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Torsten L¨ohne and Alexander V Krivov Size Distribution Abstract by sdfsb346f


Torsten L¨ohne and Alexander V Krivov Size Distribution Abstract

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									                                                                                                                                        Size and Spatial Distribution of Dust in Debris Disks
                                                                                                                                                                                                           Torsten Lohne and Alexander V. Krivov
                                                                                                                                                                                                               a                                            a             a
                                                                                                                                                                     Astrophysikalisches Institut und Universit¨ ts-Sternwarte, Friedrich-Schiller-Universit¨ t, Schillerg¨ sschen 2–3, 07745 Jena, Germany
                   since 1558                                                                                                                                                          ,,

                                                                                                                                            Size Distribution                                                                     Abstract                                                              Radial Distribution
                                                                                                                                                                                                                                                                                                                                                                                                                                                             2x105 a
                                                       Total                                                                                                                                                                                                                                                                                                                                                                                                 6x105 a
                                                                                                                                                                                                                                                                                                                                                                                                                                                             2x106 a
                                                                     10-4                                                                                                      After the early protoplanetary phase, circumstellar disks have lost much of their material or                                                                                                                                                                                 6x106 a

                                                                                  Geom. optical depth / size decade
                                         Normal hyperbolas                                                                                                                                                                                                                                               The radial profile of the normal optical depth                                   10-4
                                      Anomalous hyperbolas                                                                                                                     collected it into planets. The moderate amount of solids in these optically thin, gas-poor disks                                                                                                                                                                              2x107 a
                                           Dohnanyi (s-3.5)
                                                                                                                      As was known before, the size distribution                                                                                                                                         of the disk shows a convergence in time to-                                                                                                         6x107 a

                                                                                                                                                                                                                                                                                                                                                              Geom. optical depth
                                                                                                                      in debris disks evolves towards a wavy pat-              still covers a broad range of sizes from planets down to small dust grains. The main force                                wards a quasi-steady state, which is estab-                                                                                                         2x108 a
                                                                                                                      tern. This pattern is due to the stellar radi-           acting upon this circumstellar material and holding it on Keplerian orbits is the stellar gravity.                        lished after around 10 Ma. The final slope of
                                                                                                                      ation pressure, which poses a natural lower              Small grains below one hundred microns in size are additionally affected by stellar radiation,                            the outer part is ≈ -1.5. On timescales longer
                                                                     10-6                                             cut-off to the size of grains on bound orbits.                                                                                                                                     than 100 Ma the optical depth decreases as
                                                                                                                      Some minor peaks appear. The main contri-                weakening or overpowering the gravitational bond. Due to high relative velocities, mutual
                                                                                                                                                                                                                                                                                                         the disk’s total mass does. The radial position
                                                                                                                                                                               collisions among orbiting bodies are usually destructive. The material undergoes a collisional                                                                                                            10-5
                                                                                                                      bution to the total optical depth comes from                                                                                                                                       of the peak moves outward because the inner
                                                                                                                      grains which are 2–3 times larger than this              cascade from parent bodies to ever-smaller debris until it is lost to evaporation close to the                            material is depleted more rapidly.
                                                                                                                      limit.                                                   star or ejection out of the system.

                                                                                                                                                                               Based on this process, the size and the spatial distribution in debris disks develop charac-
                                                     6x105 a             -4                                                                                                    teristic features even if there is nothing but statistically distributed parent bodies and grains.                                                                                                                                      ring
                                                     2x106 a         10
                                                                                  Geom. optical depth / size decade

                                                                                                                      The evolution in time of the size distribution           The Vega disk is used as numerical example.                                                                                                                                                               10-4
                                                     6x106 a
                                                     2x107 a                                                          shows the timescales at which the wavy pat-                                                                                                                                        The final radial slope is basically independent

                                                                                                                                                                                                                                                                                                                                                              Geom. optical depth
                                                     2x108 a                                                                                                                                                                                                                                             of the disk’s initial set-up. The optical depth is
                                                     2x109 a         10-5                                             tern evolves for a modeled ring-only disk for
                                                                                                                      Vega at a distance of 100 AU to the star. For                                                                                                                                      ruled by small but bound fragments produced
                                                                         -6                                           the population of grains smaller than about                                                               The Model                                                                in the densest parts adjacent to the inner gap.
                                                                                                                      100 µm it does not take more than 1 Ma                                                                                                                                             These fragments have extended orbits due
                                                                                                                      to reach a quasi-steady state. This time de-                                                                                                                                       to radiation pressure and cloak the dust and
                                                                                                                      pends on the total amount of material and on                                                               The modeled axisymmetric, gas-poor                                      parent body population of the outer regions.
                                                                                                                      the distance to the star, as well. The steady                                                              Vega disk is optically thin (τ    10−4). It                                                                                                                                                                             ring only
                                                                                                                                                                                                                                 has an assumed moderate opening an-                                                                                                                                                                                    r-4 profile
                                                                                                                      state is merely a slowly changing state.                                                                                                                                                                                                                                                                                          r-2 profile
                                                                     10-8                                                                                                                                                        gle (< 20° and an inner gap (80 AU).

                                                                                                                                                                                                                                                                               1300 AU
                                                                                                                                                                                                                                                                                                                                                                                                                                              low ecc., r-2 profile
                                                                                                                                                                                                                                 Initially ring-like configurations and se-
                                                                                                                                                                                                                                                                                                                                                                                                                       100                                             1000
                                                                                                                                                                                                                                 tups with different outer slopes are con-
                                                         rock                                                                                                                                     Inner gap                                                                                                                                                                                                            Distance to star [AU]
                                              rock, circ. orb.           -4                                                                                                                                                      sidered. The sizes of the dust and parent
                                                                                  Geom. optical depth / size decade

                                                           ice                                                                                                                                                                   bodies range from 0.1 µm to hundreds of
                                                                                                                                                                                               Possible ring of
                                                                                                                      This convergence also depends on the disk’s                                                                meters. The only cause for evolution is        Vega’s disk at 70 µm
                                                                                                                                                                                                parent bodies
                                                                                                                                                                                                                                 mutual collisions.                            (Su et al., ApJ, 2005)
                                                                     10-5                                             state of dynamical excitation and on the me-

                                                                                                                      chanical properties of the material. The com-
                                                                                                                      parison of the distributions resulting from dif-                                1. The rate of collisions of
                                                                     10                                                                                                                               bodies depends on the cross         3.   According to their opti-
                                                                                                                      ferent grain types indicates, that a realistic
                                                                                                                      mixture of different populations would prob-                                    section, the number densities       cal properties and their sizes,                                This example of the distribution of cross sec-
                                                                                                                      ably weaken the waviness. The cut-off would                                     and the relative velocity. It is    the grains respond to radia-                                   tion area over the grid of orbits shows the
                                                                                                                                                                                                      calculated for every pair of        tion pressure. Here, we as-                                    dominant role of particles on highly eccentric                                                          rb.
                                                                                                                      be less sharp.                                                                                                                                                                                                                                                                Anom. hype
                                                                                                                                                                                                      colliders over a grid of orbits     sume perfect absorbers.                                        orbits. The backbone of the distribution is pro-
                                                                     10-8                                                                                                                             and masses.                                                                                        duced mainly in the denser ring at 80–120 AU
                                                                                                                                                                                                      2.    High velocities and in-       4.    The smaller the frag-                                    and spreads outward from there. For disks

                                                                                                                                                                                                      elastic collisions lead to frag-    ments, the more extended                                       optically as thin as the one considered, the


                                                                                                                                                                                                                                                                                                                                                                                                                                                    . hy
                                                                                                                                                                                                      mentation. Relative velocities,     their orbits, or they are even                                 abundance of β-meteoroids is low, because
                                                                               Mass density / size decade [g/cm3]

                                                                                                                                                                                                      masses, and material proper-        blown out as β-meteoroids.                                     they leave the disk in 1000 years.

                                                                                                                      The size of the particles mainly responsible
                                                                                                                                                                                                      ties determine the size distri-     They move along effective Ke-
                                                                                                                      for the observed fluxes determines the to-
                                                                     10-20                                                                                                                            bution of the fragments.            plerian trajectories.
                                                                                                                      tal mass of observable dust. An extrapola-
                                                                     10-21                                            tion from this value needs to be based on the
                                                                                                                      according mass distribution. Here total disk
                                                                     10-22                                            masses for largest bodies of 300 m in radius
                                                                                                                      were derived to lie at around 1 earth mass in                                                            Conclusions
                                                                                                                      order to produce half a lunar mass of grains
                                                                     10-24                                            smaller than 1.5 mm in radius.
                                                                         -25                                                                                                   The size distribution of material in a debris disk converges towards a quasi steady-state wave
                                                                                                                                                                                                                                                                                                        Long-term Evolution
                                                                                                                                                                               pattern, whose quantitative characteristics depend on material properties and the distribution                                                                                                            2.5
                                                                                                                                                                               of parent bodies’ orbits and relative velocities. It peaks at radii of ≈3 times the blow-out size,                                                                                                                                                              extrap. f. 200 Myr
                                                                                                                                                                                                                                                                                                         The simulated total mass over time, which                                                                                             extrap. f. 600 Myr
                                                                                                                      The lifetimes of larger dust grains and parent           for Vega: ≈ 10 µm.                                                                                                        reduces due to blow-out of small fragments,                                     2.0                                                  extrap. f. 2000 Myr

                                                                                                                                                                                                                                                                                                                                                              Disk mass [earth masses]
                                                                     10                                               bodies are limited by mutual collisions. This                                                                                                                                      is compared with curves for the behavior of
                                                                                                                      timescale peaks at around 10 µm, reaches                 The geometrical optical depth is dominated by small grains on wide orbits, and thus, the                                  ideal disks in relative equilibrium at 200, 600,
                                                                     107                                              a minimum at around 100 µm, and rises                                                                                                                                                                                                                              1.5
                                                                                                                                                                                                                                                                                                         and 2000 Ma following M = M0/[1 + XM0(t −
                                                                                  Lifetime [a]

                                                                                                                                                                               steady-state radial profile in the outer part cannot be steeper than τ ∝ r−1.5. A steeper
                                                                                                                      to larger bodies, where it exceeds 100 Ma.                                                                                                                                         t0)] and is essentially inversely proportional to
                                                                     106                                                                                                       profile would indicate a deviation from the steady state.
                                                                                                                      For less dense disks with the same amount                                                                                                                                          the time. But the disk thins out faster in the                                  1.0
                                                                                                                      of small grains, but where the parent ma-                                                                                                                                          inner part than in the outer, and thus, devi-
                                                     2x105a                                                           terial is not strongly confined to a ring, the            The mass in parent bodies required to produce the observed amount of small dust in the                                    ates from a strictly steady state, making X
                                                     2x106a          104                                              whole disk’s half-life against mass depletion            Vega system is found to lie at reasonable 1 to 10 earth masses. A sufficient dust production                               decrease. The largest bodies in these runs                                      0.5
                                                     6x106a                                                           reaches some billion years.
                                                     2x107a                                                                                                                    can be sustained on timescales of 100 Ma to 1 Ga.                                                                         measure 10 km in diameter.
                                                     2x108a          103
                                                     2x10 a                                                                                                                                                                                                                                                                                                                              0.0
                                                                                                                                                                                                                                                                                                                                                                                                0         500                   1000                  1500             2000
10-1   100   101    102   103   104    105    106    107     108   109
                          Grain radius [µm]                                                                                                                                  Based on                                          ¨                        ˇ    ´
                                                                                                                                                                                          A LEXANDER V. K RIVOV, TORSTEN L OHNE , M IODRAG S REM CEVI C: Dust Distributions in Debris Disks:                                                                                                                                  Time [Ma]
                                                                                                                                                                                          Effects of Gravity, Radiation Pressure, and Collisions. In: Astronomy & Astrophysics (2006, in press)

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