PDF Slides - UMBC

Meeting #4 

PHYS 304 

Meeting 04

[Image Credit: Bodmas.org] 

NCP 

Zenith 

Hour Angle h 

…from the definitions 

An object’s hour angle 

(& hence transit or culmination) 

can be calculated from 

RA (α) if LST (Θ) known: 

ϒ 

RA α 

h = Θ − α 

…or of course the LST (Θ) can be calculated trivially 

if the RA (α) of objects passing through an 

observer’s N-S meridian is known at a given instance… 

Ian M. George PHYS 304 (2011 Fall) Meeting 04

[Image Credits: Stellarium] 

… right now 

(~11:00 EDT, Sept 9 th ) 

@UMBC 

(lat ϕ~40 O ) 

objects with 

α~9h 00m 

are culminating (looking South) 

ie. Θ~9h 00m 

… so γ (α=0 O , δ=0 O ) passed through 

our N-S meridian ~2am (local) 

this morning 

with a (max) altitude a max ~50 O 

… by coincidence 

Uranus is currently at (α~0 O , δ=0 O ) 

Ian M. George 

PHYS 304 (2011 Fall)

[Image Credits: Stellarium] 

… from current LST (Θ~9h 00m) 

… know γ (α=0 O , δ=0 O ) passes through 

our N-S meridian ~2am (local) 

around this time of year 

… thus can easily determine Altair 

(α~20h, δ=+8 O ) 

ie. with an RA 

~ 20 hours “after” γ 

or ~ 4 hours “before” γ 

culminates at ~10pm (local) 


…relationship between systems can be expressed as.. 

Hour Angle 

(in degrees etc) 

cosh = − tanδ tanφ + 

declination 

sin a 

cosδ cosφ 

altitude of object 

geogrpahic 

latitude of 

observer 

…you will be given this expression if you need it! 

…objects “rise” and “set” when a = 0 O of course 

…hence when 

cosh = − tanδ tanφ 

…so given δ and ϕ can calculate h for rise and set 

…and given α can use Θ = h + α 

…to calculate the sidereal time 

of the rise or setting 


… Altair (α~20h, δ=+8 O ) from UMBC (lat ϕ~40 O ) today/tonight 

Source will rise or set when cosh = − tanδ tanφ ~ - 0.12 

( ) 

h = cos −1 − 0.12 

so Sidereal time of rise and set given by 

ie Θ r/s ~ 26.5hr or ~13.5 hr 

ie. 

but we also know that currently (~11:30 EDT) 

Θ ~ 9.5hr 

~ +/- 97.4 O 

which is equivalent to ~ +/- 6.5 hr 

Θ = h + α 

..hence Altair will cross local horizon 

~4.0 hr and ~17 hr in the future 

ie at ~3:30pm (EDT) this afternoon [when actually still light of course] 

& ~4:30am (EDT) tomorrow morning 

Also Note: ~10pm culmination [found previously] 

half way between these times (of course!) 


… again its usually best to choose a coordinate system 

that best suits ones specific needs 

… hence there are other coordinate systems which are 

sometimes more appropriate 

… a moderately common examples are 

The Ecliptic Coordinate System 

and the 

The Galactic Coordinate System 

both are again spherical systems 

just with different Origins and 

different “special directions” 


Origin: Sun 

“Special Direction #1” – Ecliptic Plane 

“Special Direction #2” – ϒ 

Spatial coordinates: 

Ecliptic Latitude (β) angle “up” from the ecliptic 

plane 

Ecliptic Longitude (λ) angle around the ecliptic plane 

from ϒ (counterclockwise) 

… useful for objects in the Solar system 


Origin: Sun 

“Special Direction #1” – Galactic Plane 

“Special Direction #2” – direction of the Galactic Center 

Spatial coordinates: 

Galactic Latitude (b) angle “up” from the galactic 

plane 

Galactic Longitude (l) angle around the galactic plane 

from G.C (counterclockwise) 

… useful for objects in (duh!) the Galaxy 


… so say you have.. 

Target position in 

Coord System #1 

(eg (α,δ) in Eq.Sys @ defined time) 

and 

Your Position 

(eg long, lat & local time on Earth) 

Coordinate 

Transformations 

Any 

Necessary 

Corrections 

Target position in 

Coord System #2 

(eg (alt,az) in Horiz.Sys ) 

at desired time 


… which corrections are really necessary depends upon 

the accuracy one requires 

(often all the corrections certainly are not always required since other 

effects can dominate the accuracy that can be achieved) 

1) Shape of the Earth [text Sect.2.2] 

Earth is not a perfect sphere 

radius towards equator ~ 21.4 km larger than that to poles 

thus corrections to account for observer’s pos n req d 

2) Parallax & Aberration due to Orbit & Spin of the Earth 

[text Sect.2.9,10] 

Nearby objects will appear to move against more 

distant objects due to annual & diurnal parallax 

Special relativity: aberration due to Earth’s motion (

… and this is a biggie … 

3) Precession of the Earth’s spin axis [text Sect.2.2] 

Since Earth’s shape is an oblate spheroid 

and it’s spin axis is tilted to the ecliptic 

(and also at ~5 O to the orbital plane of the moon) 

there are gravitational torques on the Earth 

due to the Sun & Moon (& the other planets) 

…eg. Sun trying to pull equatorial bulge into plane of the ecliptic 

… result is the direction of the spin axis moves 

around on a circular path every ~26,000 years 

“precession” (~50”/yr clockwise) 


Tilt angle stays (approx) constant 

as a result …. 

the North Celestial Pole 

moves in a circle 

(of “radius” ~23.45 O ) 

“around” the sky 

obviously 

the Celestial Equator 

also moves 

[Image Credit: Pearson Education] 


Celestial North “moves” against the sky with time 

Celestial Equator “moves” against the sky with time 

… so the location the Vernal Equinox 

“moves” against the sky with time 

But these directions form the basis of our whole 

Equatorial Coordinate System (α,δ) !!! 

Solution: 

Quote (α,δ) at a precisely defined time 

- and make corrections for the date & time 

of the desired observation 

[see text pp38-40 for precession matrix calcs] 


For the last few years… 

The precisely defined time is 

noon (UTC) on 2000 Jan 01 

(ie. JD 2,451,545.0 – more of this later) 

Thus one often sees “(J2000)” after equatorial coords 

(and some other time-dependent quantities) 

Example: 

the (α,δ) of the Galactic Center 

are α = 266.405100 O , δ = -28.936175 O (J2000) 

… which as noted already is more often written as 

17h 45m 37.224s, -28 O 56’ 10.23” (J2000) 

Right Ascension 

in time units 

Declination 

in angular units 


… this can be quite a biggie too for some stars … 

4) Proper Motion of Objects [text Sect.2.2] 

No object in the Universe is “stationary” 

(& “stationary” can’t be defined on its own anyhow of course) 

… thus even after correcting for the spin & orbit of the Earth 

all objects are moving with respect to our chosen 

coordinate system 

(unless it was the basis of our coordinate system of course) 

Proper Motion is the term used to describe the 

“velocity” of objects “on the plane of the sky” 

it is almost always quoted in arcsec/yr (few “/yr for nearest starts) 

obviously given the direction of motion 

is trivial to correct (α,δ) (J2000) for observation date 


… for ground-based observations … 

5) Refraction caused by the Atmosphere [text Sect.2.9] 

… can approximate atmosphere as a set of layers - see Fig 2.20 

each with different index of refraction n i 

n 2 

sin z 2 

= n 1 

sin z 1 

…etc 

but n i depends on density (temperature, pressure) etc 

so current conditions: difficult to predict exactly… 

… simple parameterizations often used 

to get initial, best-guess estimates 


[Image Credits: Mike-Willis.com, aoptics.co.uk, asterism.com] 

… for “typical” conditions, towards 

the horizon effect leads to ~34’ 

increase in apparent altitude 

(so Sun has already set 

when appears to set) 


A final problem for ground-based observations 

(again esp in the optical band) 

is again due to the atmosphere. 

The atmosphere can be thought of consisting of many 

packets or bubbles of air in motion. 

… many of these will be along the line of sight to a target, 

with slightly different optical characteristics & motions 

… this results in the image of the target appearing to 

move slightly & vary in brightness on short timescales 

(…why stars “twinkle”) 

In “good seeing” the effect is < 1arcsec 

but in “bad seeing” conditions the effect can be many 10s of arcsec 


[Image Credit: Michael Richmond] 

Turbulent “bubbles” of air 

@ different heights in the 

Atmosphere 

… each with slightly different 

optical characteristics 

& motion 

apparent 

position 

time 

Can result in the apparent position of 

the target moving on detector (in 2-D) 

on short timescales 


… even spin axis wobbles (v.slightly) on short timescales 

[Image Credits: http://hpiers.obspm.fr/] 


Obviously we all know the units: 

seconds, minutes, hours, months, years etc 

And that 1 second is a fundamental SI units 

defined to be 9,192,631,770 periods of the radiation emitted by transitions 

between the two hyperfine levels of the ground state of a 55 Cs 133 atom 

… and that 1 minute is 60 seconds, 1 hour is 60 minutes… 

… but what exactly do we mean by a “day”, 

a “month” 

& a “year” ?!? 

- remember, due to the corrections mentioned previously, 

we need to make sure we precisely define these… 


there are 

24 hours in a “Day”, 

(for historical reasons) 

between 28 & 31 Days in a Month 

and 

12 months in a Year 

which turns out to be 

365 days in a Year, and 366 days in a Leap Year 

which occurs every 4 years, 

when the year is exactly divisible by 4, 

except for the “0 th year” of each century, 

unless that the “0 th year” is itself exactly divisible by 4 

… why so “messy”..?!! 


Additional Slide [not shown in class] 

While we have a fundamental unit of time for science 

- the second which is now well defined [& accepted] 

via an atomic transition (hyperfine levels of grd state of 55 Cs 133 ) 

[& have atomic clocks accurate to 1 part in 10 14 per day] 

.. this unit is not that practical for “everyday” use 

- desire & practical usefulness to divide “time” 

into ‘bite-sized’ units (hours, days, years etc…) 

BUT 

also want these ‘bite-sized’ units to be related to astronomy! 

… for practical reasons we want the units we’ve named 

“days” & “years” to be related to the Sun. 

[this is very convenient in everyday life!] 

..and we want to have these ‘bite-sized’ units 

to be integer multiples of each other [60s in an “hour” etc] 

[this is also very convenient in everyday life!] 



..obviously Physics neither ‘knows’ or ‘cares’ about human convenience! 

… eg. there is no reason why the Earth 

has to spin an integer (eg exactly 365 “days”) number of times 

every complete orbit of the Sun (ie 1 “year”) 

… this and other effects make keeping 

practical “wall-clock time” (UTC) 

in agreement with ‘celestial time’ a challenge 

… and numerous time systems can be defined 

all of which are based upon practical need 

Dennis Hopper: Apocalypse Now: 

You can't travel in space, you can't go out into space, you 

know, without, like, you know, uh, with fractions, okay? 

…Disagree – there are a lot of fractions in space 


Sidereal Time is based upon the rotation of the Earth 

1 sidereal day is defined to be the time taken for a non-solar system 

object to return to exactly the same azimuth 

(in the horizontal Coord system) 

… hence LST discussed previously.. 

Solar Time is based upon the rotation of the Earth 

and its orbit around the Sun 

1 solar (or synodic) day is defined to be the time taken for the 

Sun to return to exactly the same az (in horizontal Coord sys) 

A Solar Day is 24 hrs (on average!) 

A Sidereal Day is 23 hrs 56 mins 4.09 seconds (on average!) 


Solar Day is longer 

than Sidereal Day 

since Earth has to 

rotate an extra ~1 O 

Distant 

star 

Sun 

Earth 

1 Sidereal year 

is 365.2564 d 




View from “above” 

the ecliptic 

Sun 

Earth 

Solar Day is longer 

than Sidereal Day 

since Earth has to 

rotate an extra ~1 O 

obviously! 

360 O orbit of Earth 

takes ~365 days 

hence Earth moves 

~1 O per day 

around orbit 

.. But Note: 

fact that Earth’s spin is in same sense 

as Earth’s orbital motion 

(ie counterclockwise when view from “above”) 

24 × 60 

also that ~ 4 min 

365 


ut Solar Time is actually a bit tricky … 

- Earth’s orbit is an ellipse 

so speed is not constant throughout orbit 

- Sun moves on ecliptic, not celestial equator 

so dα/dt ≠ constant 

Solution: introduce an hypothetical “Mean Sun” 

which moves around the around the celestial 

equator at a constant angular velocity 

…hence 24x60x60 = 86,400 s 

in a Mean Solar Day throughout the year 

[even this actually changes slowly with time!] 



…Sundial will show “real” Local Solar Time 

….Correction to Mean Solar Time given by the “equation of time”… 

[Image Credit: HartRAO] 

….number of minutes 

to be subtracted from 

Local Solar Time 

to obtain 

Mean Solar Time 

Note 

Tilt: 6 month period 

Orbit: 12 month period 



is actually not that easy in detail …. 

- Earth’s rotation rate is slowly decreasing 

- Precession of the spin axis 

- plus additional wobbling due to “Nutation” 

(caused by precession of Moon’s orbit around the ecliptic every 18.6 yrs) 

- and other effects 

Hence even Sidereal Time is not constant with “time” 

due to all these effects 

but corrections can be calculated & applied 

Mean Sidereal Time (“UT0” & variations) 

often used 

eg. “Greenwich Mean Sidereal Time” 

or “Universal Time” 



Universal Time (UT0) is based on Mean Sidereal Time 

as specified at Greenwich (GMST) 

Terrestrial Time (TT) is based upon atomic clocks, and 

corrects for relativistic effects due 

to motion of the Earth. 

Julian Date (JD) is number of days from noon UT 

starting from noon Jan 1 st , 4713BCE (!) 

(this class held on JD ~2455814.152778) 

Modified Julian Date (MJD), for convenience, is simply 

= JD – 2400000.5 (begins at midnight) 

(so class held on MJD ~55813.652778) 



a Julian Year is 365.25 days (SI abbreviation ‘a’) 

a Tropical Year 

is the time taken for the Sun to move from 

one Vernal Equinox to the next 

ie. RA of Sun increases by 24.00000… hr 

365.2422 days long (J2000 each with 86400s) 

…used as the basis of civil calendar corrections in the west 

…but tropical day actually getting longer 1.5 ms /century 

a Sidereal Year 

is the time taken for the Sun to make 

one revolution wrt background stars 

365.2564 days long (J2000 each with 86400s) 

(longer due to precession – also varies) 



Coordinated Universal Time (UTC) is related to 

atomic-clock time (TAI) and is ‘corrected’ 

to take into account all effects 

Leap seconds 

added as required 

to keep difference 

between 2 systems 

< ± 0.9s 

Usual ‘practical’ time system used 

(on wall clocks etc) 



[Image Credits: http://hpiers.obspm.fr/] 



… message to the 

worldwide 

“keepers of time”…. 





vote in 2012 Jan! 



… “Months” based on the Moon’s orbit around the Earth 

But just as for Earth- Sun system … 

… have to distinguish 

Synodic Month (from new moon to next new moon) 

29.53059 d 

from 

Sidereal Month (from fixed location on sky, back to same location) 

27.32158 d 

Earth’s rotation, orbit & Moon’s orbit 

also why high tides are 24hr 50min apart 

(not 24hrs exactly) 



..in everyday life… 

..hence 12:00 “Zulu” 



.. Not adopted by all states (or countries) 

.. Not adopted on same date(s) by all countries(!) 

.. “daylight saving time” 

eg. 

EST (UTC-5 hrs) 

vs EDT (UTC-4 hrs) 

etc 

.. USA: EDT starts second Sunday in March 

EST starts first Sunday in November 

Europe: Daylight saving Time 

stats last Sunday in March 

ends last Sunday in October etc etc 



Choice of best Spatial & Temporal Coords to use 

depends upon application & desired accuracy 

Know the definitions of the (alt, az) and (α,δ) 

spherical coord systems 

Know the definitions of (and differences between) 

the Mean Sidereal & Mean Solar Days/Years 

Be able to explain (in a sentence or two) the various 

effects which lead to corrections having to be 

made to these coordinate systems for very 

high-accuracy work. 



Astronomical objects are named in variety of ways: 

Proper Nouns: eg. the planets, many bright stars 

(from mythology or historical translations) 

Constellation appreciations: eg. αCyg, βCyg, γCyg, δCyg, … etc 

Discoverer(s): eg. Halley’s comet, Comet Hale-Bopp 

Catalog Entry: eg. NGC 4151, M 31, UGC 454 … etc, etc etc ! 

Celestial Coords: eg. J00424433+4116074 

Date-related Names: eg. GRB 090410, 

SN 1987a 

RA & dec 

in J2000 coords 

or simply the Year 

UTC 

YYMMDD

PDF Slides - UMBC

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?