1. Coordinate Systems: Alt, El, Az, Zenith Angle ElevationorAltitude Zenith Angle

2. Coordinate Systems: Alt, El, Az, Zenith Angle Azimuth

3. Celestial Sphere

4. Coordinates on Celestial Sphere

5. One hemisphere… view from inside

6. Map from inside of Cel. Sphere, looking out

7. Map from inside of Celestial Sphere of radio sky

8. Zero of R.A. coordinate system tied to intersection of Ecliptic Plane with Equatorial Plane The Vernal Equinox

9. Moon’s orbit inclined ~5 deg w.r.t. plane of ecliptic:

10. after ~4.65 (=18.6/4) years, ….

11. after ~9.3 years,…

12. after 18.6 years… Moon orbit has regressed full turn…

13. N.P. Vernal Equinox RA=0 observer’s location Local Siderial Time = LST 6 hours 12 hours 18 hours

14. RA Dec

15. N.P. Vernal Equinox observer’s location LST 6 hours 12 hours 18 hours RA HA HA = Hour Angle

16. N.P. Vernal Equinox observer’s location LST 6 hours 12 hours 18 hours RA HA HA = Hour Angle HA = LST - RA LST = HA

17. N.P. Vernal Equinox RA=0 LST 6 hours 12 hours 18 hours Sun Solar Time = hours since midnight

18. N.P. Vernal Equinox RA=0 GST 6 hours 12 hours 18 hours Sun Greenwich Time …. becomes UTC = Coordinated Universal Time Greenwich

19. N.P. Vernal Equinox RA=0 observer’s location Local Siderial Time = LST 6 hours 12 hours 18 hours

20. One hemisphere: future North Poles

21. Zero of R.A. coordinate system tied to intersection of Ecliptic Plane with Equatorial Plane The Vernal Equinox

22. (at equator, 4 minutes of time = 1 degrees) motion of RA=0, Decl=0 (J2000) in past and future Epochs… Current epoch called `epoch of date’

23. Spherical Trigonometry & Spherical triangles sides are portions of great circles

24. PCD is a spherical triangle PAB is NOT a spherical triangle

25. A c

26. Spherical triangles sin A sin B sin C sin a sin b sin c == cos c = cos a cos b + sin a sin b cos C

28. Radio Astronomy Overview Radio `source’ Goals of telescope: maximize collection of energy (sensitivity or gain) isolate source emission from other sources… (directional gain… dynamic range) Collecting area

29. LOFAR “elements”

30. thought experiment… 2 wires out (antennas are “reciprocal” devices… can receive or broadcast)

31. Black Body oven at temperature = T thought experiment …

32. Black Body oven at temperature = T thought experiment … R

33. Black Body oven at temperature = T thought experiment … R … wait a while… reach equilibrium… at T warm resistor delivers power P = kT B (B = frequency bandwidth; k = Boltzmann Const)

34. R real definition… warm resistor produces P = kT B = P a = kT a B temp = T Measure Antenna output Power as “T a ” = antenna temperature TaTa

35. Radio `source’ Collecting area  Reception Pattern or Power Pattern

36. Radio `source’ Collecting area If source with brightness temperature T b fills the beam (reception pattern), then T a = T b (!! No dependence on telescope if emission fills beam !!)

37. receiver “temperature” … quantify Receiver internal noise Power as “T r ” = “receiver temperature” TaTa Ampl, etc T r +T a Ampl, etc Real electronics adds noise …treat as ideal, noise-free amp with added power from warm R

38. “ system temperature” … quantify total receiver System noise power as “T sys ” T sys +T a Ampl, etc RMS fluctuations =  T  T = (fac) T sys /(B t int ) 1/2 Fac ~ 1 – 2 B = Bandwidth, Hz t int = integration time, seconds

39. Radio `point source’ Collecting area Power collected = S A eff B/2 S = flux density (watts/sq-m/Hz) [ 1 Jansky = 1 Jy = 10 -26 w/sq-m/Hz ] A eff = effective area (sq-m) B = frequency bandwith (Hz) T a = S A eff /2k