A&A 452, 701-707 (2006)
T. Minato1 - M. Köhler1 - H. Kimura2 - I. Mann1 - T. Yamamoto2
1 - Institut für Planetologie, Westfälische Wilhelms-Universität, Wilhelm-Klemm-Str. 10, 48149 Münster, Germany
2 - Institute of Low Temperature Science, Hokkaido University, Kita-ku Kita-19 Nishi-8, Sapporo 060-0819, Japan
Received 25 December 2005 / Accepted 27 January 2006
Aims. Dust-plasma interaction may play a dominant role in the dynamics of dust particles around young main-sequence stars. Circumstellar dust is expected to be an aggregate consisting of small grains. The momentum transfer cross section (MTCS) for an aggregate is a key quantity for determining the lifetimes of circumstellar dust disks.
Methods. We formulate the MTCS of fluffy dust aggregates and propose an algorithm for computing the MTCS. We compare the magnitude of the plasma and photon Poynting-Roberson effects (PR effects).
Results. We find an empirical formula that approximates numerical results well for the MTCSs. A comparison of the magnitudes of the PR effects shows that the lifetime of debris dust around young main-sequence stars is shorter by orders of magnitude than the lifetime estimated by considering the photon PR effect alone. Brief discussion is given on the plasma PR effect for dust debris disks around young main-sequence stars.
Key words: stars: winds, outflows - solar wind - meteors, meteroids - interplanetary medium - Kuiper Belt - circumstellar matter
Dust particles orbiting a star spiral into the star owing to the drag by stellar photons that are absorbed and scattered by the dust. This is called the Poynting-Robertson effect (PR effect). Stellar winds striking dust also decrease its orbital angular momentum, which we call the plasma PR effect analogous to the photon PR effect. Both photon and plasma PR effects play a role in limiting the lifetime of dust around main-sequence stars. An evaluation of the plasma PR lifetimes requires knowledge of interaction between dust and stellar wind ions, in other words, the momentum transfer cross section (MTCS). Stellar wind ions are not perfectly absorbed by a dust particle, but penetrate through it if their momentum is sufficiently large. This reduces the MTCS of the dust as demonstrated by Minato et al. (2004) who take penetration of stellar wind ions into account for a study of the plasma PR effect. Clearly, realistic estimates of the MTCS are the key to better understanding the lifetimes of dust particles (see Plavchan et al. 2005).
In the previous studies of the plasma PR effect, dust is assumed for simplicity to be a spherical grain (Gustafson 1994; Burns et al. 1979; Mukai & Yamamoto 1982; Fahr et al. 1995; Minato et al. 2004). However, cometary dust is often assumed to be fluffy aggregates consisting of submicron grains. These aggregates are in fact observed as interplanetary dust particles (e.g., Brownlee 1978). One expects that dust in the disks around main-sequence stars also has a fluffy structure, which increases the geometric cross sections of particles compared with a spherical dust of the same volume. In consequence, this may increase their MTCSs. If this is the case, the orbital angular momentum of dust aggregates is more efficiently lost compared to that of spherical dust particles. The importance of the plasma PR effect also depends on the stellar-wind parameters. Recently, Wood et al. (2002) have shown that the stellar winds of solar-like stars become weaker with the age of the star. Therefore, the plasma PR effect may even play a dominant role in determining the lifetimes of dust particles around young main-sequence stars (Mukai et al. 2004).
In this paper, we study the plasma PR effect for dust aggregates and its consequences on their lifetimes in disks around main-sequence stars. We first derive the MTCSs for dust aggregates and then calculate the stellar wind pressure and plasma PR drag forces to compare with the radiation pressure and photon PR drag forces. We finally discuss the results in terms of lifetimes for dust around main-sequence stars with a variety of ages.
|Figure 1: A schematic view of a dust aggregate irradiated by stellar wind ions. The shadow on the xy plane expresses the geometrical cross section of the aggregate.|
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It is easy to see that in the case of a spherical grain
Eq. (6) leads to the MTCS given
in our previous study (Minato et al. 2004). We take the cylindrical
coordinates in which the origin is placed at the center of the grain and
the z axis is parallel to the wind velocity. The thickness of the
grain along a path of a stellar wind ion incident with impact parameter
is given by
|Figure 2: Models of dust aggregates.|
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We take two kinds of model dust aggregates BPCAs and BCCAs
(e.g., Mukai et al. 1992). Typical dust aggregates of BPCA and BCCA
are shown in Fig. 2.
The BPCAs are rather compact aggregates formed via ballistic
particle-cluster aggregations, whereas the BCCAs are sparse ones formed
via ballistic cluster-cluster aggregations.
The fractal dimension is
for the BPCAs
for the BCCAs (Meakin 1984; Mukai et al. 1992).
For BCCAs and BPCAs composed of a large number of monomers (),
their geometrical cross sections
are related to the
fractal dimension as
We calculate the MTCSs averaged over aggregate orientations, namely,
The numerical procedure is as follows:
|Figure 3: Momentum transfer cross section averaged over aggregate orientations, , for ballistic cluster-cluster aggregates (BCCAs; left) and ballistic particle-cluster aggregates (BPCAs; right) versus the number of monomers N. The vertical axis shows divided by the geometrical cross section of a sphere of the same volume as the aggregate. The lines indicate the cross section given by the empirical formula Eq. (15). In the figure, is the radius of volume-equivalent spheres and X=2a/l(p0) is the size parameter of monomers, where a is the monomer radius and l(p0) is the projected range of stellar wind ions.|
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Figure 3 shows the MTCSs averaged over orientations, , calculated numerically for the BPCAs and the BCCAs as a function of the number of monomers N. The cross section is normalized by the geometrical cross sections of spheres with the same volumes as the aggregates, where . In the cross section of the aggregate, two effects appear: (1) the geometrical cross section larger than that of a compact sphere; (2) the penetration of stellar wind ions through the aggregates (Minato et al. 2004). The first effect increases cross section , whereas the second one decreases . For an aggregate composed of a small number of monomers (i.e., small N) of small radius (i.e., a/l(p0) < 1), is less than the geometrical cross section of a sphere of the same volume as is seen from Fig. 3. This is because the penetration effect dominates the increase in the geometrical cross section. For an aggregate of high N, on the other hand, becomes larger than because of the effect in the increase in the geometrical cross section. For , the ratio tends to a constant of for BPCAs, while tends to be proportional to N1/3 for BCCAs. The behavior of for large N can be understood in the following way. For , the cross section is approximated to be the geometrical cross section, i.e. , where D is fractal dimension of the aggregate. Since , one has for . Thus, for BPCAs for which , and for BCCAs for which .
We find that
are approximated well by an empirical
formula similar to Eq. (9):
We consider solar wind as an example of stellar winds. We use the same solar-wind parameters as those used in Minato et al. (2004): for the radial velocity, with for the radial density distribution, and the number ratio for protons and particles to be 1 : 0.05, while neglecting heavier particles (Neugebauer 2001; Phillips et al. 1995). These parameter values reflect ecliptic conditions, which are appropriate for the majority of interplanetary dust.
We take silicate (MgFeSiO4) and carbon as plausible dust materials. The data of the stopping power of carbon used here are the same as in Minato et al. (2004). For silicate, for which there are no data, we calculate the stopping power with the use of the Bragg's rule (Bragg & Kleeman 1905). This rule estimates the stopping power as a sum of the stopping powers of atoms composing the solid. The error of the Bragg's rule for SiO2 is less than 50% for protons of energies 1-10 keV (Bauer et al. 1998). Table 1 lists the projected ranges and stopping cross sections of protons and -particles with initial kinetic energy 1 keV/amu for silicate (MgFeSiO4) and carbon.
Table 1: The projected range li and the stopping cross section S(1)i of protons () and -particles () with initial kinetic energy 1 keV/amu for silicate (MgFeSiO4) and carbon.
|Figure 4: The ratio of the solar wind pressure to gravitational attraction by the sun for aggregates: a) silicate BCCAs (ballistic cluster-cluster aggregates), b) silicate BPCAs (ballistic particle-cluster aggregates), c) carbon BCCAs, d) carbon BPCAs. The solid and dashed curves are the ratios with the monomer radius of 0.1 and 0.01 m, respectively. The dotted line indicates the ratio for a spherical dust taking into account the effect of passage of incident ions through the dust grains. Here, is the radius of volume-equivalent spheres.|
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Figure 4 shows the ratio of solar wind pressure to gravitational attraction acting on dust particles around the sun. The values are much smaller than unity and thus the solar wind pressure is negligibly small compared to the gravitational attraction. For a star of mass-loss rate , the -value is scaled by multiplying , if the energy distribution of the stellar wind is similar to that of the sun, where is the mass loss rate of the sun. The stellar wind pressure is ineffective unless , which is implausible even for young solar-like stars (Wood et al. 2005).
|Figure 5: Ratio of the plasma PR drag to the photon PR drag for silicate and carbon aggregates consisting of 0.1 m-radius monomers. The right axis indicates the ratio of the stellar wind pressure to the radiation pressure. Solid, dash-dotted, and dotted lines are the results for ballistic particle-cluster aggregates (BPCAs), ballistic cluster-cluster aggregates (BCCAs), and spheres, respectively.|
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|Figure 6: Falling time for ballistic particle-cluster aggregates (BPCAs; solid lines) and ballistic cluster-cluster aggregates (BCCA; dash-dotted lines) and for spherical particles (dotted lines) from 50 AU from a central star. The left and right axes indicate the falling times for the mass loss rates of 1 and , respectively. The dashed line is the falling time for spherical particles due to the photon PR effect under the solar luminosity of (see the left axis). Note that the range of luminosities of young main-sequence stars is limited around .|
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Figure 6 shows the time required for dust aggregates consisting of 0.1 m monomers at 50 AU from the central star to fall to the star due to the plasma PR effect. The falling time for a spherical grain is shown for comparison. Also shown is the falling time for a spherical grain due to the photon PR effect. Figure 6 shows that the falling time is shorter for the aggregates than that for a spherical particle because the ratio is higher for the aggregates than for the spherical grains. It should be noted that, for stars with high mass loss rate and luminosity of such as young main-sequence stars, the falling time due to the plasma PR drag is shorter by orders of magnitude than the one caused by the photon PR drag for any size. This is more pronounced if the monomer size is smaller than 0.1 m.
In the zodiacal dust cloud of our own solar system, mutual collisions have a negligible effect compared to the photon PR effect, except for very large grains of mass greater than 10-8 kg (Grün et al. 1985). Under the current solar-wind condition, the plasma PR effect is known to be comparable to the photon PR effect (Mukai & Yamamoto 1982). We have shown that the fluffy structure of aggregates enhances the relative importance of the plasma PR drag. This drag becomes even stronger than the photon PR drag for the aggregates composed of small monomers ( m). For the young sun, the mass loss rate is estimated to be two orders of magnitude higher than for the present sun, while the luminosity does not change significantly with time(Wood et al. 2002, 2005). As a consequence, the plasma PR effect should have been the dominant process in removing dust particles orbiting the young sun (e.g., Gyr). The result of the short falling time due to the plasma PR effect requires a high dust production rate to maintain the zodiacal dust cloud around the young sun.
We make a sketch for the dominant regions of the plasma PR drag, the photon PR drag, and the mutual collisions in terms of the mass loss rate of a star and the total dust mass in its circumstellar disk (see Fig. B.1). In massive debris disks like Vega-type stars, collisional destruction plays a major role in removing dust particles from the disks, owing to their high concentrations in space. The collisional lifetimes for dust in the observed debris disks are typically four orders of magnitude shorter than their photon PR lifetimes (Dominik & Decin 2003). The plasma PR effect is also negligible for the removal of dust particles in the massive disks, unless the flux ratio of the stellar wind to the stellar radiation exceeds ten thousand times the ratio for the sun. Observations show that the total dust mass decreases with the ages of stars, thus dust particles are less frequently destroyed by collisions at the late stages of the disk evolution (e.g., Metchev et al. 2004). Because collisional lifetimes are inversely proportional to the total disk dust mass (see Eq. (B.3)), the total dust mass will reach a value in which the collisional destruction no longer plays a role compared to the photon PR effects. Even at such late stages of the disk evolution, the plasma PR effect might still be insignificant because the mass loss rate of the star also decreases with the stellar age. However, there should be main-sequence stars whose disk luminosities are so low that observations with currently available instruments have not been successful enough so far to detect. For low-mass disks around young stars, we expect the near-future detection of such faint dust clouds where the collisional destruction is insignificant and the plasma PR effect may dominate the dust removal process.
We would like to thank Munetaka Ueno and Brian E. Wood for useful communications. This research is supported by the German Aerospace Center DLR (Deutschen Zentrum für Luft- und Raumfahrt) under the projects "Kosmischer Staub: Der Kreislauf interstellarer und interplanetarer Materie'' (RD-RX-50 OO 0101-ZA), "Mikro-Impakte'' (RD-RX-50 OO 0203), "Rosetta: MIDAS, MIRO, MUPUS'' (RD-RX-50 QP 0403), and by MEXT Japan, Grant-in-Aid for Scientific Research on Priority Areas, "Development of Extra-Solar Planetary Science'' (#16077203).
For practical purposes, we give empirical formulae for ,
geometrical cross sections of the BPCAs and BCCAs averaged over their
Figure A.1 shows
for three aggregates formed numerically by
random ballistic particle-cluster or cluster-cluster collisions.
By making a fit to 's, we obtain the following empirical formula for .
For BPCAs, the
is approximated by
|Figure A.1: Geometrical cross sections of aggregates, , relative to the sum of monomer cross sections, . Mean values are plotted by symbols, their standard deviations are given by error bars, and fitting curves are indicated by dotted lines. Squares: ballistic cluster-cluster aggregates (BCCAs); Circles: ballistic particle-cluster aggregates (BPCAs).|
|Figure B.1: Dust-removal processes in the phase space for the mass loss rate of a star and the total dust mass of its circumstellar disk . The dominant process is determined by the shortest lifetime among Eqs. (B.1)-(B.3).|
The dominant process is determined by the shortest lifetime among Eqs. (B.1)-(B.3). Figure B.1 illustrates the dominant regions of the above-mentioned processes in the phase space for the total dust mass and the stellar mass loss rate. The boundaries of the three dominant processes are determined by equating Eqs. (B.1)-(B.3). Note that the boundaries are independent of and , although the absolute values of , , and depend on them. Nominal values such as r=50 AU and the assumption of are used to determine the boundaries among the processes. The figure shows the dominant process for dust around the sun and Eridani whose mass loss rate is given by Wood et al. (2002, 2005). The total mass of dust in the Kuiper belt is estimated by Backman et al. (1995) and by Yamamoto & Mukai (1998), who include production of dust particles by interstellar dust impacts on Kuiper belt objects. When the sun was young at its t=0.7 Gyr age, both the mass loss rate and the total dust mass were higher than the current values as expected from the mass loss history of solar-like stars ( ) given by Wood et al. (2005) and the collisional evolution ( ) or PR drag evolution ( ) of dust given by Dominik & Decin (2003). The total dust mass of Eri varies from one estimate to another within the range for typical debris disks (Schütz et al. 2004; Greaves et al. 2005; Sheret et al. 2004; Greaves et al. 1998). Also plotted in Fig. B.1 as a dashed line is the detection threshold of currently available top-class instruments (see, e.g., Metchev et al. 2004; Beichman et al. 2005).