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

2. 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

3. Muinonen et al. 2002 Photometric & Polarimetric Phase Effects

4. Albedo & Phase Function (1)   phase angle,  = (2    )  scattering angle,   (solar) elongation Sun de Pater & Lissauer 2001

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

6. 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!)

7. 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

8. 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. 1989 (Asteroids II)

9. 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. 1989 (Asteroids II)

10. Obtaining H & G (1) Observation (  reduced mag) data V i (  i ) and errors  i Least-squares solution Buktikan… Bowell et al. 1989 (Asteroids II)

11. Obtaining H & G (2) a 1 and a 2 are of order 10 -0.4H, 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. 1989 (Asteroids II) Phase integral q = 0.290 + 0.684 G

12. H & G Error Analysis Magnitude residuals  m(  i ) is the calculated magnitude drop from zero phase angle Then, Bowell et al. 1989 (Asteroids II)

13. 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)

14. 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

15. Examples Karin cluster interloperCometary asteroid