Week 2. Problem Set 1 is due Thursday via Collab. Moon awareness Weather awareness

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1 Week 2 Lab 1 observa.ons start real soon (in progress?) Prelab done? Observa.ons should wrap up this week. Lab 2 + Prelab 2 will be out next week early and observa.ons will follow Lab 1 write-up guidance is available on the course schedule page Evening sessions NEXT week will focus on wri.ng up (and formalng) results) NO evening sessions this week as your evening.me is devoted to Lab Problem Set 1 is due Thursday via Collab An assignment slot is there on the assignments page. Submit your assignment in PDF format! Problem set 2 will follow closely on Problem Set 1 Moon awareness Weather awareness APOD

2 Where were we: R.A. and Dec

3 Equatorial Coordinates

4 Right Ascension q In a sense, R.A. marks the passage of (sidereal) time on the sky. q As time passes different (increasing) R.A. coordinates are overhead. q If 8 hours R.A. is overhead right now, 9 hours R.A will be overhead in an hour (of sidereal time, to be exact ). q 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. q Right Ascension (as well as Longitude) needs an arbitrary zeropoint (Greenwich for Earth longitude, the First Point of Aries on the sky). q This celestial reference point is the intersection celestial equator and ecliptic at of the location of the Sun at the Spring Equinox.

5 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.

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

7 Reminder: 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

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

10 Accessible Hour Angles vs. Declination Celestial equator (circumpolar) As seen from the Northern Hemisphere. a star well south of the celestial equator may rise at H.A. = -3 hours (above horizon for only 6 hours) a star well north of the celestial equator may rise at H.A. = -9 hours (above horizon for 18 hours)

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

12 Accessible Hour Angles vs. Declination Celestial equator (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

13 Accessible Hour Angles Solar Edition Celestial equator (circumpolar) When the Sun lies on the celestial equator (the Equinox) 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 equator in declination. Accessible hour angles are then heavily latitude dependent

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

15 Precession of the Equinox q The loca.on of the crossing points of the eclip.c on the celes.al equator depend on the direc.on of the Earth s rota.on axis. q 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. q The pole star changes substan.ally over.me as a result. q So does the loca.on of the Vernal Equinox on the eclip.c.

16 Precession of the Equinox q The loca.on of the crossing points of the eclip.c on the celes.al equator depend on the direc.on of the Earth s rota.on axis. q 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. q The pole star changes substan.ally over.me as a result. q So does the loca.on of the Vernal Equinox on the eclip.c.

17 Precession of the Equinox q The loca.on of the crossing points of the eclip.c on the celes.al equator depend on the direc.on of the Earth s rota.on axis. q The shie can be substan.al (from the point of view of a telescope with a limited field of view) even over a decade. q Since it takes 26,000 years for the pole to complete a precession cycle the loca.on of the vernal equinox moves (westward) by about an hour of right ascension every 1000 years.

18 The Precession of the Equinox

19 Precession's Consequence Stellar celes.al coordinates must be constantly updated to account for precession. Telescopes are.ed to the Earth and point rela.ve to the Earth s pole and equator. 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 equinox of the coordinates. Star catalogs must be.ed to a par.cular equinox. Historically the default equinox 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 equinox) it is likely that catalog coordinates will s.ck to J for centuries to come.

20 Other Consequences 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 Charlooesville in 10,000 years.

21 Mid-Week 2 Update Observa.ons for Lab 1 should be complete? By the end of the week? It will be clear Friday night 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 a week from Monday. Problem set 2 will be available Friday. Lab 2 (and prelab) should be posted by Monday evening.

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24 Es.mate LST at the current.me The Sun lies at the Vernal Equinox (0h RA) on March 20. The Sun lies on the Meridian at Noon by defini.on, so the sidereal.me at local solar noon on the Spring Equinox is 0:00. You were asked to calculate the hour angle of Vega (RA = 18h 36m) at 11:30 EST. On the Spring equinox the sidereal.me at that.me would be 23:30. However, that would be true at the center of the.me zone, which is 75 degrees longitude. Charlooesville is 78 degrees longitude, or 3 degrees west of the center of the.mezone. 15 degrees of longitude corresponds to one hour (360/24 hours = 15 degrees/ hour), so 3 degrees is 1/5 of an hour or 12 minutes. West means earlier so the true local sidereal.me on the equinox would be 23:18. The calendar date was Jan 30, so 49 days before the spring equinox. The sidereal clock runs 3m 56 seconds fast per day. On Jan 30 a sidereal clock will be 49*3.93 = 3 hr and 12 min behind the spring equinox clock 23:18 3:12 = 20:06 LST at 11:30 a.m. on Feb 29 for longitude 78 degrees. HA = Sidereal.me RA so 20:06 18:36 = +01:30

25 Coordinates: Equinox vs. Epoch Both equinox and epoch refer to reference.mes. Equinox refers to the posi.on of the vernal equinox 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 mo2on and parallax. Specific example: The Hipparcos Star Catalog a precision astrometric catalog specifies the equinox posi.on for stars 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 equinox 2000.

26 Conver.ng Between Spherical Coordinate Systems Precession is a prime example of the need to transform between spherical coordinate systems with different poles.

27 Conver.ng Between RA/Dec and Alt/Azimuth The quan..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 quan..es. The Meridian H is hour angle P is the equatorial pole δ is the declination a is altitude Z is the zenith A is azimuth φ is the latitude

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

29 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

30 Equatorial to Alt/Az Coordinate Conversion The quan..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 quan..es, par.cularly the ones containing al.tude, a, and azimuth, A. The Meridian H is hour angle P is the equatorial pole δ is the declination a is altitude Z is the zenith q is the parallactic angle A is azimuth φ is the latitude

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

32 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 equatorial pole δ is the declination a is altitude Z is the zenith q is the parallactic angle A is azimuth φ is the latitude

33 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 selng 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 equals the secant of the zenith angle for a plane parallel atmosphere. Since the atmosphere aoenuates starlight, knowing the airmass is cri.cal to precision stellar photometry. * horizon z altitude one atmosphere airmass = sec(z) = 1 / cos(z)

34 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 equal 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)

35 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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37 However, Plane Parallel is Plenty Good Enough