The Umov effect for remote sensing of cosmic dust

The Umov

effect provides an excellent example of using polarimetry

for remote sensing.

In application to planetary regoliths, it describes a relation between maximum of positive polarization Pmax

and the geometric albedo

of the surface A.

N. Umov

(1846-1915)

Umov

formulated the law as follows:

The brighter powder, the lower its linear polarization

N. Umov, Phys. Zeits. 6, 674-676 (1905)

However, Umov

did not specify neither type of albedo

nor the phase angle of

linear polarization measurements.

Only much later (Markov, 1958), the Umov’s

finding was refined. In particular, the type of albedo

has been specified: the normal

albedo

or the geometric albedo. Also, it was found that one needs to consider the maximum of the linear polarization.

Figure shows the angular profiles of the degree of linear polarization for two lunar terrains measured at 0.6 μm: curve (a)

–

region in Oceanus

Procellarum

(A=6.3%) and curve (b)

–

highland near

crater Crüger

(A=11.7%).

adapted from Dollfus

and Bowell, A&A 10, 29–53 (1971)

In middle of sixtieth, there were taken a few attempts to quantify the relation between the geometric albedo

A and

maximum of linear polarization Pmax

(e.g., Avramchuk, 1964; Clarke, 1965):

a linear dependence between log(A) and log(Pmax

) was found.

In order to understand why is that, one needs to consider the definition for the degree of linear polarization P:

||

||

IIII

P+−

=⊥

⊥

Here, the denominator presents the total intensity of the scattered light. However, this intensity is proportional to the geometric albedo

A:

||IIA +∝ ⊥

Taking into account the last relation, one can take a logarithm from both parts in the definition for P:

It is important to notice that the term f is not a constant, but can be a complicated function of albedo

and other parameters,

describing regolith.

Therefore, in general, the relationship is not trivial.

)log()log()log( || AIIP −−≈ ⊥

A necessary condition for maximum of the linear polarization is a minimum for the first term on right. Denoting this parameter with f, one can express an approximate relation between Pmax

and A in very general form as follows:

f)log()log( ≈+ AP

An example of the relation between log(Pmax

) and log(A).

-2 -1.5 -1 -0.5log(A)

0

0.5

1

1.5

2lo

g(P m

ax)

22 various sites on the Moon

λ=0.42 μmλ=0.65 μm

Data adapted from Shkuratov

et al., Icarus

95, 283–299 (1992)

Sometimes, it is difficult to observe a target at exact backscattering (i.e., at α=0°). Then, the geometric albedo

is

being related to a small angle near 0°.

By analysis of photo-polarimetric

observations of the Moon, one can derive a quantitative relationship

between the geometric

albedo

A at α=5°

and maximum of linear polarization Pmax

as follows:

Here, A and Pmax

are measured in natural units

(not per cent!), constants a = 0.724 ±

0.005 and b = –1.81 ±

0.02 (Dollfus

and

Bowell, A&A, 10, 29–53 (1971)).

It is important to remember that

the given relationship was obtained in orange light (at λ

= 0.6 μm). Similar relationships at

other wavelengths can be found, for instance, in Shkuratov and Opanasenko, Icarus 99, 468–484 (1992)

bAaP =+ )log()log( max

In order to understand the mechanism governing the Umov effect, we have to consider evolution of the Stokes vector S

while light experiences a multiple scattering in random media.

Let us assume that the Mueller matrix

describing light scattering by a constituent particle,

takes a form as follows:

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

−

=

4434

3433

2212

1211

2

0000

0000

)(1

MMMM

MMMM

kRM

The first act of scattering of unpolarized

light is expressed as:

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

∝

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⋅

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

−

=⋅=

00

0001

0000

0000

)(1 12

11

4434

3433

2212

1211

21 M

M

MMMM

MMMM

kRincSMS

However, the formalism

of Mueller matrices and Stokes vectors is strongly related

to the chosen scattering plane.

When considering double light scattering from a pair of particles that are not oriented in the same scattering plane, we have to adjust a Stokes vector (or, equally, Mueller matrix) to a consequence of different scattering planes.

For instance, within double light scattering, number of such adjustments is of three:

(1) from the principal scattering plane to local one related to the particle 1;

(2) from local scattering plane related to particle 1 to that one is related with particle 2;

(3) from local scattering plane related to particle 2 back to the principal scattering.

source of light

particle 1

detector

particle 2

Scheme explaining changes in the scattering plane.

Adjustment of a given Stokes vector (or, the same, Mueller matrix) from one scattering plane to another one can be done with so-called rotation matrix 4 x 4. It takes form as follows:

Here, φ

is angle between normal vectors to scattering planes.

Thus, in general, double light scattering is being described as follows:

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

ϕϕ−ϕϕ

=

100002cos2sin002sin2cos00001

O

incpp SOMOMOS ⋅⋅⋅⋅⋅= →→→ 1121222

Obviously, rotations do not affect the first parameter in the Stokes vector (i.e., intensity of light), but they distribute the second parameter partially to the third and fourth ones.

However, the last two parameters

are washed out

when averaging

light-scattering properties over random orientations of

particles pair.

This reduces

“the total amount”

of polarization produced by a randomized group of the scatterers

as compared to a single-

scattering particle.

In ground-based astronomical observations, the usage of the Umov

effect is restricted by the maximal phase angle

that can

be achieved by a target.

For example, for the heliocentric distance of Mars (1.5 AU), it is approximately 48°; whereas, in the case of the main belt asteroids (2.5 AU), it is approximately 23.5°.

As consequence, in the most of cases, one cannot observe the maximum of linear polarization.

The problem can be solved by using a slope of polarization curve h instead of maximum of linear polarization Pmax

.

An obvious necessary condition to make such replace possible is as follows: negative and positive polarization branches have to be caused by the same physical mechanism.

0 60 120 180phase angle, deg

Line

ar P

olar

izat

ion,

%

αmin

αmaxPmin

Pmaxαinv

h

An example of the relation between log(h) and log(A).

Figure is adapted from Geake and Dollfus, MNRAS 218, 75–

91 (1986)

An important feature in diagram log(h)–

log(A) is

that the inverse correlation does not hold in the case of very dark surface A ≤

5%.

This feature could be explained by relatively strong contribution of the first order of scattering.

In general, the relationship between the geometric albedo

and slope of the linear polarization curve takes a form as follows:

Here, A is measured in natural units

(not per cent!); whereas, slope h –

in percent per degree.

The relationship was calibrated by Zellner

et al. (1977; LPSC VIII, 1091–1110) by the laboratory measurements of meteorite samples. It was found that a = –0.93 and b = –1.78.

Later on, Lupishko

and Mohamed (1996; Icarus

119, 209–213) through analysis of data for 127 various asteroids, corrected the constants as follows: a = –0.98 and b = –1.73.

Note also that the geometric albedo

derived from polarimetric observations (either h or Pmax

) very often is referred to as the polarimetric

albedo

and denoted in scientific literature as pv.

bhaA += )log()log(

Interestingly to remind that the inverse correlation between log(Pmax

) and log(A) holds through all the values of the geometric albedo

A.

-2 -1.5 -1 -0.5log(A)

0

0.5

1

1.5

2

log(

P max

)

22 various sites on the Moon

λ=0.42 μmλ=0.65 μm

This fact is an extremely important since it makes possible an extension of the Umov

effect to the

case of single-scattering particles.

It would be very useful for cometary

applications

because many of comets are indeed reachable at large phase angles.

In what follows, we will consider an extension of the Umov

law to the case of single-scattering irregularly shaped particles

comparable with wavelength.

We study six different types of particle morphology:

Computations of light scattering were carried out with the discrete dipole approximation (DDA).

Light scattering by particles comparable with wavelength depends

on size parameter x and refractive index m.

In the case of agglomerated debris particles, we consider 15 various values of refractive index m; whereas, other five types

of

particle morphology have been studied only at three refractive indices

m=1.313+0i, 1.6+0.0005i, and 1.5+0.1i.

Size parameter

quantifies the rate of particle radius r to wavelength λ: x=2πr/λ

(r is radius of circumscribing sphere).

For agglomerated debris particles, we vary x from 2 up to 40. The upper limit of x depends on the refractive index (this information is summarized on the next slide).

In the case of other five types of particles, size parameter x was varied from 2 to 14, for all refractive indices.

We classify all the refractive indices

into two categories

with weak absorption

Im(m)≤0.02 and high absorption

Im(m)>0.02.

When studying a possible extension of the Umov

law for the case of single-scattering particle, we have to choose a type of albedo.

In general, there are two options:

(1) Like in the case of regolith, the geometric albedo

A. In the case of single-scattering small particles, it is defined as ratio

A=πM11

(0)/(k2G)

Here, M11

(0) is the total intensity Mueller matrix element at backscattering, k –

wavenumber, and G –

the geometric cross-

section of the particle.

(2) Single-scattering albedo

ω, which is defined as follows:

ω=Csca

/Cext

Here, Csca

and Cext

are cross-sections for scattering and extinction.

Results for agglomerated debris particles and the single-scattering albedo

ω

Results for agglomerated debris particles and the geometric albedo

A

The non-linearity in the diagram log(Pmax

)–

log(A) mainly is caused by contribution of small particles with x<14. So, it is of interest to limit data points by this condition.

One can also consider a further limitation of data points by condition x=14 only.

Since the data points corresponding to x<14 are the main reason for non-linearity in the diagram log(Pmax

)–

log(A), it is of interest to study other particle morphologies in this range of x.

Though all the morphologies

reveal qualitatively similar behavior, there are certain deviation

between data points corresponding to

different particle types. Averaging

over particle types linearizes substantially the diagram for weakly absorbing particles.

So far, we were considering model particles of a fixed size.

However, in the most of cosmic and terrestrial applications, dust is polydisperse, so dust particles follow some size distribution.

For instance, according to in situ measurements, cometary

dust particles reveal a power law size distribution:

r–a

The power index a was found to be varied from 1.5 to 3.4. However, some works predict that a may be as high as 4.

For agglomerated debris particles

with 10 refractive indices

and wide range of size parameter x (see Table 1), we average light-

scattering properties

over size.

Diagrams log(Pmax

)–

log(A) for agglomerated debris particles averaged over size.

An example of remote sensing

of dust in cometary

circumnuclear haloes (Pmax

≈12%) with the Umov

law.

One can estimate

the geometric albedo

A of dust particles forming cometary

circumnuclear

haloes as A = 0.1 –

0.2. This

value is a few times larger than A averaged over the entire coma. Simultaneously, the power law distribution index a is found to be a = 1.9 –

2.5, which is well-consistent with findings

of in situ measurements carried out by VeGa

–

1 and 2.

Literature:

1.

Bohren

and Huffman, Absorption and scattering of light by small particles (Wiley, 1983)

2.

Dollfus

and Bowell, A&A 10, 29–53 (1971)

3.

Geake and Dollfus, MNRAS 218, 75–91 (1986)

4.

Shkuratov and Opanasenko, Icarus 99, 468–484 (1992)

5.

Lupishko

and Mohamed, Icarus

119, 209–213 (1996)

6.

Zubko et al., Icarus 212, 403–415 (2011)