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Doppler shifts in zodiacal light
Planet. Printed
Space Sci., Vol. 35. No. 8, pp. 1021-1027, in Great Britain.
1987
DOPPLER
6
SHIFTS IN ZODIACAL D. C. HIRSCH1
The University
of Kansas,
00324633/87 $3.00+0.00 1987 Pergamon Journals Ltd.
LIGHT
D. B. BEARD
and
Lawrence,
KS 66045, U.S.A
(Received in final form16 February 1987) Abstract-Doppler shifts in zodiacal light are calculated for various eccentricities, dust sizes, and assumptions about radiation pressure. The purpose is to determine what effects in spectra might be observed which would enhance our understanding of the origin and lifetimes of interplanetary dust. The solar absorption line half-width in the scattered light is a much more difficult measurement to make, but it is a better indication of orbital eccentricity than the first moment of the line can ever be. While the first moment of the line (which is close to the displacement of the maximum of the line) is similar for both orbital eccentricity and radiation pressure and difficult to sort out from other effects like radial outflow, the line half-width tends to broaden most for elliptical orbits at all elongations especially in the Gegenschein (backward scattering angle) where all other effects (excepting radial outflow which reddens the line) cause no change in the original solar line.
1. INTRODUCTION
In orbiting the sun the velocity of interplanetary dust with respect to Earth varies depending on its orbital radius and direction of sight ; if in a noncircular orbit its radial velocity with respect to the sun will not be zero as well. Thus the Fraunhofer absorption lines in the solar spectrum will appear Doppler shifted when the spectrum of the zodiacal light is observed. Observing this is difficult and error prone through Earth’s atmosphere against a background of stellar light sources. Nevertheless, measuring Doppler shifts in the zodiacal light is a means of determining the orbital characteristics of the dust, eccentricity, effect of radiation pressure, and radial outflow, having important implications on dust origin, lifetimes, rate of supply and loss. Ingham (1963) calculated the distortion of the HP line in light scattered from dust in circular orbits observed at 30” elongation. Ring et al. (1964), Daehler et al. (1968) Hindle et al. (1968) and Reay and Ring (1968) observed the H/I line despite the severe interference of emission in Earth’s atmosphere. James (1969) calculated the distortion of the 5 183.6 8, magnesium line for circularly orbiting dust at 30” elongation which avoided the emission feature and was easier to observe. Since the magnesium line was complex and in any case washed out in the scattered spectrum he calculated the distortion of an idealized wedge-shaped line for elongation angles from 10” to 40”. Banderman and Wolstencraft (1969) calculated the displacement of the line from several effects including noncircular orbits with a unique eccentricity 0 < E < 1. Reay (1969) calculated displacements
resulting from radiation pressure effects. Vanysek and Harwit (1970) found that using Mie scattering in place of isotropic scattering did not affect the calculated displacement. James and Smeethe (1970) observed the 5184 magnesium line at elongations between 28” and 40” in the evening and morning sky and concluded that the dust was in direct orbits about the sun and not significantly about Earth. Hicks et al. (1974) observed the first moment of the line at a larger range of elongations and Rodriguez and Magro (1978) calculated line displacements as a function of many distribution parameters. Fried (1978) reported a very controversial observation of the first moment of the line leading to the conclusion that significant amounts of dust were in hyperbolic orbits. East and Reay (1984) observed the first line moment which disagreed significantly from Fried’s observations. For almost 20 y, investigations of Doppler shifts in zodiacal light have concentrated on line displacements or first moments because the detailed spectra are difficult to observe and tedious to calculate. Still, other changes in the spectra, outstanding enough to be observed, are worth studying to see what may be learned. A wedge-shaped source line as used by James (1969) is simpler and yet sufficient a reliable indicator of line broadening in view of the blurring of detail in line form to serve our purpose. We used a wedgeshaped line in studying the effect of radial outflow, radiation pressure, and orbital eccentricity of the dust on line width and line displacement at elongation angles from 10” to 170”. We find that in the Gegenschein (elongation angle 180’) the line width is significantly broadened (of the order twice as wide) by 1021
D. C. HIRSCHI and D. B. BEARD
1022
introducing noncircular orbits, not broadened at all by radiation pressure effects, and not broadened much but made asymmetrical by a significant radial outflow. The three perturbations affect the line in different amounts at other elongation angles, but the various effects can be separated out by observations at all elongation angles.
2.
CALCULATION
LINE
A particle in an elliptical orbit in the ecliptic plane about the sun has a speed towards or away from Earth of V=
V,cos++V,sint+-V,sin.s
(1)
where VR is the particle’s radial velocity from the sun, V, is the particle’s azimuthal velocity, V, is Earth’s azimuthal velocity, $ is the angle between Earth and sun at the particle, and E is the elongation angle, the angle between sun and particle at Earth. From the law of sines sin*
= sine/R
(2)
where R is the heliocentric distance of the particle expressed in astronomical units. For a particle in an elliptical orbit with eccentricity e and semi-major axis a and Earth in a circular orbit V, = (K/RM)“*(2-R/a-((a/R)(l
g
of (3) with respect to e is R/u
= (K/MR)“*(2-(a/R)(l
+cos$)ue/R
-e2))‘12(1
-(Ku/M)e(l-e2)-“2sinsR-Z.
(4)
As Haug (1958) has shown, the number of particles between R and R+ dR is given by
OF A DOPPLER SHIFl- AFFECTED
SOLAR ABSORPTION
derivative
-e*))‘l*
V, = (Ka/M)“*(l-e2)“*RP’ J’, = (KIM) ‘1’ = 30km s- ’
n(R)dR
- (R%V&’
dR =
(~/R)~~‘~(2-R/u-((a/R)(l-e*))-“~dR where z is the orbital period. and e
For a distribution
n(u)n(e)dude
n(R) =
ss (uR)“‘(2-R/u-(u/R)(l
The total intensity of scattered sight A is given by I=
in a
-e*))‘/*’
light along the line of
s
n(R)dA R*
n(u) n(e) da de dA (uR)3~2R2(2-R/u-(u/R)(1-e2))‘~2’
(5)
The variable A is the particle to Earth distance and from the law of cosines can be expressed in terms of Randa A=cossk(R*-sin*&)‘/*
where K is the gravitational constant and M is the mass of the sun ; a is in astronomical units. Scattered light arriving at Earth experiences two Doppler shifts, the first because of the radial velocity of the particle, the second because of the velocity component of the dust along the line of sight with respect to Earth. The Doppler shift in wavelength is given approximately by AI = (v/c),%.
from which we may find the convenient dA=
substitution
(l-sin*~/R*))‘~*dR.
(6)
The total intensity from a source J(I) at a wavelength I(1 + D/c) is given by (4), (5) and (6) Z(n( 1+ D/c)) = n(u) n(e) $.Z(I)
da dR
Let D = v = (An/@.
e can be obtained from (3) as a function of a, R, D and a. If J(I) is a wedge type source as a function of 1
Then D=
V,(l+cos$)+VOsinsR-l-V~sins
D = (K/MR)“*(2-R/u-(a/R)(l + (Ku/M) ‘/2(1- e’) ‘P sin ER-
-e’))“*(l
+cost,b)
’ - (K/M) “‘sin E (3)
where (3) can be rearranged to express the eccentricity e in terms of independent variables R, a, D and E. The
J(n) = Z,(l-llzo-11/0.9),
Lo-O.9
< a < &+0.9
where I, is a constant, the depth of the line at 1,. The light intensity of I.,( 1 + d/c), where d&/c is any given wavelength displacement from the center of the line I,, is
Doppler
shifts in zodiacal
1023
light
Z(d) = n(a) n(e) Z,(l - ]I, -L]O.9) ’ (ur)312R2(2-R/a-((a/R)(l-eeZ))‘~2(1-sin2~/R2)’12 x $dudRdi where e =
(7)
f (a, R, D, E) ; de/dD is given in (4) ; and D = &(l +d/c)-1.
-75
(8) FIG.
Similarly for circularly orbiting dust we change from the independent variable R (in place of e for elliptical orbits) to D and obtain for a wedge-shaped source line Z(d) =
2cz, 3(K/M)“* X
1.
so
MEAN
loo
150
200
250
SHIFT OF THE DOPPLER-SHIFTED
DUE TO AN OUTWARD
RADIAL
FLQW
300
350
LINE PROFILE
OF PARTICLES.
The solid line is for a heliocentric distribution, n(R) - R - I ’ ofcircularly orbiting dust shown for comparison. The dashed line illustrates the difference when 20% of the dust has a radial outflow of 8 km SC’. Shift in kilometers per second towards Earth vs elongation angle.
sin E
s
A~+‘.9R”*n(R)(l-I&-1//0.9)
,+o P
i(l-sin*~/R*)“*
3. RESULTS
d2
(9)
where R is given by R = (l+M”*(K”*sin~))‘D))*~~
(10)
and D by (8). For comparison purposes since so much study has been given to circular orbits in the literature, we show the predictions for circularly orbiting dust in all tables and graphs. For circular orbits of particles small enough for radiation pressure to affect them the gravitational attraction of the sun is weakened by an amount tic, where tl is the radiation pressure and c is the velocity of light. Hence, the velocity of the dust becomes V, = ((K/M)“*--c)/(R)“*; (10) must be replaced R = {[(k/M)“*
LL
0
by
-ctc]/[(K/M)‘/*
For that part of the dust propelled by an excess of radiation pressure D = V,(l +cos$)-
+ D cosec E]} 2/3. away from the sun (3) would become
V, sina
= V,[l+(l-sin*~/R*)“‘]-(K/M)“*sina and n(R) = N0Vi’Rm2 from which the line profile is readily obtained the circularly orbiting dust with D substituted as the independent variable : R = V,sin&{2VR[D+(K/A4)‘12sina]
as for for R
In carrying out the calculations we discovered that the results depended very little on n(R) which has been variously reported as an inverse power of R between one and two from interpretations of zodiacal light observations (e.g. Allen, 1946 ; Beard, 1959; Blackwell and Ingham, 1961; Calbert and Beard, 1972). We used a distribution of Y- I,5 and r-* interchangeably. We used in all cases a wedge-shaped source line of 5 184 and 0.9 A width at half maximum. The details of a more realistic Fraunhofer absorption line will be washed out in being scattered from an orbiting dust particle and in any case the detailed line shape is unobservable with current observational capability. For the distribution in eccentricity we used n(e) - e - ‘, constant, and e with increasing effect. The particle size distribution we chose emphasized radiation pressure effects with n(s) - sp4 and sm6 both of which amounted to a delta-function at the minimum particle size assumed. Radial outflow has been suggested by East and Reay (1984) to account for the asymmetric shift of the line maximum in going from the evening to the morning sky. As an example of this we assumed a radial outflow of 20% of the dust of 8 km s-‘. Presumably the source of this dust is small particles which the solar wind and insolation have reduced to a size where radiation pressure exceeds gravitational attraction to the sun. As large as our chosen values were they had a barely discernable effect on the shift of the line maximum as shown in Fig. 1 and on the line shape as shown in Fig. 2. Even at an elongation angle of 10” (evening sky) where the effect is maximal the asymmetry introduced by our choice of fraction of dust involved and speed in the line is too slight to be dis-
1024
D. C. HIRXHI
FIG. 2. LINE PROFILEOF A WEDGE-SHAPEDSOURCE SHOWN AS A SOLIDLINEPEAKINGAT 5184.000AWITHAWIDTH ATHALF MAXIMUMOF 0.900A. The Doppler-shifted line at 10” elongation angle for circularly orbiting dust having a heliocentric distribution of R-' is shown as a dashed line. The line profile at 10” elongation when 20% of the dust has a radial outflow of 8 km s- ’ is shown as a dotted line.
and D. B. BEARD
FIG.~.~AMEAS
FIG.~ ATELONGATION
FIG. ~.SAMEAS FIG.~ ATELONGATION
FIG. 3. LINE PROFILE OF A WEDGE-SHAPED SOURCE LIP (SOLID LINE) WHEN VIEWED AT 10" ELONGATION ANGLE FOR CIRCULARLYORBITINGDUST [n(R) - R-*1(DASHEDLINE)AND \KHENTHEDUSTISSUBJECTEDTORADIATIONPRES~URE(DOTTED LINE). The size distribution of the dust is n(s) N sm4.
cerned. A larger fraction of dust or higher speed of outflow would have to be assumed to result in a discernable asymmetry in the Gegenschein line profile. Such a large efflux of dust would require a large influx to maintain equilibrium-much larger than any source heretofore hypothesized. Except for radial outflow, the morning and evening sky were asymmetric in their effect on the line profile. That is, a shift to the red in the evening sky would be matched by a shift to the blue in the morning sky and vice versa. The radiation pressure has a marked effect on the position of the line. Radiation pressure reduces the effect of gravitational attraction to the sun and consequently the circular speed of the dust particles in their orbits. The effect on line shifts for a few representative elongation angles is illustrated in Figs 3-6 and further indicated in Table 1. The effect of radiation pressure on circularly orbiting dust is to shift the maximum of
FIG.~.SAME AS FIG. 3 ATELONGATION
ANGLE 50".
ANGLE~OO.
ANGLE 170”.
the line to longer wavelengths, except for the Gegenschein (180” elongation) where no Doppler effect is possible in the scattered light. Radiation pressure has no effect on the line half-width. The effect of most interest to us is the particle eccentricity. It has long been assumed that interplanetary dust is mainly in circular orbits because of the Poynting-Robertson Effect, a relativistic effect of radiation pressure which preferentially causes particles with nonzero eccentricity to spiral into the sun faster than those with zero eccentricity (Wyatt and Whipple, 1950). Furthermore, as an example to treat, circularly orbiting dust is easier to use in Doppler shift calculations. We are uncertain of what effect the PoyntinggRobertson Effect has on interplanetary dust for we do not know what the radiation pressure is. The assumptions frequently made are (1) that the dust scattering cross-section is geometrical, ns*, where s is the particle radius, but this assumption cannot be pushed all the way down to where Mie scattering takes
Doppler shifts in zodiacal light
1025
TABLE 1. DISPLACEMENTS AND LINE WIDTHS OF ZODIACAL LIGHT FROM CIRCULARLY ORBITING DUST WITH AND WITHOUT RADIATION PRESSURE
Elongation angle 10 50 90 130 170
Line displacement (A)
With radiation pressure
Half-width (A)
-1.00 -0.14 +0.06 +0.10 +0.03
-0.35 -to.17 +0.20 + 0.20 +0.02
1.17 1.Ol 1.07 1.05 0.97
With radiation pressure 1.18 1.06 1.09 1.00
0.96
Source line was 5184 A with a half-width of 0.900 A. n(R) - R-*; n(S)= sm4.
5182.0 5181.0 5114.0 5185.0 5186.0
FIG. 7.
LINE PROFILE OF A WEDGE-SHAPED SOURCE LINE (SOLID LINE) WHEN VIEWED AT 10" ELONGATION ANGLE FOR CIRCULARLY ORBITING DUST Lo - R~'.Z](DOTED LINE) AND FOR ELLIPTICALLY ORBITING DUST [n(e)-CONSTANT] (DASHED LINE).
over (Beard, 1984) or to smaller wavelengths where Rayleigh scattering is appropriate and (2) that the scattering, depending on s4, decreases even faster than the decrease in particle mass, s3, and attraction to the sun. The dust will, in any case, eventually disappear because of instantaneous evaporation within about four solar radii from the sun or because of erosion by the solar wind (Calbert and Beard, 1972). It is quite possible that for particle sizes where Mie scattering is important the scattering cross-section will be large and the particle will be blown away (Beard, 1984) whenever it becomes eroded to a critical size. Jupiter should have the same effect on orbiting dust that it has on comets. Oort (1950) Opik (1932) and van Woerkem (1948) showed long ago that in being perturbed by Jupiter, comets penetrating within the orbit of Jupiter would have their eccentricity changed so that their orbits would have uniformly distributed aphelia. The effect of a distribution in eccentricity on the Doppler-shifted spectra of the dust is illustrated in Figs 7-10 and Table 2 for a few representative elongation angles. As in the case for radiation pressure the effect of finite eccentricity is to redden the line in the evening sky (shift it to the blue in the morning sky) except in the Gegenschein where there is no shift at all.
FIG. &SAME
o&l.O
AS FIG.~ ATELONGATIONANGLE
5162.0 5181.0 5184.0 5115.0 5186.0
FIG. 9. SAME AS F1c.7 ATELONGATION
o&.O
30".
ANGLE
90”.
5182.0 5181.0 5184.0 5185.0 5116.0
FIG. ~O.SAMEASFIG.
7 ATELONGATIONANGLE
170".
What distinguishes the spectra for elliptically orbiting dust from the spectra for circularly orbiting dust experiencing radiation pressure is the line width, especially in the Gegenschein. The line half-width is significantly broadened at all elongation angles including the Gegenschein where a line broadening from circularly orbiting dust with or without radiation pressure does not occur. Radial outflows might
1026
D. C. HIRSCHI and
D. B. BEARD
TABLE 2. LINE DISPLACEMENTS ANDHALF-WIDTHSFOR ELLIPTICALLYORBITINGDUST Line displacement Elongation angle
n(e) = 6(e) (circle)
10 30 50 70 90 110 130 150 170
-0.96 -0.37 -0.12 0.00 f0.08 +0.12 +0.12 + 0.09 +0.03
Half-width
Constant
I/e -0.79
e
-0.61 -0.29 -0.04 +0.08 +0.18 +0.21 +0.18 f0.12 +0.04
-0.08 +0.14 +0.14 -to.04
-0.46 -0.02 +0.22 +0.21 +0.09
6(e)
l/e
Constant
e
1.24 1.12 1.08 1.06 1.08 1.06 1.00 1.02 1.03
1.63
2.02 1.65 1.41 1.40 1.45 1.59 1.68 1.73 1.73
2.17
1.25 1.30 1.37 1.30
1.59 1.68 1.82 1.96
Source line 518 8, with 0.90 8, half-width. n(R) = Rm I.‘. broaden
the line a little in the Gegenschein
case the significant effect
from
noncircular
than a constant
dependence lations. enhanced
or decreased
dust velocity
eccentricity. and eccentric
orbits
because
the effect of radiation radius
particle
pressure
where the force
Table
pressure.
is shown
twice
pressure
is
FIG. 11. EFFECT OF RADIATION PRESSURE ON CIRCULAR AND ELLIPTICAL ORBITS.
Line profile of a wedged-shaped source line (solid line) for circularly orbiting dust subject to radiation pressure (dotted line) and elliptically orbiting dust subject to radiation pressure (dashed line). Heliocentric distribution is n(R) - R ‘.5 ; particle size distribution is n(s) = s-~; eccentricity distribution is n(e) = constant ; and elongation angle is 30”.
the minithe critical
of radiation
in Figs
o1i%.O 51BP.0 51(13.0 5184.0 51115.0 5116.0
We varied
attraction
The effect of mixing radiation orbits
The line is
of radiation
by making
equals the force of gravitational of elliptical
for the
of the dust overall
of the dust particles
radius
per-
sky) and the line half-width
the speed
lessened by the effect of radiation mum
orbits
is reduced.
to the red as a result
(in the evening
is reduced
was
increased
the effect of eccentricity
in eccentric
more
orbits
as the distribution
pressure
turb the dust motion,
pressure
were used in the calcu-
with increasing
If both radiation
displaced
on eccen-
like an inverse
the effect of elliptical
or diminished
this other
a more eccen-
dependence
distribution
on eccentricity
As expected,
Distributions
of eccentricity,
like a linear
or a less eccentric
but in this
would distinguish
orbits.
function
tric distribution tricity,
asymmetry
pressure to the sun.
with the effect 11 and
12 and
3. %l.O
4. SUMMARY Aside
from
measurements
the line profile informative particles
at small
the Gegenschein. with present gration
effect
zodiacal
Nevertheless,
pressure
it requires against
to make long inte-
a troublesome
the line half-width
separates
of interplanetary
of radiation
of
light is in the line width of because
a
FIG. 12,s~~~ ASFIG. 11 ATELONGATION
clearly
measurement
of
show up in the Gegenschein. of outflowing
in the Gegenschein
quantities
(East and Reay,
1984) as an asymof the line some idea
towards
longer
of the eccentricity
wavelengths.
vations
Once
has been obtained
ments in the Gegenschein
effects
not
If measurable
they will also be apparent
center
of radiation
does
dust appear
metry in the line profile and a displacement
dust from the obscuring which
ANGLE 150".
the most
It is a difficult measurement
equipment
unambiguously
eccentricity
angles
of
to make on the velocity
times of a faint source
background. and
elongation
measurement
causing
of the first moment
5182.0 5111.0 5184.0 5185.0 51116.0
pressure
can
of line displacements
from measure-
an indication be obtained
of the effect from
or first moments
obserat other
Doppler
shifts in zodiacal
1027
light
TABLE 3. LINE DISPLACEMENTS AND HALF-WIDTHSFOR ELLIPTICALLY ORBITING DUST
Elongation angle 30” 150”
WITH
1
VARYING
AMOUNTS
Line displacements 2
+0.09 f0.22
-0.12 +0.16
OF RADIATION
3 -0.29 f0.12
1 1.08 1.08
PPZSSURE
Half-width 2 1.30 1.37
3 1.65 1.73
Minimum dust size is critical radius for column 1, twice critical radius for column 2, and no radiation pressure at all for column 3. Source line 5184 A with 0.90 8, half-width. n(R) = R-l.‘, n(s) = sm4, n(e) = constant.
elongations and comparing them with what might be expected from dust with the observed distribution in eccentricity. Zodiacal light particles are presumably dust released along cometary orbits. If the particles are sufficiently large that radiation pressure is negligible the Poynting-Robertson Effect will not change their orbits and their distribution will resemble that of periodic comets concentrated in the plane of the ecliptic, in direct orbits, and with uniform distribution in aphelia, all as a result of Jupiter’s gravitational perturbation of their orbits. Observations of zodiacal light spectra could check this out. That the particles are all in direct orbits has been confirmed (cf. James and Smeethe, 1970). Acknowledgements-One of us (DBB) would like to thank N. K. Reay for his interest in this work and the S.R.C. of the United Kingdom for early support.
REFERENCES Allen, C. W. (1946) The spectrum of the corona at the eclipse of 1940 October 1. Mon. Not. R. astr. Sot. 106, 137.
Banderman, L. W. and Wolstencraft, R. D. (1969) Interpretations of Doppler shifts in the zodiacal light spectrum. Nature 221, 251. Beard, D. B. (1959) Interplanetary dust distribution. Astrophys. J. 129,496. Beard, D. B. (1984) Infrared coronal polarization and size of interplanetary dust. Astron. Astraphys. 132, 317. Blackwell, D. E. and Ingham, M. F. (1961) Observations of the zodiacal light from a very high altitude station. III The disturbed zodiacal light and corpuscular radiation. Mon. Not. R. astr. Sot. 122, 143. Calbert, R. and Beard, D. B. (1972) The F and K components of the solar corona. Astrophys. J. 176,497. Daehler, M., Mack, J. E., Stoner, J. O., Clarke, D. and Ring, J. (1968) Measurements of the HP profile in the zodiacal light spectrum. Planet. Space Sci. 16, 795. East, I. R. and Reay, N. K. (1984) The motion of inter-
planetary dust particles. 1 Radial velocity measurements on Fraunhofer line profiles in the zodiacal light spectrum. Astron. Astrophys. 139, 512. Fried, J. W. (1978) Doppler shifts in the zodiacal light spectrum. Astron. Astrophys. 68, 259. Haug, V. (1958) Uber die Haufigkeitsverteilung der Bahnelemente bei den interplanetaren Staubteilchen. Z. Astrophys. 44, 71. Hicks, T. R., May, B. H. and Reay, N. K. (1974) An investigation of the motion of zodiacal dust particles. 1 Radial velocity measurements on Fraunhofer line profiles. Mon. Not. R. astr. Sot. 166,439. Hindle, P. H., Reay, N. K. and Ring, J. (1968) H/I radiation in the spectrum of the night sky. Planet. Space Sci. 16, 803. Ingham, M. F. (1963) The profile of an absorption line in the spectrum of the zodiacal light. Mon. Not. R. astr. Sot. 126, 377. James, J. F. (1969) Theoretical Fraunhofer line profiles in the spectrum of the zodiacal light. Mon. Not. R. astr. Sot. 142,45. James, J. F. and Smeethe, M. J. (1970) Motion of the interplanetary dust cloud. Nature 227, 588. Oort, J. H. (1950) The structure of the cloud of comets surrounding the solar system, and a hypothesis concerning its origin. Bull. astr. huts Neth. 11, 91. Opik, E. J. (1932) Note on stellar perturbations of nearby parabolic comets. Proc. Am. Acad. Sci. 67, 169. Reay, N. K. (1969) Zodiacal dust particles : some comments on recent evidence concerning their motion. Nature 224, 54. Reay, N. K. and Ring, J. (1968) Radical velocity measurements and the zodiacal light spectrum. Nature 219, 710. Ring, J., Clarke, D., James, J. F., Daehler, M. and Mack, J. E. (1964) Profile of the H/I line in the spectrum of zodiacal light. Nature 202, 167. Rodriguez, G. L. and Magro, C. S. (1978) The Doppler shift from zodiacal light. Astron. Astrophys. 64, 161, van Woerkem, A. J. (1948) On the origin of comets. Bull. astr. Insts Neth. 10, 1. Vanysek, V. and Harwit, M. (1970) Possibility of a terrestrial component in the Doppler shifted zodiacal light. Nature 225, 1231. Wyatt, S. P. and Whipple, F. L. (1950) The PoyntingRobertson effect on meteor orbits. Astrophys. J. 111, 134.
Space Sci., Vol. 35. No. 8, pp. 1021-1027, in Great Britain.
1987
DOPPLER
6
SHIFTS IN ZODIACAL D. C. HIRSCH1
The University
of Kansas,
00324633/87 $3.00+0.00 1987 Pergamon Journals Ltd.
LIGHT
D. B. BEARD
and
Lawrence,
KS 66045, U.S.A
(Received in final form16 February 1987) Abstract-Doppler shifts in zodiacal light are calculated for various eccentricities, dust sizes, and assumptions about radiation pressure. The purpose is to determine what effects in spectra might be observed which would enhance our understanding of the origin and lifetimes of interplanetary dust. The solar absorption line half-width in the scattered light is a much more difficult measurement to make, but it is a better indication of orbital eccentricity than the first moment of the line can ever be. While the first moment of the line (which is close to the displacement of the maximum of the line) is similar for both orbital eccentricity and radiation pressure and difficult to sort out from other effects like radial outflow, the line half-width tends to broaden most for elliptical orbits at all elongations especially in the Gegenschein (backward scattering angle) where all other effects (excepting radial outflow which reddens the line) cause no change in the original solar line.
1. INTRODUCTION
In orbiting the sun the velocity of interplanetary dust with respect to Earth varies depending on its orbital radius and direction of sight ; if in a noncircular orbit its radial velocity with respect to the sun will not be zero as well. Thus the Fraunhofer absorption lines in the solar spectrum will appear Doppler shifted when the spectrum of the zodiacal light is observed. Observing this is difficult and error prone through Earth’s atmosphere against a background of stellar light sources. Nevertheless, measuring Doppler shifts in the zodiacal light is a means of determining the orbital characteristics of the dust, eccentricity, effect of radiation pressure, and radial outflow, having important implications on dust origin, lifetimes, rate of supply and loss. Ingham (1963) calculated the distortion of the HP line in light scattered from dust in circular orbits observed at 30” elongation. Ring et al. (1964), Daehler et al. (1968) Hindle et al. (1968) and Reay and Ring (1968) observed the H/I line despite the severe interference of emission in Earth’s atmosphere. James (1969) calculated the distortion of the 5 183.6 8, magnesium line for circularly orbiting dust at 30” elongation which avoided the emission feature and was easier to observe. Since the magnesium line was complex and in any case washed out in the scattered spectrum he calculated the distortion of an idealized wedge-shaped line for elongation angles from 10” to 40”. Banderman and Wolstencraft (1969) calculated the displacement of the line from several effects including noncircular orbits with a unique eccentricity 0 < E < 1. Reay (1969) calculated displacements
resulting from radiation pressure effects. Vanysek and Harwit (1970) found that using Mie scattering in place of isotropic scattering did not affect the calculated displacement. James and Smeethe (1970) observed the 5184 magnesium line at elongations between 28” and 40” in the evening and morning sky and concluded that the dust was in direct orbits about the sun and not significantly about Earth. Hicks et al. (1974) observed the first moment of the line at a larger range of elongations and Rodriguez and Magro (1978) calculated line displacements as a function of many distribution parameters. Fried (1978) reported a very controversial observation of the first moment of the line leading to the conclusion that significant amounts of dust were in hyperbolic orbits. East and Reay (1984) observed the first line moment which disagreed significantly from Fried’s observations. For almost 20 y, investigations of Doppler shifts in zodiacal light have concentrated on line displacements or first moments because the detailed spectra are difficult to observe and tedious to calculate. Still, other changes in the spectra, outstanding enough to be observed, are worth studying to see what may be learned. A wedge-shaped source line as used by James (1969) is simpler and yet sufficient a reliable indicator of line broadening in view of the blurring of detail in line form to serve our purpose. We used a wedgeshaped line in studying the effect of radial outflow, radiation pressure, and orbital eccentricity of the dust on line width and line displacement at elongation angles from 10” to 170”. We find that in the Gegenschein (elongation angle 180’) the line width is significantly broadened (of the order twice as wide) by 1021
D. C. HIRSCHI and D. B. BEARD
1022
introducing noncircular orbits, not broadened at all by radiation pressure effects, and not broadened much but made asymmetrical by a significant radial outflow. The three perturbations affect the line in different amounts at other elongation angles, but the various effects can be separated out by observations at all elongation angles.
2.
CALCULATION
LINE
A particle in an elliptical orbit in the ecliptic plane about the sun has a speed towards or away from Earth of V=
V,cos++V,sint+-V,sin.s
(1)
where VR is the particle’s radial velocity from the sun, V, is the particle’s azimuthal velocity, V, is Earth’s azimuthal velocity, $ is the angle between Earth and sun at the particle, and E is the elongation angle, the angle between sun and particle at Earth. From the law of sines sin*
= sine/R
(2)
where R is the heliocentric distance of the particle expressed in astronomical units. For a particle in an elliptical orbit with eccentricity e and semi-major axis a and Earth in a circular orbit V, = (K/RM)“*(2-R/a-((a/R)(l
g
of (3) with respect to e is R/u
= (K/MR)“*(2-(a/R)(l
+cos$)ue/R
-e2))‘12(1
-(Ku/M)e(l-e2)-“2sinsR-Z.
(4)
As Haug (1958) has shown, the number of particles between R and R+ dR is given by
OF A DOPPLER SHIFl- AFFECTED
SOLAR ABSORPTION
derivative
-e*))‘l*
V, = (Ka/M)“*(l-e2)“*RP’ J’, = (KIM) ‘1’ = 30km s- ’
n(R)dR
- (R%V&’
dR =
(~/R)~~‘~(2-R/u-((a/R)(l-e*))-“~dR where z is the orbital period. and e
For a distribution
n(u)n(e)dude
n(R) =
ss (uR)“‘(2-R/u-(u/R)(l
The total intensity of scattered sight A is given by I=
in a
-e*))‘/*’
light along the line of
s
n(R)dA R*
n(u) n(e) da de dA (uR)3~2R2(2-R/u-(u/R)(1-e2))‘~2’
(5)
The variable A is the particle to Earth distance and from the law of cosines can be expressed in terms of Randa A=cossk(R*-sin*&)‘/*
where K is the gravitational constant and M is the mass of the sun ; a is in astronomical units. Scattered light arriving at Earth experiences two Doppler shifts, the first because of the radial velocity of the particle, the second because of the velocity component of the dust along the line of sight with respect to Earth. The Doppler shift in wavelength is given approximately by AI = (v/c),%.
from which we may find the convenient dA=
substitution
(l-sin*~/R*))‘~*dR.
(6)
The total intensity from a source J(I) at a wavelength I(1 + D/c) is given by (4), (5) and (6) Z(n( 1+ D/c)) = n(u) n(e) $.Z(I)
da dR
Let D = v = (An/@.
e can be obtained from (3) as a function of a, R, D and a. If J(I) is a wedge type source as a function of 1
Then D=
V,(l+cos$)+VOsinsR-l-V~sins
D = (K/MR)“*(2-R/u-(a/R)(l + (Ku/M) ‘/2(1- e’) ‘P sin ER-
-e’))“*(l
+cost,b)
’ - (K/M) “‘sin E (3)
where (3) can be rearranged to express the eccentricity e in terms of independent variables R, a, D and E. The
J(n) = Z,(l-llzo-11/0.9),
Lo-O.9
< a < &+0.9
where I, is a constant, the depth of the line at 1,. The light intensity of I.,( 1 + d/c), where d&/c is any given wavelength displacement from the center of the line I,, is
Doppler
shifts in zodiacal
1023
light
Z(d) = n(a) n(e) Z,(l - ]I, -L]O.9) ’ (ur)312R2(2-R/a-((a/R)(l-eeZ))‘~2(1-sin2~/R2)’12 x $dudRdi where e =
(7)
f (a, R, D, E) ; de/dD is given in (4) ; and D = &(l +d/c)-1.
-75
(8) FIG.
Similarly for circularly orbiting dust we change from the independent variable R (in place of e for elliptical orbits) to D and obtain for a wedge-shaped source line Z(d) =
2cz, 3(K/M)“* X
1.
so
MEAN
loo
150
200
250
SHIFT OF THE DOPPLER-SHIFTED
DUE TO AN OUTWARD
RADIAL
FLQW
300
350
LINE PROFILE
OF PARTICLES.
The solid line is for a heliocentric distribution, n(R) - R - I ’ ofcircularly orbiting dust shown for comparison. The dashed line illustrates the difference when 20% of the dust has a radial outflow of 8 km SC’. Shift in kilometers per second towards Earth vs elongation angle.
sin E
s
A~+‘.9R”*n(R)(l-I&-1//0.9)
,+o P
i(l-sin*~/R*)“*
3. RESULTS
d2
(9)
where R is given by R = (l+M”*(K”*sin~))‘D))*~~
(10)
and D by (8). For comparison purposes since so much study has been given to circular orbits in the literature, we show the predictions for circularly orbiting dust in all tables and graphs. For circular orbits of particles small enough for radiation pressure to affect them the gravitational attraction of the sun is weakened by an amount tic, where tl is the radiation pressure and c is the velocity of light. Hence, the velocity of the dust becomes V, = ((K/M)“*--c)/(R)“*; (10) must be replaced R = {[(k/M)“*
LL
0
by
-ctc]/[(K/M)‘/*
For that part of the dust propelled by an excess of radiation pressure D = V,(l +cos$)-
+ D cosec E]} 2/3. away from the sun (3) would become
V, sina
= V,[l+(l-sin*~/R*)“‘]-(K/M)“*sina and n(R) = N0Vi’Rm2 from which the line profile is readily obtained the circularly orbiting dust with D substituted as the independent variable : R = V,sin&{2VR[D+(K/A4)‘12sina]
as for for R
In carrying out the calculations we discovered that the results depended very little on n(R) which has been variously reported as an inverse power of R between one and two from interpretations of zodiacal light observations (e.g. Allen, 1946 ; Beard, 1959; Blackwell and Ingham, 1961; Calbert and Beard, 1972). We used a distribution of Y- I,5 and r-* interchangeably. We used in all cases a wedge-shaped source line of 5 184 and 0.9 A width at half maximum. The details of a more realistic Fraunhofer absorption line will be washed out in being scattered from an orbiting dust particle and in any case the detailed line shape is unobservable with current observational capability. For the distribution in eccentricity we used n(e) - e - ‘, constant, and e with increasing effect. The particle size distribution we chose emphasized radiation pressure effects with n(s) - sp4 and sm6 both of which amounted to a delta-function at the minimum particle size assumed. Radial outflow has been suggested by East and Reay (1984) to account for the asymmetric shift of the line maximum in going from the evening to the morning sky. As an example of this we assumed a radial outflow of 20% of the dust of 8 km s-‘. Presumably the source of this dust is small particles which the solar wind and insolation have reduced to a size where radiation pressure exceeds gravitational attraction to the sun. As large as our chosen values were they had a barely discernable effect on the shift of the line maximum as shown in Fig. 1 and on the line shape as shown in Fig. 2. Even at an elongation angle of 10” (evening sky) where the effect is maximal the asymmetry introduced by our choice of fraction of dust involved and speed in the line is too slight to be dis-
1024
D. C. HIRXHI
FIG. 2. LINE PROFILEOF A WEDGE-SHAPEDSOURCE SHOWN AS A SOLIDLINEPEAKINGAT 5184.000AWITHAWIDTH ATHALF MAXIMUMOF 0.900A. The Doppler-shifted line at 10” elongation angle for circularly orbiting dust having a heliocentric distribution of R-' is shown as a dashed line. The line profile at 10” elongation when 20% of the dust has a radial outflow of 8 km s- ’ is shown as a dotted line.
and D. B. BEARD
FIG.~.~AMEAS
FIG.~ ATELONGATION
FIG. ~.SAMEAS FIG.~ ATELONGATION
FIG. 3. LINE PROFILE OF A WEDGE-SHAPED SOURCE LIP (SOLID LINE) WHEN VIEWED AT 10" ELONGATION ANGLE FOR CIRCULARLYORBITINGDUST [n(R) - R-*1(DASHEDLINE)AND \KHENTHEDUSTISSUBJECTEDTORADIATIONPRES~URE(DOTTED LINE). The size distribution of the dust is n(s) N sm4.
cerned. A larger fraction of dust or higher speed of outflow would have to be assumed to result in a discernable asymmetry in the Gegenschein line profile. Such a large efflux of dust would require a large influx to maintain equilibrium-much larger than any source heretofore hypothesized. Except for radial outflow, the morning and evening sky were asymmetric in their effect on the line profile. That is, a shift to the red in the evening sky would be matched by a shift to the blue in the morning sky and vice versa. The radiation pressure has a marked effect on the position of the line. Radiation pressure reduces the effect of gravitational attraction to the sun and consequently the circular speed of the dust particles in their orbits. The effect on line shifts for a few representative elongation angles is illustrated in Figs 3-6 and further indicated in Table 1. The effect of radiation pressure on circularly orbiting dust is to shift the maximum of
FIG.~.SAME AS FIG. 3 ATELONGATION
ANGLE 50".
ANGLE~OO.
ANGLE 170”.
the line to longer wavelengths, except for the Gegenschein (180” elongation) where no Doppler effect is possible in the scattered light. Radiation pressure has no effect on the line half-width. The effect of most interest to us is the particle eccentricity. It has long been assumed that interplanetary dust is mainly in circular orbits because of the Poynting-Robertson Effect, a relativistic effect of radiation pressure which preferentially causes particles with nonzero eccentricity to spiral into the sun faster than those with zero eccentricity (Wyatt and Whipple, 1950). Furthermore, as an example to treat, circularly orbiting dust is easier to use in Doppler shift calculations. We are uncertain of what effect the PoyntinggRobertson Effect has on interplanetary dust for we do not know what the radiation pressure is. The assumptions frequently made are (1) that the dust scattering cross-section is geometrical, ns*, where s is the particle radius, but this assumption cannot be pushed all the way down to where Mie scattering takes
Doppler shifts in zodiacal light
1025
TABLE 1. DISPLACEMENTS AND LINE WIDTHS OF ZODIACAL LIGHT FROM CIRCULARLY ORBITING DUST WITH AND WITHOUT RADIATION PRESSURE
Elongation angle 10 50 90 130 170
Line displacement (A)
With radiation pressure
Half-width (A)
-1.00 -0.14 +0.06 +0.10 +0.03
-0.35 -to.17 +0.20 + 0.20 +0.02
1.17 1.Ol 1.07 1.05 0.97
With radiation pressure 1.18 1.06 1.09 1.00
0.96
Source line was 5184 A with a half-width of 0.900 A. n(R) - R-*; n(S)= sm4.
5182.0 5181.0 5114.0 5185.0 5186.0
FIG. 7.
LINE PROFILE OF A WEDGE-SHAPED SOURCE LINE (SOLID LINE) WHEN VIEWED AT 10" ELONGATION ANGLE FOR CIRCULARLY ORBITING DUST Lo - R~'.Z](DOTED LINE) AND FOR ELLIPTICALLY ORBITING DUST [n(e)-CONSTANT] (DASHED LINE).
over (Beard, 1984) or to smaller wavelengths where Rayleigh scattering is appropriate and (2) that the scattering, depending on s4, decreases even faster than the decrease in particle mass, s3, and attraction to the sun. The dust will, in any case, eventually disappear because of instantaneous evaporation within about four solar radii from the sun or because of erosion by the solar wind (Calbert and Beard, 1972). It is quite possible that for particle sizes where Mie scattering is important the scattering cross-section will be large and the particle will be blown away (Beard, 1984) whenever it becomes eroded to a critical size. Jupiter should have the same effect on orbiting dust that it has on comets. Oort (1950) Opik (1932) and van Woerkem (1948) showed long ago that in being perturbed by Jupiter, comets penetrating within the orbit of Jupiter would have their eccentricity changed so that their orbits would have uniformly distributed aphelia. The effect of a distribution in eccentricity on the Doppler-shifted spectra of the dust is illustrated in Figs 7-10 and Table 2 for a few representative elongation angles. As in the case for radiation pressure the effect of finite eccentricity is to redden the line in the evening sky (shift it to the blue in the morning sky) except in the Gegenschein where there is no shift at all.
FIG. &SAME
o&l.O
AS FIG.~ ATELONGATIONANGLE
5162.0 5181.0 5184.0 5115.0 5186.0
FIG. 9. SAME AS F1c.7 ATELONGATION
o&.O
30".
ANGLE
90”.
5182.0 5181.0 5184.0 5185.0 5116.0
FIG. ~O.SAMEASFIG.
7 ATELONGATIONANGLE
170".
What distinguishes the spectra for elliptically orbiting dust from the spectra for circularly orbiting dust experiencing radiation pressure is the line width, especially in the Gegenschein. The line half-width is significantly broadened at all elongation angles including the Gegenschein where a line broadening from circularly orbiting dust with or without radiation pressure does not occur. Radial outflows might
1026
D. C. HIRSCHI and
D. B. BEARD
TABLE 2. LINE DISPLACEMENTS ANDHALF-WIDTHSFOR ELLIPTICALLYORBITINGDUST Line displacement Elongation angle
n(e) = 6(e) (circle)
10 30 50 70 90 110 130 150 170
-0.96 -0.37 -0.12 0.00 f0.08 +0.12 +0.12 + 0.09 +0.03
Half-width
Constant
I/e -0.79
e
-0.61 -0.29 -0.04 +0.08 +0.18 +0.21 +0.18 f0.12 +0.04
-0.08 +0.14 +0.14 -to.04
-0.46 -0.02 +0.22 +0.21 +0.09
6(e)
l/e
Constant
e
1.24 1.12 1.08 1.06 1.08 1.06 1.00 1.02 1.03
1.63
2.02 1.65 1.41 1.40 1.45 1.59 1.68 1.73 1.73
2.17
1.25 1.30 1.37 1.30
1.59 1.68 1.82 1.96
Source line 518 8, with 0.90 8, half-width. n(R) = Rm I.‘. broaden
the line a little in the Gegenschein
case the significant effect
from
noncircular
than a constant
dependence lations. enhanced
or decreased
dust velocity
eccentricity. and eccentric
orbits
because
the effect of radiation radius
particle
pressure
where the force
Table
pressure.
is shown
twice
pressure
is
FIG. 11. EFFECT OF RADIATION PRESSURE ON CIRCULAR AND ELLIPTICAL ORBITS.
Line profile of a wedged-shaped source line (solid line) for circularly orbiting dust subject to radiation pressure (dotted line) and elliptically orbiting dust subject to radiation pressure (dashed line). Heliocentric distribution is n(R) - R ‘.5 ; particle size distribution is n(s) = s-~; eccentricity distribution is n(e) = constant ; and elongation angle is 30”.
the minithe critical
of radiation
in Figs
o1i%.O 51BP.0 51(13.0 5184.0 51115.0 5116.0
We varied
attraction
The effect of mixing radiation orbits
The line is
of radiation
by making
equals the force of gravitational of elliptical
for the
of the dust overall
of the dust particles
radius
per-
sky) and the line half-width
the speed
lessened by the effect of radiation mum
orbits
is reduced.
to the red as a result
(in the evening
is reduced
was
increased
the effect of eccentricity
in eccentric
more
orbits
as the distribution
pressure
turb the dust motion,
pressure
were used in the calcu-
with increasing
If both radiation
displaced
on eccen-
like an inverse
the effect of elliptical
or diminished
this other
a more eccen-
dependence
distribution
on eccentricity
As expected,
Distributions
of eccentricity,
like a linear
or a less eccentric
but in this
would distinguish
orbits.
function
tric distribution tricity,
asymmetry
pressure to the sun.
with the effect 11 and
12 and
3. %l.O
4. SUMMARY Aside
from
measurements
the line profile informative particles
at small
the Gegenschein. with present gration
effect
zodiacal
Nevertheless,
pressure
it requires against
to make long inte-
a troublesome
the line half-width
separates
of interplanetary
of radiation
of
light is in the line width of because
a
FIG. 12,s~~~ ASFIG. 11 ATELONGATION
clearly
measurement
of
show up in the Gegenschein. of outflowing
in the Gegenschein
quantities
(East and Reay,
1984) as an asymof the line some idea
towards
longer
of the eccentricity
wavelengths.
vations
Once
has been obtained
ments in the Gegenschein
effects
not
If measurable
they will also be apparent
center
of radiation
does
dust appear
metry in the line profile and a displacement
dust from the obscuring which
ANGLE 150".
the most
It is a difficult measurement
equipment
unambiguously
eccentricity
angles
of
to make on the velocity
times of a faint source
background. and
elongation
measurement
causing
of the first moment
5182.0 5111.0 5184.0 5185.0 51116.0
pressure
can
of line displacements
from measure-
an indication be obtained
of the effect from
or first moments
obserat other
Doppler
shifts in zodiacal
1027
light
TABLE 3. LINE DISPLACEMENTS AND HALF-WIDTHSFOR ELLIPTICALLY ORBITING DUST
Elongation angle 30” 150”
WITH
1
VARYING
AMOUNTS
Line displacements 2
+0.09 f0.22
-0.12 +0.16
OF RADIATION
3 -0.29 f0.12
1 1.08 1.08
PPZSSURE
Half-width 2 1.30 1.37
3 1.65 1.73
Minimum dust size is critical radius for column 1, twice critical radius for column 2, and no radiation pressure at all for column 3. Source line 5184 A with 0.90 8, half-width. n(R) = R-l.‘, n(s) = sm4, n(e) = constant.
elongations and comparing them with what might be expected from dust with the observed distribution in eccentricity. Zodiacal light particles are presumably dust released along cometary orbits. If the particles are sufficiently large that radiation pressure is negligible the Poynting-Robertson Effect will not change their orbits and their distribution will resemble that of periodic comets concentrated in the plane of the ecliptic, in direct orbits, and with uniform distribution in aphelia, all as a result of Jupiter’s gravitational perturbation of their orbits. Observations of zodiacal light spectra could check this out. That the particles are all in direct orbits has been confirmed (cf. James and Smeethe, 1970). Acknowledgements-One of us (DBB) would like to thank N. K. Reay for his interest in this work and the S.R.C. of the United Kingdom for early support.
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