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Conditions for Pulsar Planet Formation Michele Horner1 Conditions for Pulsar Planet Formation Michele Horner Department of Astronomy University of Virginia May 14, 2021 This thesis is submitted in partial completion of the requirements of the BS Astronomy-Physics Major. Draft version May 14, 2021 Typeset using LATEX twocolumn style in AASTeX63 Conditions for Pulsar Planet Formation Michele Horner1 1The University of Virginia Department of Astronomy 1. ABSTRACT The exact mechanisms of pulsar planet formation are largely unknown, and the existence of the planets This paper outlines the conditions for planet forma- themselves is observed to be extremely rare. Estimates tion around pulsars. A common model proposes the fall- from surveys show that only about 5% of pulsars may back of mass after an initial supernova explosion, which host planets.(4) There are two primary theories that ex- occurs as a reverse shock propagates the inward accre- plain their existence. The first proposes that these plan- tion of a fraction of bound material back onto the surface ets survive the supernova explosion, and the subsequent of the star. The initial fallback mass spreads and si- formation of a neutron star. It is extremely unlikely multaneously decreases in surface density, during which that a preexisting planet of a near-Earth mass would planet formation may take place at sufficiently cool tem- survive such an event, however, this case may poten- peratures. The existence of these conditions, and the tially explain the existence of the exceptionally high- availability of material to form planets, depends on the mass planet PSR B16020-26 b, which exists at a ∼20 time since the initial fallback as well as the fallback ra- AU distance from its central star. A second, and more dius from the central star. By modeling the profile of a likely theory, is that the fallback material from the ini- pulsar accretion disk and using solar estimates for dust tial explosion forms an accretion disk with the materials settling time and planetary isolation mass, we may pre- to form planets. The neutron stars themselves can be dict the conditions for the evolution of planetesimals in classified into one of the four categories: young radio a pulsar system. pulsars, millisecond radio pulsars, thermally emitting dim, isolated, neutron stars, or accreting neutron stars. Semi- The type of star, and its surrounding environment, has Planetary major Axis a crucial effect on the formation and composition of the Pulsar Object (AU) Mass surrounding planets.(1) Three of the first pulsar planets which have been detected orbit around the radio pul- PSR B1620- PSR B1620- sar PSR B1257+12. Two of the three have near-Earth 26 26 b 23 2.5 MJ masses and semi-major axes of .3-.4 AU. These planets PSR PSR are believed to have formed from the accretion of matter B1257+12 B1257+12 A .19 0.020 ME from a companion star. The location and composition of PSR PSR such planets is first determined by the accretion process, B1257+12 B1257+12 B .36 4.3 ME specifically the disk formation and composition follow- PSR PSR ing a supernova. Over time, this disk thins, evolves, and cools. In order for dust within a disk to sublimate, B1257+12 B1257+12 C .46 3.90 ME the blackbody temperature of material around a neu- PSR PSR tron star must cool to the sublimation temperature for B0943+10 B0943+10 b 1.8 2.8M J specific solids, ∼ 2000 K for silicates. This blackbody PSR PSR temperature is determined by distance from the central B0943+10 B0943+10 c 2.9 2.6 MJ source and the luminosity of the central source: PSR PSR 10.26 1.97±0.19 B0329+54 B0329+54 b ±0.07 ME GMM_ L = (1) PSR J2322- PSR J2322- 0.7949 r 2650 2650 b 0.0102 MJ This value is typically ∼ 1038 erg s−1, given an accretion −8 −1 Table 1. List of Known Pulsar Planets. rate of & 10 M y . 2. INTRODUCTION 3. FALLBACK 3 After a supernova explosion, although most mate- place. The formation process of a disk may be either rial is ejected from the system, about 0.001-0.1 M may passive or active. (2) In this paper, we focus on pas- remain bound and fall back toward the star. sive accretion, during which the disk evolution is driven The radius of the initial fallback disk is described by the luminosity of the central source. Calculations for by equation(3): source brightness, source temperature, and disk temper- ature estimate that a thin passive disk absorbs about GMt2 a quarter of incident flux from its central star. The r = ( )1=3 (2) π2 disk vertical profile consists of a surface layer, which ab- where t, the the timescale of this event, is a few hours, sorbs and re-radiates incident energy from the star, and a cooler inner layer, which intercepts that re-radiated and the fallback mass would range from 0.001-0.1 M . This fallback radius is calculated to be 108 cm. energy, and emits it as thermal energy.(2) From this radius, the inner mass moves inward to- In addition to central source luminosity, the accre- ward the star and accretes onto the central source, simul- tion rate and fluid transport within a disk are heavily taneously decreasing in mass. As the inner mass loses dependent on the viscosity, or the internal friction of angular momentum, the total angular momentum of the fluid material, within that disk. The time scale of disk system is conserved. The outer mass gains angular mo- evolution decreases with higher viscosity. Commonly, mentum and spreads outward. The angular momentum viscosity is characterized by the mean free path of par- of the system relates to the mass of the disk by: ticles within a fluid: p J = M(t) GMa(t) (3) ν ∼ csλ (5) where a is the radius of the disk. Planet formation must However, the observed formation rates of disks cannot occur in the outwardly expanding parts of the disk, out- be explained by this molecular viscosity alone, which im- side the fallback radius of 108 cm. (7) The mass of the plies the existence of a much greater viscosity, or turbu- disk varies with radius as: lent viscosity, as proposed by Shakura Sunyaev (1973). As the hydrodynamic stability of a disk is disrupted 1 M ∼ p (4) by magnetorotational instability, perturbations within a the disk occur and increase exponentially. This process creates a high, turbulent viscosity which sufficiently in- So, if an initial fallback mass of 0.01 M is found at a radius of 108 cm, the mass at a radius of .5 AU would be creases the rate of angular momentum transport. The around the size of ∼5 Earth masses. Considering plan- equation for this turbulent viscosity is: ets discovered at this distance are several Earth masses ν = αc h (6) in size, this would imply a near 100% efficiency rate of s planet production,(6) which may explain why the ob- where c is the speed of sound, and α is a constant served occurrence of pulsar planets is so unlikely. s & 0.01-0.1. This equation can be subsequently be used to determine viscosity as a function of radius and sur- 4. INITIAL STAGES OF ACCRETION DISK face density. A profile of the initial make up of accre- FORMATION tion disks can be determined using viscosity, as well as The initial stages of planet production begin with the different opacity values for different ranges of radii the formation of a protoplanetary disk around a star within the disk(6). The opacities of a circumpulsar disk during its evolution. An initial core within a molecular differ from those of a protosolar disk through the char- cloud precedes star formation. The large angular mo- acteristic nucleation process of the disk grains, as well mentum (J∼1054g2 cm s−1) of that core is consistent as the low ratio of gas to dust initially present in the with the angular momentum of molecular gas in 10-102 disk. In a pulsar system, the materials for planet pro- AU Keplerian orbit around a Solar mass star. Through duction come from stellar remnants, and the layers of these phenomena, large disks may form around these heavy elements within an original star, rather than the evolving stars. In astronomical observation,the presence molecular clouds that precede solar systems. The size of of an initial disk is characterized by excesses of infrared these initial grains may be about 10 times larger than and ultra-violet radiation in the Spectral Energy Dis- those in protosolar disks. Additionally, the concentra- tribution from that disk, respectively emitted from the tion of heavy elements, such as C, O, or Si may much hot dust around a star, and the high temperature re- be higher. The opacities throughout a pulsar disk can gions on the star itself where the gas accretion takes be differentiated using separate laws, boundaries, and 4 constants for different regions. These various regions include electron scattering, free-free/bound-free absorp- tion, areas of intermediate temperatures, and lower tem- perature areas composed of grains and ice grains. These opacity laws can be applied to solve for temperature, viscosity, scale height, and surface density over a disk. The fourth opacity law specifically accounts for temper- atures within the disk bounded from above by T=3000 K and below by T=4.6×103ρ1=15, at which solid mate- rials may sublimate. 5. SURFACE DENSITY The surface density of a given part of a disk is de- pendent on both radius and time.
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