Week 2. Lab 1 observa.ons in progress. Problem Set 1 is due Wednesday via Collab. Moon awareness Weather awareness. Prelab done?

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1 Lab 1 observa.ons in progress Prelab done? Week 2 Observa.ons should wrap up this week. Lab + Prelab 2 will be out next week early and observa.ons will follow Lab 1 write-up guidance will be forthcoming Problem Set 1 is due Wednesday via Collab An assignment is there on the assignments page. Problem set 2 will follow closely on Problem Set 1 Moon awareness Weather awareness

2 R.A. and Dec

3 A Review Perspective on Euatorial Coordinates

4 Right Ascension In a sense, R.A. marks the passage of (sidereal) time on the sky. As time passes different (increasing) R.A. coordinates are overhead. If 8 hours R.A. is overhead right now, 9 hours R.A will be overhead in an hour. Since stars rise in the east and set in the west, R.A. must increase toward the east (left as you are facing south in the northern hemisphere) and decrease toward the west. Right Ascension (as well as Longitude) needs an arbitrary zeropoint (Greenwich for Earth longitude, the First Point of Aries on the sky). This celestial reference point is the intersection celestial euator and ecliptic at of the location of the Sun at the Spring Euinox.

5 The Sun and the Celestial Sphere Right Ascension reuires a zeropoint. The Sun s path amongst the stars provides the reference. As the Earth orbits the Sun we see the Sun in different locations against the backdrop of stars. ² The Sun s path amongst the stars (which is the Earth s orbital path as seen from the Sun) is called the Ecliptic. ² The constellations through which the Ecliptic passes are the constellations of the Zodiac. The Sun obscures your birthsign on your birthday.

6 The Sun s apparent path around the sky, inclined to the celes.al euator by the 23.5 degree.lt of the Earth, crosses the celes.al euator at two points the Fall and Spring Euinox. The Spring Euinox marks the zero-point of Right Ascension

7 Right Ascension and the Sun The R.A. of the Sun determines what portion of the sky is accessible, for example at Sunset (R.A. s several hours greater than than of the Sun). The R.A. of the Sun is 0h 0m 0.00s at the moment of the Spring Euinox ( h at the Fall Euinox.) At the Spring Euinox 12 hours R.A. is high in the midnight sky (opposite the Sun). At the Fall Euinox 0 hours is overhead at midnight. At the Spring Euinox 6 hours is on the meridian at Sunset. Each day the position of the Sun advances 3 m 56 s in R.A. (3m 56 seconds is 24 hours divided by days in a year) Consider the Sun on the first day of Spring at 0h 0m 0s R.A. and consider a star at 2h 0m 0s R.A. at the same declination as the Sun That star will set 2 hours after the Sun on the first day of Spring. A day later that star will set 1 hour 56m 4s after the Sun (~4 minutes earlier) because the Sun s R.A. is creeping up on the star s. In a month the star will be hidden behind the sun.

8 Rising and Setting of the Stars The stars rise and set approximately 4 minutes earlier each day, accumulating to 2 hours earlier in the course of a month. A star that transits the meridian at 9 p.m. tonight will do so at 7 p.m. one month from now.

9 Es.mate LST at the current.me Sidereal.me at Noon on the Spring Euinox is 0 hours; Fall Euinox is 12 hours. It was about 12:10 EST when you calculated. The calendar date was Jan 26, so days since the last Euinox. The sidereal clock has run 3m 56 seconds fast per day since then. At 12:10 EST it will be about hours * 3.93 min LST = hours However, L in LST stands for Local Solar.me and Standard.me agree only on meridians spaced 15 degrees from Greenwich. Charloeesville is at longitude 78.5 degrees = 3.5 degrees * 4 minutes west of the EST meridian of 75 degrees so Local Sidereal Time is 14 minutes earlier.

10 Solar vs. Sidereal Time A Sidereal clock tracks star time the clock keeps 24 hour time but completes a 24 hour cycle in 23h 56m 4s A conventional clock can be made into a Sidereal clock by adjusting it to run fast by about 4 minutes a day. What s the use of a Sidereal clock? Ø The time read by a Sidereal clock corresponds to the meridian of Right Ascension that is overhead at that instant. Ø You can set a sidereal clock by observing a star of known R.A. crossing the meridian At Noon on the Spring Euinox a sidereal clock will read approximately 0 hours (why approximately??) Running fast by 3.9 minutes a day, a Sideral clock accumulates a full hour in about two weeks (2 hours a month). One month after the euinox a Sidereal clock will read 2 hours at Noon.

11 Hour Angle It is useful to have a measure of how far a star is from transi.ng the meridian. The Hour Angle denotes the -hh:mm:ss un.l transit or the +hh:mm:ss since transit for a given star. The Hour Angle is simply calculated as the difference between the star s R.A. and the current Sideral.me. Hour Angle (H.A.) = Sidereal Time - Right Ascension This formula gives nega.ve H.A. for R.A. s greater than the current LST (that should make sense to you). A star whose Right Ascension matches the Sidereal Time is on the meridian; H.A. = 0 A star s airmass is a func.on of hour angle, reaching a minimum when H.A. = 0

12 Hour Angle and The Meridian Ø Ø Hour angle is the time until (or time since) a star reaches the Meridian. Hour angle is the difference between the right ascension of a star and the local sidereal time.

13 One Simple Connec.on/Defini.on The current Sidereal Time euals the Hour Angle of the Vernal Euinox It also euals the Right Ascension of the Meridian. Celestial Sphere Review

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15 Accessible Hour Angles vs. Declination Celestial euator (circumpolar) A star on the celestial euator (declination = 0) rises at H.A. = -6.0 hours and sets at +6.0 hours regardless of the latitude of the observer.

16 Accessible Hour Angles vs. Declination Celestial euator (circumpolar) As seen from the Northern Hemisphere. a star well south of the celestial euator may rise at H.A. = -3 hours a star well north of the celestial euator may rise at H.A. = -9 hours

17 Accessible Hour Angles vs. Declination For an observer on the Euator all stars are accessible from H.A. = -6 to 6 regardless of their declination.

18 Accessible Hour Angles vs. Declination Celestial euator (circumpolar) For a Northern Hemisphere observer. Sufficiently far north on the celestial sphere stars never set below the horizon. These circumpolar stars are accessible at all hours angles (-12h to +12h) Conversely there are inaccessible stars below a certain negative declination

19 Accessible Hour Angles Solar Edition Celestial euator (circumpolar) When the Sun lies on the celestial euator (the Euinox) days are 12 hours long since the Sun is visible from H.A. = -6 to +6. The Sun can get as far as +/ degrees from the celestial euator in declination. Accessible hour angles are then heavily latitude dependent

20 Solar Apparent Motion at Different Declination Short days Low solar elevation Long days High solar elevation

21 Precession of the Euinox The loca.on of the crossing points of the eclip.c on the celes.al euator depend on the direc.on of the Earth s rota.on axis. Due to Solar and Lunar.des the Earth s.lted rota.on axis precesses in a circle of radius 23.5 degrees with a period of 26,000 years. The pole star changes substan.ally over.me as a result.

22 Precession of the Euinox The loca.on of the crossing points of the eclip.c on the celes.al euator depend on the direc.on of the Earth s rota.on axis. Due to Solar and Lunar.des the Earth s.lted rota.on axis precesses in a circle of radius 23.5 degrees with a period of 26,000 years. The pole star changes substan.ally over.me as a result.

23 Precession of the Euinox The loca.on of the crossing points of the eclip.c on the celes.al euator depend on the direc.on of the Earth s rota.on axis. Since the intersec.on point at the Spring Euinox defines 0h of Right Ascension and since the pole defines 90 degrees declina.on a star s coordinate shils over.me due to precession substan.ally so over decades or centuries

24 The Precession of the Euinox

25 Precession's Conseuence Stellar celes.al coordinates must be constantly updated to account for precession. Telescope control systems automa.cally precess coordinates so that the telescope correctly points to the of date posi.on of the star given proper input of current date, R.A., Dec, and euinox of the coordinates. Star catalogs must be.ed to a par.cular euinox. Typically the default euinox changes every 50 years as even over this.mescale the coordinate change can become significant. For the star Vega the coordinates are 18:36: :47:01.9 J (J for Julian) 18:35: :44:24.7 B (B for Besselian) small differences, but large compared to many instrument fields-of-view. Now in the computer age (and given the juicy J round number euinox) it is likely that catalog coordinates will s.ck to J for centuries to come.

26 Other Conseuences of Precession Different Stars are circumpolar at different.mes years ago the Big Dipper was circumpolar at our la.tude. Stars that currently never rise above our Southern horizon will be visible. è The Southern Cross will be visible from Charloeesville in 10,000 years.

27 mid-week 2 Observa.ons for Lab 1 should be complete in the next 24 hours Lab 1 write-up guidance is available on the Course Schedule page and will be a con.nuing theme through next week s evening sessions. The due date for Lab 1 will be next Friday. Problem set 2 will be available later today. Lab 2 (and prelab) should be posted by Monday. Out of necessity we are going to re-organize into three observing groups.

28 Coordinates: Euinox vs. Epoch Both euinox and epoch refer to reference.mes. Euinox refers to the posi.on of the vernal euinox in the sky at the specified.me and is thus the term associated with the precession of coordinates. Epoch also refers to a specific.me, but references effects that physically change a star s posi.on on the celes.al sphere over.me rela.ve to other stars such as proper mo/on and parallax. Specific example: The Hipparcos Star Catalog a precision astrometric catalog specifies the euinox posi.on for star in epoch If the star is distant and has no significant/known proper mo.on then the epoch coordinate will be the same as the epoch coordinate for euinox 2000.

29 Precession's Conseuence Stellar celes.al coordinates must be constantly updated to account for precession. Telescope control systems automa.cally precess coordinates so that the telescope correctly points to the of date posi.on of the star given proper input of current date, R.A., Dec, and euinox of the coordinates. Star catalogs must be.ed to a par.cular euinox. Typically the default euinox changes every 50 years as even over this.mescale the coordinate change can become significant. For the star Vega the coordinates are 18:36: :47:01.9 J (J for Julian) 18:35: :44:24.7 B (B for Besselian) small differences, but large compared to many instrument fields-of-view. Now in the computer age (and given the juicy J round number euinox) it is likely that catalog coordinates will s.ck to J for centuries to come.

30 RA/Dec to Alt/Az Conversion The uan..es we know: Hour angle (via R.A. and sidereal.me), Declina.on, and La.tude 2 sides and one interior angle. That s enough to calculate the other uan..es. The Meridian H is hour angle P is the euatorial pole δ is the declination a is altitude Z is the zenith A is azimuth φ is the latitude

31 Law of Sines Recall Triangle Rules for Plane Geometry Law of Cosines

32 Great circles on a sphere intersect at (spherical) angles. The intersection of three great circles defines a spherical triangle (with a sum of interior angles greater than 180 degrees) The side s lengths (a,b,c which are actually angles seen from the center of the sphere) and the intersection angles (A,B,C) are related by the formulae at right which look a whole lot like their planar cousins. Geometry on a Sphere Transforming between Hour Angle, Dec and Altitude/ Azimuth is just a matter of identifying the sides From Chromey Chapter 3

33 Euatorial to Alt/Az Coordinate Conversion The uan..es we know: Hour angle (via R.A. and sidereal.me), Declina.on, and La.tude 2 sides and one interior angle. That s enough to calculate the other uan..es, par.cularly the ones containing al.tude, a, and azimuth, A. The Meridian H is hour angle P is the euatorial pole δ is the declination a is altitude Z is the zenith is the parallactic angle A is azimuth φ is the latitude

34 Euatorial to Alt/Az Coordinate Conversion cosc = cosa *cosb + sin a *sinb*cosc The Meridian H is hour angle P is the euatorial pole δ is the declination a is altitude Z is the zenith A is azimuth φ is the latitude

35 Calcula.ng Airmass Airmass, the number of atmospheric thicknesses light traverses on the way from the star to the observer, depends only on the al.tude of the target, a. The Meridian Note the utility of viewing a set of targets in Xephem/ Stellarium in an Alt/Az view. You immediately see the airmass of all of the targets. H is hour angle P is the euatorial pole δ is the declination a is altitude Z is the zenith is the parallactic angle A is azimuth φ is the latitude

36 Zenith Angle and Airmass The complement of the al.tude angle, the zenith angle, measures the angular separa.on of a star from the point overhead. A star that is just rising or serng has a zenith angle of z=90. A star overhead has a zenith angle of zero. The airmass of a star measures the number of atmospheric thicknesses a star's light is passing through on its way to the observer and euals the secant of the zenith angle for a plane parallel atmosphere. Since the atmosphere aeenuates starlight, knowing the airmass is cri.cal to precision stellar photometry. * horizon z altitude one atmosphere airmass = sec(z) = 1 / cos(z)

37 Airmass Facts and Figures A star at the zenith has airmass = 1 A star 45 degrees from the zenith has airmass = 1.41 = 1/cos(45) A star 30 above the horizon, 60 degrees from zenith, has airmass = 2 = 1/cos(60) In general astronomers try to conduct observa.ons as close to the zenith as possible and at worst at airmass ~ 2. Stars are at their smallest airmass when they transit the meridian. This value depends only on the star s declina.on and the observers la.tude. A star with declina.on eual to your la.tude passes overhead (minimum airmass=1) A star far south on the celes.al sphere (e.g. dec = -30) never gets to low airmass. * horizon z altitude one atmosphere airmass = sec(z) = 1 / cos(z)

38 Plane Parallel Approximates the Spherical Atmosphere Very much not to scale Angle c is the altitude Angle b is the zenith angle, z

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40 However, Plane Parallel is Plenty Good Enough

41 Airmass Curves

42 Which Way is the Earth Turned: Apparent Solar Time The direction on the celestial sphere toward which the Earth is turned at your location (your local up ) has everything to do with what is up in the sky. The Sun is also an important factor as it obscures the stars when it is up. Apparent Solar Time is literally that. Measure the position (to be exact the Hour Angle of the Sun and add 12) of the Sun in the sky to determine Earth's rotation angle relative to the Sun. Each longitude has its own Apparent Solar Time. Note: The Earth rotates and revolves around the Sun counterclockwise when viewed from the North Pole.

43 A Sundial at the South Pole An intui.ve example of an euatorial sundial Eually spaced divisions as the rota.on of the Earth turns the dial uniformly with.me. You can translate this dial anywhere on the Earth s surface as, as long as you don t tip it, and it will work the same way. Said another way, keep the axis of the sundial pointed toward the pole and the shadow plane parallel to the Earth s euator.

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46 Horizontal (Garden Variety!) Sundials Take an euatorial sundial, but.p the readout so that it is flat on the ground. The gnomon remains pointed at the pole. Geometrical projec.on based on la.tude defines the paeern. Hour divisions are no longer eual.

47 Apparent/Local Solar Time vs. Mean Solar Time Local Solar Time is sundial.me. Noon = the.me when the Sun crosses the Meridian Because of the Earth s ellip.cal orbit (more exactly because of the varying angular speed as the Earth moves around its ellip.cal orbit (Kepler s 2 nd Law in ac.on) the.me of Solar Noon drils throughout the year. The obliuity of the Earth is also a factor in this dril. Mean Solar Time averages the day length throughout the year. There are four dates during the year where mean = apparent solar.me Apparent - Mean The Euation of Time The offset between mean and apparent solar time

48 The Analemma

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