Observational Astronomy 2017 Part 2 Prof. S.C. Trager

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Positional Astronomy Observational Astronomy 2017 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. The Celestial Sphere

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

Instead, 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 coordinates

Nearly 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! Equatorial coordinates +90º

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

–90º Equatorial coordinates +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 Equatorial coordinates +90º The 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 Equatorial coordinates +90º The 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. –90º Equatorial coordinates +90º The 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 Equatorial coordinates +90º

Because 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 Equatorial coordinates +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 The local equatorial system

The 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 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 system

In 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! Equatorial coordinates

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 coordinates

It 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) Galactic coordinates

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 Galactic coordinates

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

The system is precisely deﬁned by the direction of the North Galactic Pole (NGP): h m ↵NGP(B1950) = 192.25 = 12 49

NGP(B1950) = +27.4 = +27240 and by the Galactic longitude of the North Celestial Pole: lNCP = 123 Galactic coordinates

The ﬁrst set of coordinates α=12h49m NCP l=123º implies that the celestial δ=27.4º and galactic equators are tilted by 90º–27.4º=62.6º

line of nodes These two great circles cross at two nodes, and the line of nodes that l=33.0º connect them is the axis α=18h49m that transforms one plane to the other Galactic coordinates

α=12h49m NCP l=123º The equators cross at δ=27.4º

lnode = 123 90 = 33 B1950 h m h h m ↵node = 12 49 +6 = 18 49 line of nodes for the ascending node the extra 90º angles in l and α shift from the l=33.0º NCP and the galactic α=18h49m equator to the nodes Galactic coordinates

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

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

To transform from l,b to α,δ use

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

Ecliptic 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. Motions of stars, apparent and real

Let’s use the horizon A zenith B

coordinate system and NCP see how stars move on δ polar axis θ the sky during a night. rise B Consider a northern set hemisphere site like celestial equator 53º N S Groningen (θlat≈+53º) W looking east Motions of stars, apparent and real Stars with

✓lat 90 < < 90 ✓lat A zenith B

rise from the east and set in NCP the west, just like the Sun δ polar axisrise θB Stars with

set > 90 ✓lat celestial equator 53º N S (like star A) are always W looking east above the horizon and just circle around the NCP Motions of stars, apparent and real Stars with

0 < < 90 ✓lat, A zenith B

like star B, rise north of NCP east, moves towards the δ south until they transit the polar axisrise θB meridian (directly south of

set the zenith), then move celestial equator westwards and set north of 53º N S west. They cross the W looking east meridian at a maximum altitude ✓ = 90 ✓ + B lat Motions of stars, apparent and real Stars with

0 < < 90 ✓lat, A zenith B

like star B, rise north of NCP east, moves towards the δ south until they transit the polar axisrise θB meridian (directly south of

set the zenith), then move celestial equator westwards and set north of 53º N S west. They cross the W looking east meridian at a maximum altitude ✓ = 90 ✓ + B lat Parallax

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

Consider 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. Then if r (=1 AU) is the radius of the Earth’s orbit, we r ﬁnd = tan rad d because ϖ is clearly small; then converting to seconds of arc, = 206265 rad 206265 Deﬁning 1 AU such that d = AU and 1 parsec as the distance at which a star would have a parallax of 1″: 1 pc = 206265 AU = 3.086 1013 km = 3.26 light years The distance to a star with observed parallax ϖ″ is 1 then d = pc 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 Aberration of starlight

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

Consider... a stationary telescope

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

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

ecliptic Aberration

The maximum effect — for objects perpendicular to the Earth’s orbit (the ecliptic) — is 6 1 vEarth 2.979 10 cm s 4 ✓ = ⇥ =0.994 10 rad = 20.500 aber,max ⇡ c 2.998 1010 cm s 1 ⇥ ⇥ 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

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″) 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

Ecliptic pole

NCP in 12885 years NCP today

Because the Moon’s orbit

is almost in the plane of ﬁxed stars ﬁxed stars the ecliptic, and because 23.45º it is so close, it combines with the Sun (and Jupiter, Earth also in the ecliptic plane) track of Sun (ecliptic plane) celestial equator today to exert a torque that

results in a 25770-year celestial equator in 12885 years precessional period ﬁxed stars ﬁxed stars Precession

Ecliptic pole

NCP in 12885 years NCP today

ﬁxed stars ﬁxed stars This causes the equatorial 23.45º coordinate system to move (precess) with Earth respect to the track of Sun (ecliptic plane) celestial equator today background stars

celestial equator in 12885 years

ﬁxed stars ﬁxed stars Precession

Ecliptic pole Note that the Earth’s orbit NCP in 12885 years NCP today is fixed in space, and therefore so is the ecliptic fixed stars fixed stars 23.45º As Earth precesses away from its current position, the celestial equator starts Earth track of Sun (ecliptic plane) to slip to the west, and so celestial equator today the intersection of the celestial equator and the celestial equator in 12885 years ecliptic — the vernal fixed stars fixed stars equinox — does as well

Precession

Ecliptic pole

NCP in 12885 years NCP today Since the vernal equinox defines the zeropoint of right ascension, (fixed) stars will have fixed stars fixed stars increasing right ascension as 23.45º the Earth precesses (and their declinations will also change) Earth

This is the precession of the track of Sun (ecliptic plane) equinoxes and is why we must celestial equator today always specify a date — an epoch — when giving the celestial equator in 12885 years equatorial coordinates of an ﬁxed stars ﬁxed stars object Precession

Ecliptic pole

NCP in 12885 years NCP today

ﬁxed stars ﬁxed stars The rate of this precession 23.45º of the equinoxes is

Earth 360 360000/ / 25770 yr = 50.300/yr ⇥ track of Sun (ecliptic plane) celestial equator today or 42′ in 50 years

celestial equator in 12885 years

ﬁxed stars ﬁxed stars 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 motions

Refraction: 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

GIF 178

mean [ α, δ ], FK4, mean [ α, δ ], FK4, mean [ α, δ ], FK5, any equinox no µ,anyequinox 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 µ parallax parallax

FK5, J2000, current epoch, geocentric

light deﬂection annual aberration precession-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 Coordinates Star positions are published or catalogued using one of the mean [ α, δ ]systemsshownatthetop. 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 International Celestial Reference Frame (in the optical, the Hipparcos catalogue). The ﬁgure relates a star’s mean [ α, δ ]totheactualline-of-sighttothestar.Notethatfortheconventionalchoicesof equinox, 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. The calendar and seasons

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 seasons

Julius 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 seasons

The 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). The calendar and seasons

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 time

A 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 time +90º A 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 Time: solar time

Because 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 time

The equation of time: analemma of the Sun

Picture taken at the same Mean Solar Time Time: solar time

But 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 time

Because 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. 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/ d h h 365 x 24 ) east of the vernal equinox, so α⊙≈18.7 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 time

Terrestrial 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) 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 — 26 April 2017 at 10h 00m 00s UT1 — is JD 2457869.916667 The beginning of each Julian day is deﬁned to be at noon in Greenwich, 12h UT1 Julian Date

The 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)