- Email: [email protected]
Surface albedo of cometary nucleus
Adv. Space Res. Vol. 9, No. 3, pp. (3)77—(3)80, 1989 Printed in Great Britain. All rights reserved.
0273—1177/89 $0.00 + .50 Copyright © 1989 COSPAR
SURFACE ALBEDO OF COMETARY NUCLEUS Tadashi Mukai and Sonoyo Mukai Kanazawa institute of Technology, Nonoichi, Ishikawa 921, Japan
ABSTRACT Direct images of the nucleus of cornet Halley have revealed an extremely structured and dark surface. This low albedo arises from, at least, two plausible possibilities, i.e. (1) absorption of incident light by non—volatile crust of dark material, and (2) multiple reflection of incident light due to the roughness of the nucleus surface, with a dimension of the order of visual wavelength or greater. We have investigated a variation of the albedo on the illuminated disk of cometary nucleus covered with rough surface consisting of absorbing material. It is found that the albedo of the nucleus significantly decreases as the degree of roughness increases. In addition, the albedo near the limb is roughly one order of magnitude smaller than that in the central part of the illuminated disk. Our treatment is applicable to examine the micro—structure on the surface of the airless body by remote sensing. INTRODUCTION The observations of the nucleus of cornet Halley/1,2/ indicate that the nucleus is covered with a rough surface with very low albedo, i.e. the visual geometric albedo of the nucleus is about 0.04. In addition, it is reported/3/ that the geometric albedo of the nucleus near the edge of the terminator is small(i.O.OO5) than that as a whole(”.O.04). Since the Moon shows a similar decrease of the albedo near the terminator, it is suggested/4/ that a small scale roughness, i.e. a dimension of the order of visual wavelength or greater, exists on the surface of the cometary nucleus like that of lunar soil. On the other hand, hydrogen cyanide polymers and related compounds are proposed for dark material on the surface layer/5/. The primary purpose of our investigation is to estimate a variation of the albedo over the nucleus surface taking into account the multiple reflection of incident light due to small scale roughness. The dependences of the average albedo over the illuminated disk on the degree of roughness and on the complex refractive index m* of the surface materials are also examined. METHOD We assume that the surface of the cometary nucleus is covered with an array of randomly oriented facets, composed of the absorbing material with the complex refractive index m*. The facet has a scale slightly larger than the visual wavelength, and it reflects the light specularly. Since we neglect the geometrical features of the roughness, e.g. height, width or mutual position of the elemental facets, the geological interpretation of the surface roughness cannot be obtained. We apply the mathematical treatment presented in /6/ to examine the stochastic process of the multiple reflection of the light by the surface roughness. For the slope distribution of facets, we use a gaussian distribution as
(3)77
(3)78
T. Mukaj and S. Mukai
2}/(1T~a), P(x):exp{—(x/a) where a is a parameter to define a degree of roughness (larger a means higher roughness), and x=tan 8 (8 is a polar angle of the normal vector of the reflecting facet measured from the average surface in the reference system(see figure 1)). In our calculations, the a.lbedo is defined by a ratio of the reflected light by the rough surface to that by the lambertian surface. This is called in / 1,3/ reflectance factor or reflectivity.
1~r1ew
ctse-ve~
~
.
:0’-
-......
9.
/
Fig. 1.
Schematic description of parameters used in the text.
RESULTS AND DISCUSSION (1) Variation of albedo on the illuminated disk We will compare our calculations of the albedo with the direct measurements of the nucleus albedo of comet Halley by HMC images reported in /3,4/. For the m* of the surface materials, it is applied m*= 1 .70—0.023 i. This is obtained for organic tholins synthesized in /7/ at a wavelength of 0.56 .im. From the geometry of the Giotto encounter, a phase angle 0 becomes 107°. Figure 2 shows our computed results of the albedos on the illuminated disk of the nucleus (where the nucleus was assumed as a sphere for simplicity). We can conclude fran this figure that (i) as roughness increases, i • e• a becomes larger, the albedo at a certain point on the illuminated disk decreases. Furthermore its spatial variation becomes weak, especially in the central part of the disk when the roughness is strong. (ii) The albedo near the limb is always lower than that in the central part of the illuminated disk. However, our calculations did not find a decrease of the albedo near the edge of the terminator, although HMC images have recorded smaller albedo there/3 ,4/. (2) Albedo vs. a degree of surface roughness In general, the light multiply reflected by the surface roughness is more attenuated than the light singly reflected. This suggests that stronger roughness on the nucleus surface reduces the value of the albedo.
Surface Albedo
e~107°
,
nf
(3)79
1.70-0.0231
Fig.2. Contour maps of the albedos of the cometary nucleus at a phase angle (sun—cornet—observer angle) 0~ 107° for the surface material with the complex refractive index of m*1.7O_O.O23 i. The surface has extremely roughness toward the right figure with larger value of a degree of roughness a. To examine a dependence of the albedo on the surface roughness, we calculate the average albedo as a function of a degree of roughness a, which is obtained by integrating the albedo derived at each of the positions on the disk illustrated in figure 2, The result in figure 3 confirms the above expectation, i.e. as the roughness increases, the average albedo becomes lower.
albedo
e= 107° ~
m*=1.70~ao23f
\
004
Fig.3. Albedo as a function of a degree of surface roughness a. The albedo shown here
\\
is obtained by integrating the albedo at each of the points on the illuminated disk in figure 2. A large value of a means strong roughness. A phase angle 6 (sun—comet— observer angle) is fixed 107°and the complex refractive index of surface material m* is
‘s,,
•
0.02
1
2 Ra~gtv~ess~
3
1. Mukai and S. Mukai
(3)80
altiejo Fig.4. The average albedo over the illuminated disk as functions of a refractive index n of the surface material and its absorption coefficient k, where a degree of roughness a~2and a phase angle 0:107°
06
ao4
9 n
k
(3) Albedo vs. complex refractive index of surface material m* A dependence of the average albedo on m* is shown in figure 4, where m*:n_i k. This result is derived for a phase angle 0:1O7~after the integration of the albedo over the illuminated disk. A degree of roughness is a=2. Note that larger values of n(refractive index) and k (absorption coefficient) lead higher albedos, although a k—dependence of the albedo is very weak in, at least, 0.01~k~0.15.This comes from the fact that, in spite of the increasing absorbtivity, the surface becomes brighter reflector due to an enhancement of the external reflection, resulting fran Fresnel’s formulae generalized to the complex indices of refraction(see /8/). CONCLUSION It is found that the multiple reflection of incident light by the surface roughness with a dimension of a wavelength of interest significantly reduces the intensity of the reflected light. When one examines the surface structure of the airless body based on the reflected light, the effect of the surface roughness on the intensity of the observed light should be taken into account, as well as that of the absorptivity of the surface materials. ACKNOWLEDG~1ENTS This work was partially supported by the Grant—in—Aid for Scientific Research on Priority Area(Origin of the Solar System) of Japanese Ministry of Education, Science, and Culture (No. 63611007). REFERENCES 1. Sagdeev R.Z., Avanesov G.A., Zimari Ya.L., Moroz V.1., Tarnopolsky V.1., Zhukov B.S., Shamis V.A., Smith B. and Toth I. ESA SP—25O II, 317 (1986). 2. Keller H.U. ESA SP—278, 447 (1987). 3. Delamere W.A., Reitseina H.J., Huebner W.F., Schmidt H.U., Keller H.IJ., Schmidt W.K.H., Wilhelm K. and Whipple F.L. ESA SP—25O 11, 355 (1986). 4. Reitsema H.J., Delamere W.A. and Whipple F.L. ESA SP—278, 455 (1987). 5. Matthews C.N. and Ludicky R. ESA SP—250 II, 273 (1986). 6. Mukai S., Mukai T., Giese R.H., Weiss K. and Zerull R.H. The Moon and the Planets 26, 197 (1982) 7. Khare B.N., Sagan C., Arakawa E.T., Suits F., Callcott T.A. and Williams M.W. Icarus 60, 127 (1984). 8. Mukai T., Fechtig H., GrUn E., Giese R.H. and Mukai S. Astron. Astrophys. 167, 364 (1986).
0273—1177/89 $0.00 + .50 Copyright © 1989 COSPAR
SURFACE ALBEDO OF COMETARY NUCLEUS Tadashi Mukai and Sonoyo Mukai Kanazawa institute of Technology, Nonoichi, Ishikawa 921, Japan
ABSTRACT Direct images of the nucleus of cornet Halley have revealed an extremely structured and dark surface. This low albedo arises from, at least, two plausible possibilities, i.e. (1) absorption of incident light by non—volatile crust of dark material, and (2) multiple reflection of incident light due to the roughness of the nucleus surface, with a dimension of the order of visual wavelength or greater. We have investigated a variation of the albedo on the illuminated disk of cometary nucleus covered with rough surface consisting of absorbing material. It is found that the albedo of the nucleus significantly decreases as the degree of roughness increases. In addition, the albedo near the limb is roughly one order of magnitude smaller than that in the central part of the illuminated disk. Our treatment is applicable to examine the micro—structure on the surface of the airless body by remote sensing. INTRODUCTION The observations of the nucleus of cornet Halley/1,2/ indicate that the nucleus is covered with a rough surface with very low albedo, i.e. the visual geometric albedo of the nucleus is about 0.04. In addition, it is reported/3/ that the geometric albedo of the nucleus near the edge of the terminator is small(i.O.OO5) than that as a whole(”.O.04). Since the Moon shows a similar decrease of the albedo near the terminator, it is suggested/4/ that a small scale roughness, i.e. a dimension of the order of visual wavelength or greater, exists on the surface of the cometary nucleus like that of lunar soil. On the other hand, hydrogen cyanide polymers and related compounds are proposed for dark material on the surface layer/5/. The primary purpose of our investigation is to estimate a variation of the albedo over the nucleus surface taking into account the multiple reflection of incident light due to small scale roughness. The dependences of the average albedo over the illuminated disk on the degree of roughness and on the complex refractive index m* of the surface materials are also examined. METHOD We assume that the surface of the cometary nucleus is covered with an array of randomly oriented facets, composed of the absorbing material with the complex refractive index m*. The facet has a scale slightly larger than the visual wavelength, and it reflects the light specularly. Since we neglect the geometrical features of the roughness, e.g. height, width or mutual position of the elemental facets, the geological interpretation of the surface roughness cannot be obtained. We apply the mathematical treatment presented in /6/ to examine the stochastic process of the multiple reflection of the light by the surface roughness. For the slope distribution of facets, we use a gaussian distribution as
(3)77
(3)78
T. Mukaj and S. Mukai
2}/(1T~a), P(x):exp{—(x/a) where a is a parameter to define a degree of roughness (larger a means higher roughness), and x=tan 8 (8 is a polar angle of the normal vector of the reflecting facet measured from the average surface in the reference system(see figure 1)). In our calculations, the a.lbedo is defined by a ratio of the reflected light by the rough surface to that by the lambertian surface. This is called in / 1,3/ reflectance factor or reflectivity.
1~r1ew
ctse-ve~
~
.
:0’-
-......
9.
/
Fig. 1.
Schematic description of parameters used in the text.
RESULTS AND DISCUSSION (1) Variation of albedo on the illuminated disk We will compare our calculations of the albedo with the direct measurements of the nucleus albedo of comet Halley by HMC images reported in /3,4/. For the m* of the surface materials, it is applied m*= 1 .70—0.023 i. This is obtained for organic tholins synthesized in /7/ at a wavelength of 0.56 .im. From the geometry of the Giotto encounter, a phase angle 0 becomes 107°. Figure 2 shows our computed results of the albedos on the illuminated disk of the nucleus (where the nucleus was assumed as a sphere for simplicity). We can conclude fran this figure that (i) as roughness increases, i • e• a becomes larger, the albedo at a certain point on the illuminated disk decreases. Furthermore its spatial variation becomes weak, especially in the central part of the disk when the roughness is strong. (ii) The albedo near the limb is always lower than that in the central part of the illuminated disk. However, our calculations did not find a decrease of the albedo near the edge of the terminator, although HMC images have recorded smaller albedo there/3 ,4/. (2) Albedo vs. a degree of surface roughness In general, the light multiply reflected by the surface roughness is more attenuated than the light singly reflected. This suggests that stronger roughness on the nucleus surface reduces the value of the albedo.
Surface Albedo
e~107°
,
nf
(3)79
1.70-0.0231
Fig.2. Contour maps of the albedos of the cometary nucleus at a phase angle (sun—cornet—observer angle) 0~ 107° for the surface material with the complex refractive index of m*1.7O_O.O23 i. The surface has extremely roughness toward the right figure with larger value of a degree of roughness a. To examine a dependence of the albedo on the surface roughness, we calculate the average albedo as a function of a degree of roughness a, which is obtained by integrating the albedo derived at each of the positions on the disk illustrated in figure 2, The result in figure 3 confirms the above expectation, i.e. as the roughness increases, the average albedo becomes lower.
albedo
e= 107° ~
m*=1.70~ao23f
\
004
Fig.3. Albedo as a function of a degree of surface roughness a. The albedo shown here
\\
is obtained by integrating the albedo at each of the points on the illuminated disk in figure 2. A large value of a means strong roughness. A phase angle 6 (sun—comet— observer angle) is fixed 107°and the complex refractive index of surface material m* is
‘s,,
•
0.02
1
2 Ra~gtv~ess~
3
1. Mukai and S. Mukai
(3)80
altiejo Fig.4. The average albedo over the illuminated disk as functions of a refractive index n of the surface material and its absorption coefficient k, where a degree of roughness a~2and a phase angle 0:107°
06
ao4
9 n
k
(3) Albedo vs. complex refractive index of surface material m* A dependence of the average albedo on m* is shown in figure 4, where m*:n_i k. This result is derived for a phase angle 0:1O7~after the integration of the albedo over the illuminated disk. A degree of roughness is a=2. Note that larger values of n(refractive index) and k (absorption coefficient) lead higher albedos, although a k—dependence of the albedo is very weak in, at least, 0.01~k~0.15.This comes from the fact that, in spite of the increasing absorbtivity, the surface becomes brighter reflector due to an enhancement of the external reflection, resulting fran Fresnel’s formulae generalized to the complex indices of refraction(see /8/). CONCLUSION It is found that the multiple reflection of incident light by the surface roughness with a dimension of a wavelength of interest significantly reduces the intensity of the reflected light. When one examines the surface structure of the airless body based on the reflected light, the effect of the surface roughness on the intensity of the observed light should be taken into account, as well as that of the absorptivity of the surface materials. ACKNOWLEDG~1ENTS This work was partially supported by the Grant—in—Aid for Scientific Research on Priority Area(Origin of the Solar System) of Japanese Ministry of Education, Science, and Culture (No. 63611007). REFERENCES 1. Sagdeev R.Z., Avanesov G.A., Zimari Ya.L., Moroz V.1., Tarnopolsky V.1., Zhukov B.S., Shamis V.A., Smith B. and Toth I. ESA SP—25O II, 317 (1986). 2. Keller H.U. ESA SP—278, 447 (1987). 3. Delamere W.A., Reitseina H.J., Huebner W.F., Schmidt H.U., Keller H.IJ., Schmidt W.K.H., Wilhelm K. and Whipple F.L. ESA SP—25O 11, 355 (1986). 4. Reitsema H.J., Delamere W.A. and Whipple F.L. ESA SP—278, 455 (1987). 5. Matthews C.N. and Ludicky R. ESA SP—250 II, 273 (1986). 6. Mukai S., Mukai T., Giese R.H., Weiss K. and Zerull R.H. The Moon and the Planets 26, 197 (1982) 7. Khare B.N., Sagan C., Arakawa E.T., Suits F., Callcott T.A. and Williams M.W. Icarus 60, 127 (1984). 8. Mukai T., Fechtig H., GrUn E., Giese R.H. and Mukai S. Astron. Astrophys. 167, 364 (1986).