Penentuan Jarak dalam Astronomi

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  • Penentuan Jarak dalam AstronomiKuliah Fisika Galaksi 1 Februari 2011

  • Urutan metoda penentuan jarak berdasarkan jarak maksimal yang dapat dicapai

  • Urutan metoda penentuan jarak berdasarkan jarak maksimal yang dapat dicapaiMetoda langsung / primary method

  • Urutan metoda penentuan jarak berdasarkan jarak maksimal yang dapat dicapaiMetoda yang dikalibrasi oleh p

  • Pergeseran paralaks bintangEarthSunEarth 6 month later Apparent changeof the directionParallactic angle Nearby star Initial starpositionStar positi-on after 6monthMore distant stars Perlu diingat ! Sebenarnya kita tidak mengukur paralaks absolut dari bintang tetapi paralaks relatif, yaitu perubahan posisi bintang relatif terhadap bintang-bintang latar belakang yang lebih redup dengan anggapan lebih jauh atau galaksi yang jauh

  • Paralaks tahunan bintang disebut juga paralaks trigonometri karena gerak orbital Bumi. Bintang-bintang dekat bergerak relatif terhadap bintang yang lebih redup dalam bentuk elips, bergantung pada posisi bintang tersebut di langit. Sudut paralaksnya berkurang sebanding dengan pertambahan jarak, dan bentuk ellips-nya bergantung pada lintang ekliptika, : axes ratio b/a = sin Degrade to line for =0

  • The parallactic angle strongly depends on stars distance

    For (really) small angles, p 1/d

  • Parsec new unit of interstellar distancesParsec = PARallax + SECond of arc:

    The distance fromwhich 1 AU is seenat the angle of1 arcsec

    D(pc) = 1/p

    1 pc 206265 AU 3.0861016 m 3.26 light years

    Large distances:1 kpc = 103 pc1 Mpc = 106 pc1 Gpc = 109 pc Do we need 1 Tpc (TeraParsec) or not ?

  • Practical measurements of stellar parallaxes. I.Step 1: Choose a large set (N>>1) of faint (presumably distant) reference stars. These stars are expected to set (fixed?) local coordinate frame used to reduce most of optical aberrations and distortions and to measure shift of program stars relatively to this reference frame. Step 2: Measure Descart (x,y) coordinates of all program and reference stars on each frame (time ti, i=1,2,,K; good if K >>1).Step 3: Choose one frame (near the middle of the set) as standard frame, time labelled as t0

  • If:All reference stars were fixed (say, very distant),All optical aberrations involved were negligible,Errors due to the differences in observational conditions (temperature, focusing etc.) were absent,Measurement errors were also negligible, then all of K coordinate frames would be completely identicalThis is really not the caseThe reduction procedure is necessary

  • Our goal: put all frames together and see how program star moves among reference starst1t2tK-1tKt0Reference starsxyProper MotionParallactic wobbleacross PM vector

  • Practical measurements of stellar parallaxes. I.Step 4: Find coordinate transformations from each ith frame to standard frame using (x,y)i and (x,y)0 coordinates of N reference stars, as

    (R for reference star)

    Here Fi-to-0 is (weekly nonlinear) parametrized function (matrice type) of the coordinates (x,y)The unknown parameters can be estimated from the system of 2N conditional equations by any suitable nonlinear least-squares algorithm.Now we are ready to match all coordinate frames.

  • Transformation matrix Fi-to-0 may look as follows (taking into account frame scale differences, distortions, overall centers shift etc.): x0 a0x + axxi + bxyi + cxxi2 + dxxiyi + y0 a0y + ayxi + byyi + cyyi2 + dyxiyi +

    Here index i used for ith frame, (x,y) are the Descart coordinates of reference stars Here all as, bs etc. are unknown constants that need to be calculated by the least squares for each of ith frame (i=1, 2, , K)

  • Practical measurements of stellar parallaxes. I.Step 5: With these Fi,(K on total) recalculate the coordinates of program stars from ith frame to standard one, as

    (P for program star) Fi-to-0 are known now!Now we have K calculated coordinate pairs (x,y) for each program star reduced to standard frameStep 6: Based on telescope properties (focal length) calculate equatorial coordinates (,)i of program star(s) on each ith frame. Apparent coordinates (,)i vary with i (with time!)

  • Now, it looks like we have transferred all positions of our (moving!) program star to single frame

  • Practical measurements of stellar parallaxes. II.Apparent coordinates of program stars are changing because ofParallactic shiftStellar proper motionsSimplest model for time variations of stellar coordinates:

    Here p is the parallax, & are two components of stellar proper motion, ,i & ,i are the residuals from model adopted (considered as normally distributed random values)(1)

  • Here & are the differences of the parallax phase between two time points separated by ti=ti-t0 parallax phases

    and auxiliaryangle are calculatedfor each giventime point ti| , | < 1 ecliptic longitude0 Sun ecliptic longitude ecliptic latitude ecliptic inclination

  • 2K conditional equations (1) can be resolved for unknown parameters p by the least-squares or maximum-likelihood technique (minimizing residuals ,i ,i) for time moments (t1 t2 t3 tK), 2 K >> 3Important notes 1-2. Trignometric parallax and proper motion components are calculated simultaneously! To calculate reliable parallax from (1), you should plane your observations so that some & were of the order of 1 (unity)!

  • Important note 3. To derive precise proper motions, we need long time intervals t , typically dozens of years

    Astronomer should live long enough to use his own observations!

  • Important note 4. Parallax and proper motions are calculated relatively to reference stars which are supposed to be very distant. If it is not the case, we calculate only relative parallax & relative proper motion.We need to estimate the absolutization corrections as p = pabs prel & = abs rel

    The best way: to use objects with zero parallax and proper motions seen on the same frames: (1) compact distant galaxies or (2) quasars; (3) stars with known parallax & proper motions taken from the fundamental catalogs FK4- FK5, ICRS catalogs HIPPARCOS, TYCHO-2, Or (4) use appropriate statistical absolutization, based on Milky Way Galaxy dynamical model when (1-3) are not available

  • Measuring parallaxesis very difficult and tedious jobExtremely small anglesBefore HIPPARCOS epoch, till 1992, ~16000 stellar parallaxes for ~8100 stars have been measured with typical accuracy of 16 mas with ground-based instruments.Publications: The General Catalogue of Trigonometric Stellar Parallaxes (4th edition) by Van Altena W.F., Lee J.T., Hoffleit E.D. (Yale University Observatory, New Haven, 1995); Catalog I/238A in CDSAbsolutization: statisticalMengukur paralaks dan proper motion adalah pekerjaan yang sulit dan membutuhkan waktu yang panjang !

  • HIPPARCOS satellite on Earth and above Earth: scanning the skySynchronous observationof two stellar fields separated by the basicangle = 5800'31.25"with scanning

  • HIPPARCOS actual orbit(instead of geostationary)Perigee distance 500 kmApogee distance 36000 km

  • HIPPARCOS optical layout29-cmhalf-mirrorBeam 2Beam 1Basic angleBeam combiner unitModulating grid2 PMTs

  • Reduction to ICRF/ICRS~240 quasars (from 608) were usedObservations of stellar radio-sources both with VLBI and on-board, with HIPPARCOSMeasuring angles between HIPPARCOS stars and radio-sources with HSTResults:Final reduction accuracy ~0.6 masFinal residual rotation of HIPPARCOS coordinate system ~0.25 mas/year

  • Why 240 not 608? Quasars proper motions? No, complex jet structure!Photocentermovement reflects jetmotion that canbe masked to proper motion

  • HIPPARCOS observationsHIPPARCOS satellite was launched on August, 1989 and lived on orbit till March, 1993Active observations lasted ~37 months~118000 sample of stars complete to ~7.3m included to HIPPARCOS catalog~1 mln sample of stars complete to ~10.5m included to TYCHO catalog~2.5 mln stars later included to TYCHO-2 catalog (used also ~140 ground-based catalogs of positions and proper motions)

  • HIPPARCOS project stagesThe HIPPARCOSInput Catalog(Catherine Turon et al.,ESA SP-1136, 1992) HIPPARCOSworking catalog (after 37 month) HIPPARCOS final catalog(ESA SP-1200, 1997 )

  • Data reduction by HIPPARCOS consortiumsFAST (Fundamental Astronomy by Space Techniques) Jean Kovalevsky, 1992-1997NDAC (Northern Data Analysis Consortium) Lennart Lindegren, Erik Hog, 1992-1997Approaches differ in details but not in general conceptFinal HIPPARCOS catalog appeared on 1997----------------------------------------------Floor van Leeuwen HIPPARCOS, the new reduction of raw data, 2007 new version

  • HIPPARCOS family of catalogs

    HIPPARCOSTYCHOTYCHO-2Catalog systemICRSICRSICRSMean observationepoch & catalog epochJ1991.25J2000.0J1991.25J2000.0J1991.5J2000.0Number of stars118 2181 058 3322 539 913Limiting magnitude12.4m11.5m11.5mComplete to:7.3m10.5m11mMean accuracy of:positions< 1 mas7-25 mas7-60 masproper motions< 1 mas/year-2.5 mas/yearparallaxes 1 mas--Mean photometric errors 0.002m0.06-0.10m0.013-0.10m

  • HIPPARCOS: new revolution in astronomy~20 000 stars with parallaxes better than 10% ~30 000 other stars with parallaxes better than 20%~100 000 stars with broad-band magnitudes (close to standard B,V system) accurate within 0.002-0.003mNearly uniform covering of the sky wit