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Two-parameter Magnitude System for Small Bodies Kuliah AS8140 & AS3141 (Fisika) Benda Kecil [dalam] Tata Surya Prodi Astronomi 2006/2007

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Observing Plane The plane Sun-Object-Observer is the plane of light scattering of the radiating reaching us from the Sun via the object. It is a symmetry-breaking plane, and because of this, makes the light from the object polarized Karttunen et al. 1987

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Muinonen et al Photometric & Polarimetric Phase Effects

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Albedo & Phase Function (1) phase angle, = (2 ) scattering angle, (solar) elongation Sun de Pater & Lissauer 2001

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Albedo & Phase Function (2) Karttunen et al. 1987

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Albedo & Phase Function (3) A solar system body radiated by the Sun: F Sun = 1.36 10 3 W m -2 is the solar constant at 1 AU, r is heliocentric distance in AU, For a black body albedo A V = A IR = 0 Assuming isotropic thermal emission (not quite true!)

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when observing a target … Data: V obs V(r ; , ), dV obs Brightness at unique distances [r, ] and phase angle ( ) V obs should be a magnitude at base-level lightcurve Sometimes V(r ; , ) V( ) V obs ( ) Data set: exp. per day Time: t i ; i = 1,…,n Magnitude at base-level: V(r i ; i, i ) ; i = 1,…,n Magnitude error: i ; i = 1,…,n

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The Two Parameters: H & G (1) Reduced magnitude V obs is observed magnitude Absolute magnitude is the magnitude of a body if it is at a distance 1 AU from Earth and Sun at phase angle = 0 Standard visual Two-parameter (HG) magnitude system: G is the slope parameter the gradient of the phase curve Bowell et al (Asteroids II)

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The Two Parameters: H & G (2) Phase function l ( ); for 0 120 , 0 G 1 Simpler, more symmetric, but slightly less accurate expression Bowell et al (Asteroids II)

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Obtaining H & G (1) Observation ( reduced mag) data V i ( i ) and errors i Least-squares solution Buktikan… Bowell et al (Asteroids II)

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Obtaining H & G (2) a 1 and a 2 are of order H, which may be computationally inconvenient. If so, they may be scaled to order unity by setting m is one of the reduced magnitude V i ( i ) (for instance at smallest ) Thus, Bowell et al (Asteroids II) Phase integral q = G

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H & G Error Analysis Magnitude residuals m( i ) is the calculated magnitude drop from zero phase angle Then, Bowell et al (Asteroids II)

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Drawbacks… The H,G-magnitude system fails to fit the narrow opposition effects of E-class asteroids (Harris et al. 1986) It shows poor fits to the phase curves of certain dark asteroids (e.g., Piiroen et al. 1994, Shevchenko et al. 1996) Hapke’s photometric model (5 parameters) has photometric fits as good as the H,G-magnitude system (Verbiscer & Veverka 1995)

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Lightcurve Amplitude – Phase Angle Relation A(0°) and A( ) are, respectively, the lightcurve amplitude at zero phase angle and that at a phase angle Amplitude at zero phase angle is smaller than that at a phase angle Zappala et al. (1990) a b c

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Examples Karin cluster interloperCometary asteroid

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