

Earth





23.45ºfixed stars

fixed stars

fixed stars

fixed stars

PrecessionSince the vernal equinox defines the zeropoint of right ascension, (fixed) stars will have increasing right ascension as the Earth precesses (and their declinations will also change)

This is the precession of the equinoxes and is why we must always specify a date — an epoch — when giving the equatorial coordinates of an object

Ecliptic pole

Earth





23.45ºfixed stars

fixed stars

fixed stars

fixed stars

Precession

The rate of this precession of the equinoxes is

or 42′ in 50 years

Ecliptic pole

Earth





23.45ºfixed stars

fixed stars

fixed stars

fixed stars

360� ⇥ 360000/� / 25770 yr = 50.300/yr

By 21 March of this year (2019), the equinox had slipped by nearly 16′ to the west from its J2000.0 position…

Nutation

Nutation is often separated from precession but is actually just the small-scale wobbles around the steady precession caused by the same processes, plus some less-predictable wobbles due to ocean–crust interactions on Earth

It has an amplitude of ≈9″ with a period of 18.6 years

Two other motionsRefraction: the Earth’s atmosphere moves images, with an angle depending on altitude and wavelength, due to atmospheric refraction; we’ll come back to this later

Proper motion: stars have motions relative to very distant background objects; their projected motions on the celestial sphere (the plane of the sky) relative to the barycenter of the Solar System are called proper motions. When known, these must be taken into account when pointing to a star. An extreme example is Barnard’s Star, which has a proper motion of 10.25″ per year!

The proper motion of Barnard’s Star

The proper motion of Barnard’s Star

178

mean [α, δ ], FK4,any equinox

mean [α, δ ], FK4,no µ, any equinox

mean [α, δ ], FK5,any equinox

space motion space motion– E-terms – E-terms

precess to B1950 precess to B1950 precess to J2000+ E-terms + E-terms

FK4 to FK5, no µ FK4 to FK5, no µparallax parallax

FK5, J2000, current epoch, geocentric

light deflectionannual aberrationprecession-nutation

Apparent [α, δ ]

Earth rotation

Apparent [h, δ ]

diurnal aberration

Topocentric [h, δ ]

[h, δ ] to [Az,El ]

Topocentric [Az,El ]

refraction

Observed [Az,El ]

Figure 1: Relationship Between Celestial CoordinatesStar positions are published or catalogued using one of the mean [α, δ ] systems shown at the top.The “FK4” systems were used before about 1980 and are usually equinox B1950. The “FK5”system, equinox J2000, is now preferred, or rather its modern equivalent, the InternationalCelestial Reference Frame (in the optical, the Hipparcos catalogue). The figure relates a star’smean [α, δ ] to the actual line-of-sight to the star. Note that for the conventional choices ofequinox, namely B1950 or J2000, all of the precession and E-terms corrections are superfluous.

Time

The calendar and seasons

Even though the vernal equinox precesses, it is always the time at which the Sun crosses from the Southern to the Northern Hemisphere on the celestial sphere, marking the beginning of (Northern) spring.

Thus the right ascension of the Sun — and not its position relative to the background stars — indicates the season.


A sidereal year is the time it takes for the Sun (to appear) to complete a complete circuit of the background stars (~the constellations of the Zodiac) and return to its original position:

1.00 sidereal year = 365.2564 days

The calendar and seasonsJulius Caesar adopted what is now known as the Julian calendar in 46 BCE. The Julian calendar is tied to the seasons so that 21 March should always take place at the beginning of spring (i.e., the vernal equinox), even though the vernal equinox keeps moving with respect to the background stars.

This means that the Julian calendar is based on the tropical year, the time between successive vernal equinoxes.

The calendar and seasonsThe Julian calendar has a year of exactly 365.2500 days — but a tropical year is actually 365.242189 days, shorter than both the Julian and sidereal years!

By 1582, the vernal equinox had slipped back to 11 March. Pope Gregory XIII reset the vernal equinox back to 21 March (thus removing 10 days!) and also removed the leap days from century years not evenly divisible by 400 (e.g., 1700, 1800, 1900, 2100, but not 1600 or 2000).

That is, there were leap years (leap days) in both 1600 and 2000. The offset was made on Thursday, 4 October 1582; the next day, Friday, was 15 October 1582.


This means the Gregorian calendar is now synchronous with the tropical year to within one day in ~3000 years.

Note that only Roman Catholic countries adopted the Gregorian calendar in 1582 — it took until 1752 for England and its colonies (including the future USA) and until the Russian Revolution in 1918 for Russia to do so!

Time: sidereal time

Sidereal time: the local sidereal time is the right ascension of the meridian passing directly overhead

i.e., the right ascension of the zenith right now!

Time: solar time

Solar time is defined by the transit of the Sun through the meridian — that is, the Sun is “directly overhead” (at its highest point in the sky) at noon solar time.

Time: solar timeA solar day clearly has a variable length due to both

the eccentricity of the Earth’s orbit around the Sun — Kepler’s second law, an orbit sweeps out equal areas in equal times, means that the Earth moves faster when it’s nearer to the Sun than farther away

the tilt of the Earth’s rotational axis with respect to the ecliptic (obliquity) — the ecliptic’s projection onto the celestial equator is smaller near the equinoxes than near the solstices

Time: solar timeA solar day clearly has a variable length due to both

the eccentricity of the Earth’s orbit around the Sun — Kepler’s second law, an orbit sweeps out equal areas in equal times, means that the Earth moves faster when it’s nearer to the Sun than farther away

the tilt of the Earth’s rotational axis with respect to the ecliptic (obliquity) — the ecliptic’s projection onto the celestial equator is smaller near the equinoxes than near the solstices

♈

+90º

–90º

Time: solar timeBecause of this variation, astronomers define a mean solar time that averages the day length to a constant 24 hours

The difference between mean solar time and solar time is given by the equation of time, which shows that the length of the solar day varies by ~ ±15 minutes throughout the year

We can use this to calibrate the time given by a sundial...

Time: solar time

The equation of time: red curve

Time: solar timeThe equation of time: analemma of the Sun

Picture taken at the same Mean Solar Time

Time: solar timeBut the mean solar time is still based on the position of the observer, as it’s based on the point at which the “fictitious mean Sun” crosses the observer’s meridian

This means that two observers at slightly different locations will read different times!

The time zone system was created to deal with this problem, with the zeropoint of the system at 0º longitude — the longitude of Greenwich, UK: this is the prime meridian

Time: solar timeBecause the Sun moves along the ecliptic by 360º/365.25 days ≈ 1º/day, the Earth has to rotate 360º+1º every day to keep up with the Sun.

This results in the solar day being ~4 minutes longer than the sidereal day:

1 mean solar day = 24h 00m 00s (solar time) = 86400.0 s (solar time) ≈ 86400 s [SI]

1 sidereal day = 23h 56m 04.09s (mean solar time) = 86164.09 s (mean solar time)

which is valid even as the Earth’s rotation rate varies because the mean solar day lengthens proportionally to the sidereal day.

note also that the sidereal second is shorter than the solar second, by the ratio 0.9973

Time: solar time

Time: solar time

Time: Universal Time (1)Universal time (UT1) is based on the motion of the fixed stars but is adjusted so that it is approximately equal to the mean solar time at Greenwich — such that the “fictitious mean Sun” is on the (prime) meridian at noon and that one day equals precisely 86400 s.

Formally, 0h UT (midnight in Greenwich) on 1 January is defined to occur at Greenwich LST (=GMST) ≈ 6.7h.

1 January is 285 days after 21 March, so it’s 18.7h (285d/365d x 24h) east of the vernal equinox, so α⊙≈18.7h at noon — and midnight is 12h earlier.

Time: physical time

Note that all of these time systems so far have been angular measurements, not true physical times: sidereal, solar, mean solar, and universal time all measure the Earth’s rotation in some way.

Physical times are based on physical measurements: the SI (atomic) second is based on the hyperfine transition of 133Cs:

1.0 SI second = 9 192 631 770 cycles of 133Cs

Time: atomic time

TAI (Temps Atomique Internationale) is the current atomic time, based on the average of ~150 atomic clocks in ~30 countries, and is currently stable to ≈30 µs/century

Note that TAI and UT1 are independent: atomic clocks have shown that the Earth’s rotational period is lengthening by a variable rate of ~1.7 ms/century, as the Earth loses spin angular momentum to the Earth-Moon orbit

Note that this is longer than it seems: integrated, a year of 365.25 days is 3.1 ms longer than it began...

Time: Universal Time (2)

UTC (Universal Coordinated Time) was adopted to link the TAI and UT1 systems. It is based on TAI but includes (positive or negative) leap seconds to keep within 0.9s of UT1, so that in some years 31 December has 86399s, 86400s, or 86401s

Time: physical time

The variations in the length of the day can be large, as big as 3 ms longer than 86400 s [SI]

Time: physical timeTerrestrial time (TT) is a time standard not adjusted for variations in the Earth’s rotation. TAI is a natural choice for this, but TT was adopted before TAI and is based on “ephemeris time” (ET).

Each TT day contains exactly 86400 s [SI].

It is now defined as TT=TAI+32.184 s to match ET. TT effectively keeps track of the leap seconds inserted into UTC

There are also relativistic timescales, TDB and TCB, which try to synchronize times across the Solar System (instead of just the Earth, which TT does)

Gaia uses TCB: barycentric coordinate time

Julian Date

Julian Date (JD) is an extremely useful way of keeping track of observations made over long time periods. Julian Dates are defined as the number of (Julian) days since noon on 1 January 4713 BCE (really!)

Roughly now — 24 April 2019 at 12h 00m 00s UT1 — is JD 2458598.0

The beginning of each Julian day is defined to be at noon in Greenwich, 12h UT1

15 (indiction cycle) × 19 (Metonic cycle) × 28 (Solar cycle) = 7980 yearsindiction cycle: 15-year late-Roman tax cycle, extended into medieval EuropeMetonic cycle: common multiple of the solar year and the lunar cycle — i.e., the time it takes for the lunar cycle to appear at the same solar dateSolar cycle: the time it takes for a solar year (including leap years) to begin on the same week day

Julian DateThe Modified Julian Date (MJD) is often used (because it’s shorter!): MJD=JD–2400000.5

Note that it starts at midnight in Greenwich rather than at noon (in fact, it started precisely at 00h 00m UT on Wednesday 17 November 1858!)

Note also that J2000.0 is defined on the Julian day/year (=365.2500d exactly)/century (36525d) system and began at 12h (TDB) 1 January 2000 exactly, i.e., JD2451545.0 (TDB)

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