Sidereal Rate as a Function of (Az,El,Latitude)

Transcription

1 Sidereal Rate as a Function of (Az,El,Latitude) Jeff Mangum (NRAO) May 23, 20 Contents Introduction 2 Spherical Trigonometry 3 Derivation of (, ) 2 3. Derivation of Derivation of Summary of the (, ) Calculation Introduction While tracking a sidereal position (i.e. a position in (RA,Dec)) while monitoring (A,E) coordinates it is often useful to know what the sidereal rate is toward these (A,E) positions for a given observing site. In the following we derive the sidereal rate (, ) given only (A,E) and the observatory latitude (φ). For a very nice explanation of this an other coordinate frame conversion issues see Pointing the CBI by Martin Shepherd at mcs/cbi/pointing/. 2 Spherical Trigonometry In the following I list the spherical trigonometric relations, rules, and identities used in this analysis. First some definitions: α : Geocentric apparent right ascension δ : Geocentric apparent declination φ : Geodetic latitude H : Hour angle A : Azimuth

2 E : Elevation P : Parallactic angle τ : Local sidereal time (LST) t : Universal time (UT) From the sine formulae: and the cosine formulae: sin H cos E sin A cos δ sin P cos φ () sin δ sin φ sin E + cos φ cos E cos A (2) sin φ sin E sin δ + cos E cos δ cos P (3) sin E sin δ sin φ + cos δ cos φ cos H (4) and some analogues of the cosine formulae: 3 Derivation of (, ) cos δ cos P sin φ cos E cos φ sin E cos A (5) cos φ cos H sin E cos δ cos E sin δ cos P (6) cos E cos A sin δ cos φ cos δ sin φ cos H (7) cos δ cos H sin E cos φ cos E sin φ cos A (8) cos φ cos A sin δ cos E cos δ sin E cos P (9) cos E cos P sin φ cos δ cos φ sin δ cos H (0) From Equations, 4, and 7 we can write (A,E) as functions of α, δ, H, and φ: sin E sin δ sin φ + cos δ cos φ cos H () tan A sin H cos δ sin δ cos φ cos δ sin φ cos H (2) The easiest way to get at (, ) is to differentiate Equations and 2 with respect to hour angle (H) then multiply by the sidereal rate ( ): 2 (3) (4)

3 3. Derivation of Start by noting that: so that: sec2 A (5) sec 2 A cos2 A (6) First solve for : (sin δ cos φ cos δ sin φ cos H) 2 [ (sin δ cos φ cos δ sin φ cos H) ( sin H cos δ cos δ cos φ ( cos H cos δ + sin H sin δ ) + + sin δ sin φ cos H + cos δ sin φ sin H (sin δ cos φ cos δ sin φ cos H) 2 [ sin δ cos δ cos φ cos H + sin 2 δ cos φ sin H + cos 2 δ sin φ cos 2 H cos δ sin δ sin φ cos H sin H + cos 2 δ cos φ sin H + cos δ sin δ sin φ cos H sin H + cos 2 δ sin φ sin 2 H ) (7) Using Equation 7 and eliminating and collecting like terms, Equation 7 becomes: 3

4 { cos 2 E cos 2 A sin δ cos δ cos φ cos H + [ ( cos φ sin H sin 2 δ + cos 2 δ ) + cos 2 δ sin φ ( cos 2 H + sin 2 H )} { cos 2 E cos 2 A } sin δ cos δ cos φ cos H + cos φ sin H + cos2 δ sin φ Using Equation 0 we find that Equation 8 becomes: cos 2 E cos 2 A { cos δ cos E cos P cos 2 δ sin φ + } cos φ sin H + cos2 δ sin φ cos δ cos E cos P + cos φ sin H cos 2 E cos 2 A Now inserting Equation 9 into Equation 6: (8) (9) Using Equation : cos2 A cos δ cos P cos E + cos φ sin H cos 2 E (20) we find that Equation 20 becomes: sin H cos φ sin P cos E (2) cos δ cos P cos E + sin P cos E Now substituting Equation 22 into our equation for (Equation 3): (22) ( cosδ cos P cos E + ) sin P cos E (23) 4

5 3.2 Derivation of Start with our relation for : d(sin E) cos E [ sin φ cos δ cos E cos φ sin δ cos H cos φ cos δ sin H (24) Using the sine formula (Equation ) and Equation 0: we find that Equation 24 becomes: cos δ sin H sin A cos E (25) cos E cos P sin φ cos δ cos φ sin δ cos H (26) [ cos E [ cos P + cos φ sin A (sin φ cos δ cos φ sin δ cos H) + sin A cos E cos φ [ cos P + cos φ sin A (27) (28) 3.3 Summary of the (, ) Calculation The terms dα and are source proper motion terms in α and δ, respectively. Using the fact that H τ α, we can write and as: dτ dα (29) (30) dτ is the ratio between the rotational period of the Earth and the length of the UT day, which is rotations per UT day. Note that we need to convert these units, rotations/day, to arcsec/sec or arcsec per unit of time (such as, for ALMA, a timing event, which is 48 ms). This is done by multiplying by: arcsec sec ( rotations day ) ( 360 deg rotation ) ( day sec ) ( ) 3600 arcsec degree (3) 5

6 Using this constant and noting that for a source with no proper motion that dα 0, we find that Equations 28 and 23 become: dτ cos δ cos P cos E (32) dτ cos φ sin A (33) Now, we can write and in terms of just A, E, and φ by using Equation 5 in Equation 33 to yield the final forms for and for a source with no proper motion: dτ [ sin φ cos E cos φ sin E cos A cos E (34) dτ cos φ sin A (35) Figure shows the results of Equation 35 and Equation 35 for the ALMA site. 6

7 Figure : Differential sidereal tracking rates in arcsec per TE (TE 48 ms) for the ALMA site. Top to bottom are dr,, and. 7