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THE EFFECT OF PEBBLE ACCRETION ON A

CLOSE-CONTACT BINARY SYSTEM

Bachelor Thesis Physics and Astronomy

THOMAS JOHANNES KONIJN

UvAnetID: 10991204 Size: 15 EC

Conducted Between: March 30 - July 25, 2020

Daily supervisor: dhr. R. (Rico) Visser MSc Supervisor: dhr. prof. dr. C. (Carsten) Dominik Second examiner: dhr. dr. J.M.L.B. (Jean-Michel) Desert

Faculteit der Natuurwetenschappen, Wiskunde en Informatica UvA/VU Anton Pannekoek Institute for Astronomy

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Abstract

There are many planetary binaries in our solar system, some examples are Pluto and its moon Charon, Comet 67P and of course our Earth and Moon. These binary systems have differing characterising parameters such as mass ratio, orbital distance etc. This thesis aims to describe the physical processes of a planetesimal binary growing in a pebble accreting scenario and how these earlier named parameters evolve. First by describing the physical processes in a protoplanetary disk (Weidenschilling, 1977a). We will create an equation of motion for incoming pebbles falling into the growing planetesimals. After which we simulate many different starting positions to make sure all possible scenarios are accounted for. From here we try to look at accretion ratios and eventually make quanti-tative statements about the growth timeline of planetary binaries. This work succeeded in outlining some of the processes happening in the creation of a planetary binary. Also multiple interesting questions have arisen begging for further research.

Keywords: Planetary Binaries, Planetesimals, Pebble Accretion, Core Accretion, Pro-toplanetary Disk, Planetary Growth

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Populaire Nederlandse samenvatting

De vraag naar hoe planeten in jonge zonnestelsels ontstaan is een van de meest gestelde vragen binnen de astronomie. Zeker nu er de laatste jaren veel exoplaneten (planeten die om een andere ster dan de zon draaien) ontdekt worden is het een hot topic geworden. We kunnen nu immers veel gerichter kijken naar alle verschillende soorten exoplaneten die mogelijk zijn.

Dit onderzoek gaat over jonge planeten die zich in een binair stelsel bevinden. Dat zegt eigenlijk dat er twee grote hemellichamen, jonge planeten dus, om elkaar heen bewegen dankzij hun eigen zwaartekracht. Een mooi voorbeeld hiervan zijn de aarde en maan. Er wordt vaak gezegd dat de maan om de aarde heen draait, alleen is het eigenlijk beter om te zeggen dat ze om een gemeenschappelijk zwaartepunt draaien. Zij vormen dus een binair stelsel.

Naast de aarde en de maan zijn er nog meer binaire stelsels die we kennen in ons zonnes-telsel. Voorbeelden zijn Pluto en zijn maan Charon, waar de massa van Charon veel groter is in vergelijking tot Pluto dan bij onze aarde en maan. Of bijvoorbeeld komeet 67P die in 2014 in het nieuws kwam na de landing van Philae. Deze komeet is een prachtvoorbeeld waar de twee hemellichamen zo dicht bij elkaar kwamen dat ze elkaar nu aanraken, ze zijn dus ge¨evolueerd naar een enkel hemellichaam.

Er zijn dus vele verschillende soorten binaire systemen met verschillende eigenschappen. De centrale vraag van dit onderzoek is hoe binaire systemen zich gedragen tijdens het ontstaan van zonnestelsels en hoe deze eigenschappen zich ontwikkelen. Vooral tijdens het proces dat we “pebble accretion” noemen.

Er zijn twee leidende theorie¨en over hoe planeten ontstaan in de protoplanetaire schijf (de schijf van gas en stofdeeltjes om een jonge ster heen). Als eerste hebben we “gravitational instability”. Zonder teveel in detail te gaan zegt deze theorie dat planeten spontaan ontstaan door het in elkaar vallen van gas en stofdeeltjes in de uiterste regionen van de protoplanetaire schijf. Als tweede hebben we de “core accretion” theorie. Hier klonteren dankzij instabiliteiten in de schijf verschillende deeltjes in elkaar om zo steeds grotere objecten te maken. Zodra deze objecten rond de orde van kilometers in doorsnee zijn heeft “pebble accretion” het grootste effect op de massa opname.

Dit onderzoek heeft geprobeerd deze theorie van pebble accretion toe te passen op jonge binaire systemen. Om daarna te kijken hoe deze zich gedragen. Wat gebeurd er als we kleine aanpassingen doen op het model? Krijgen we een aarde-maan systeem? Pluto-Charon? Of zelfs een scenario zoals bij komeet 67P? Allemaal interessante vragen waar we nog geen antwoorden op hebben.

We proberen dit te beantwoorden door stenen te laten vallen op het binaire systeem, vanuit elke mogelijke denkbare manier. Om daarna te kijken naar welk gedeelte van de stenen wel en welke niet op de jonge planeten zijn gevallen. Op deze manier kunnen we kijken naar hoe deze jonge planeten groeien, maar hopelijk kunnen we ook een blik werpen op hoe de al bekende binaire systemen zich hebben ontwikkeld.

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Acknowledgements

This thesis marks the end of my bachelor’s degree here in Amsterdam. I’ve had a great ride these last few years thanks to, for the most part, my study association the NSA. A board year and committees beyond counting have given me amazing memories, best friends for life and even an amazing girlfriend. Thanks to them I was able to complete my studies here while having the best years one can hope for. I’m thrilled to stay here at Science Park for the next few years doing my master’s degree knowing the NSA hasn’t gotten rid of me yet.

To some surprise I have other friends from before my career in physics. I was lucky enough to be able to live with them during my studies and thanks to them I had the sometimes necessary distraction from the field of physics. I would especially like to thank Jelle for the amazing friendship both at home and in the lecture rooms.

Our world has been through some troubling times these last few months. As a consequence lots of people were unfortunately forced to live in uncertainty. Some of my fellow students had hard times even making sure their project was able to happen. Luckily I was able to do this research project without much problems during the way. There is one person in particular who made sure this could even be the case. I would like to express my sincere gratitude to Rico Visser for the rapid replies to my questions and the enthusiasm with which they we’re answered. Many late nights were spent discussing the various physical processes happening in deep space and I’ve always felt heard and appreciated. It was too bad we weren’t able to have these conversations in person at the coffee machine, or even in the sun accompanied with an ice cold beer.

Thanks Rico

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Contents

1 Introduction 1

1.1 Why binary systems? . . . 1

1.2 A brief summary of the theory of Pebble Accretion . . . 1

1.3 Premise of this research . . . 1

2 Getting started, an overview of the protoplanetary disk 2 2.1 A solid particle in orbit . . . 2

2.2 Dynamics of the gas . . . 2

2.3 Properties of the disk structure . . . 3

2.4 Drag laws . . . 5

2.5 Pebbles and their radial drift . . . 6

3 Pebble accretion, in the co-moving frame 7 3.1 Equation of motion of pebble . . . 7

3.2 Effect of the co-moving frame . . . 8

3.3 Computational method . . . 9

3.4 Different pebble trajectories . . . 10

4 Computing the binary system, the result 12 4.1 Dynamics of two planetesimals . . . 12

4.2 Fiducial model and its reasoning . . . 13

4.3 Graphing the model, making sense of it all . . . 13

5 Changing parameters, the effects on the model 16 5.1 Pro- & retrograde rotation . . . 17

5.2 Stokesnumber . . . 18

5.3 Planetesimal masses and their ratio . . . 19

5.4 Distance between planetesimals . . . 20

6 Discussion and errors 21

7 Summary and further research 21

References 22

Appendices 23

A Integration method 23

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1

Introduction

1.1 Why binary systems?

There are many known binary planetary systems. The most obvious one we’re living on right now, our Earth-Moon system. The Earth is approximately 81 times more massive than the Moon, while being around 60 Earth-radii apart from each other. The main theory of creation states a (proto-)planet often called Theia impacted the Earth (Halliday, 2000), subsequently creating our Moon.

There are many other planetary binaries in our solar system for which no large consensus exists on how they were created. Take Pluto and its moon Charon for instance. Pluto being approximately only 8 times more massive than Charon and orbiting at a distance of around 16 Pluto-radii this system could better be described as a real ”binary” instead of a planet-moon system.

An even more interesting case is comet 67P/Churyumov–Gerasimenko (Churyumov, 2004) where two planetary bodies are orbiting in contact. The larger lobe is around 2.3 times the mass of the smaller one. As with Pluto and Charon, for a theory about its creation we can merely guess.

All these different characteristics of planetary binaries begs the question of how new bina-ries and their parameters evolve in the stages of their creation. What will happen if they are further apart, if there is a change to their mass-ratio, if they are orbit in retrograde with respect to their central star? This work tries to answer some of those questions.

1.2 A brief summary of the theory of Pebble Accretion

Nowadays, there are two main theories of how planetary bodies form. Without going into too much detail, we first have gravitational instability. Here, planets spontaneously form due to the collapse of gas and dust in the outer rings of a protoplanetary disk (Boss, 1997). On the other hand we have the core accretion theory, where thanks to streaming instability and subsequent coagulation aggregates grow to planetesimals (kilometers). At which point Pebble Accretion is the main reason the planetesimal is accreting mass. There should still be gas in the path of the planetesimal interacting with debris, including the centimetre sized pebbles. This interaction between the pebbles and the gas makes them drift radially inwards towards the star since the drag force makes it impossible for the pebble to travel with Keplerian velocity. This drag force also slows pebbles passing the planetesimal which makes it harder for the pebbles to escape the gravitational well of the nearby mass. Therefore the accretion cross-section will be larger in comparison to existing models of planet formation and they will accrete more mass in a shorter time period.

1.3 Premise of this research

With planetesimals appearing in binary orbits with each other (Nesvorny, Li, Youdin, Simon, & Grundy, 2019), we can maybe combine the growth efficiency of Pebble Accretion to give insight in the way planetary binaries are formed. By looking at the general physics going on in a protoplanetary disk during the early life stages of a solar system and looking at its effect on a binary planetesimal system, there might be some answers to the questions mankind has had for thousands of years.

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2

Getting started, an overview of the protoplanetary disk

To make educated conclusions, we have to make reasonable assumptions regarding the protoplanetary disk. This chapter is meant to give a basic understanding of the underlying dynamics happening in a protoplanetary disk around a newly formed star.

2.1 A solid particle in orbit

Let us begin with one of the most basic calculations in classical dynamics: a particle in Keplerian orbit around the star. If we let agrav be the gravitational acceleration, r the

radial distance from the star, Mstar the mass of the star and G the gravitational constant,

we can write the differential equation of radial motion of the particle as agrav = −

GMstar

r2 = −g (1)

where g is the stellar gravity felt by the particle. If we then wanted to move to the particle’s frame of reference we add the centrifugal (pseudo-)force

d2r dt2 = − GMstar r2 + v2 r (2)

with azimuthal velocity v⊥. For circular motion around the star equation 2 should add op

to zero. We can, therefore, define Keplerian velocity vk

vk≡

r

GMstar

r .

2.2 Dynamics of the gas

In a protoplanetary disk it’s not just solid particles moving in Keplerian orbit, there is also a fair amount of gas surrounding the star in which the particles are moving through. So imagine instead of a solid particle in motion, a parcel of gas travelling around its central star. We can derive the radial acceleration balance as:

d2r dt2 = − vk2 r + v2g r + 1 ρ dP dr (3)

with gas-density ρ and pressure P . Here the first term is the gravitational acceleration from the star, the second term the centrifugal acceleration given by the rotational velocity of the gas and the last term is due to pressure given by the hydrostatic equilibrium. We know the gas is moving in a circular orbit around the star and therefore equation 3 should also add up to zero. From here we can define the residual gravity ∆g:

∆g ≡ v 2 g r − v2k r = 1 ρ dP dr (4)

Note both dPdr and ∆g are negative. If we solve for vg and Taylor expand to first order

knowing ∆g  g: vg ' vk+  ∆g 2g  vk (5)

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With equation 5 we can estimate the velocity of the gas by first relating it to vk through

the dimensionless constant η

vg = vk+ ηvk (6)

η ≡ ∆g 2g

This means particles travelling in Keplerian orbit encounter a headwind of velocity vhw =

ηvk.

2.3 Properties of the disk structure

We start by describing the properties of the protoplanetary disk in terms of convenient power laws for mass, pressure and temperature. We start out by describing the system as a two-dimensional disk (don’t worry, we’ll work out the vertical component later). For the density it is convenient to begin with a surface density profile Σ(r) and a temperature profile T (r) as described by:

Σ(r) = Σ0  r r0 −3/2 (7) T (r) = T0  r r0 −1/2 (8) With r0 = 1 AU, Σ0 = 1700 gcm−2 and T0 = 170 K (R. G. Visser & Ormel, 2016;

Weidenschilling, 1977b; Hayashi, Nakazawa, & Nakagawa, 1985). We can reasonably assume the pressure profile also obeys a power law

P (r) = P0

 r r0

−n

(9) If we take the derivative with regard to radial distance r and use the ideal gas law we can come up with another way of writing the hydrostatic equilibrium

1 ρ dP dr = − nkbT ¯ mr (10)

with kb Boltzmann’s constant and ¯m the mean molecular mass of the gas particles. To

find n we need to first look at the vertical structure of the disk. Let’s assume the disk is thin (z  r) and its own mass is negligible compared to the star. As is illustrated in figure 1 we can say of the vertical hydrostatic equilibrium

1 ρ ∂P ∂z = − GMstar r2 z r  = −Ω20z (11) With Keplerian frequency

Ω0≡

r

GMstar

r3

Since we can ignore small forces between the gas particles like the van der Waals force, we can think of the gas like a perfect gas. The equation of state of a perfect gas says pressure is written in the relation:

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Figure 1: Illustration of vertical hydrostatic equilibrium. The gas in the middle of the protoplanetary disk gives a pressure gradient upwards counteracting on the z-component of the stellar gravity g

Where cs =pkbT / ¯m is the speed of sound. Using this we can elaborate on equation 11

to make a definition for the vertical structure of the gas density: c2s ρ ∂ρ ∂z = −Ω 2 0z ∂ ln ρ ∂z = −  Ω2 0 c2 s  z ln ρ = ln ρ0− z2 2H2 ρ(z) = ρ0e− 1 2( z H) 2 (13) With scale height H ≡ cs/Ω0. Here we see the vertical density distribution is given by

a Gaussian. We can integrate this over the complete z-axis to receive the total surface density Σ = Z ∞ −∞ ρ0e− 1 2( z H) 2 dz = √ 2πρ0H (14)

If we recall the power law of equation 7 and the r-dependencies of cs and Ω0 we can say

about the density at z = 0

ρ0(r) =

Σ(r) √

2πH(r) ∝ r

−114 (15)

And with this we can calculate the r-dependency of P by manipulating equation 10 and then integrating over r

dP dr ∝ r

−17 4

P (r) ∝ r−134 (16)

Now we can safely say the value for n = 134 in equation 9. Using this and equation 12, the pressure P neatly falls away when we derive P with respect to r and make an expression

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for the residual gravity ∆g dP dr = − 13P 4r ∆g = 1 ρ dP dr = − 13c2s 4r (17)

after this we can give an estimation of η η = ∆g 2g = −13c 2 s 8vk2 (18)

Note that when we compute the headwind vhw = ηvk, we lose all dependency on r.

Therefore we can say the headwind is constant at all radial distances. For the remainder of this thesis we will be working with a headwind of vhw = −32 ms−1 (R. Visser, Ormel,

Dominik, & Ida, 2020)

2.4 Drag laws

As we can see solid particles, or pebbles, moving in orbit in a protoplanetary disk feel a drag force due to the headwind. Let’s assume these pebbles are spherical with radius s. We can describe the drag force as:

FD= CDπs2ρ

∆v2

2 (19)

Where CD is the dimensionless drag coefficient which depends on the Reynolds number

(Whipple, 1972).

When s is very small (e.g. dust particles) it is reasonable to assume that it will be moving parallel with the gas, hence it feels no headwind and there is no drag acting on it. Likewise it is a logical expectation that when s becomes very large (e.g. dwarf planets and larger) the pebbles have become large enough that the drag force has almost no effect on the pebble due to its already large momentum. There is a point in between where the effect of the drag force is highest. There is a manner to measure the effect of the drag, something called the stopping time ts.

ts ≡

mv FD

Where m is the mass of the particle and ts the amount of time it takes for v to be reduced

by a factor e thanks to a constant drag force FD.

Weidenschilling states two different regimes as to which the particle can be in (Weidenschilling, 1977a). First we have the Epstein regime for s < 94λ, where λ is the mean free path of a gas particle in the surrounding gas. In like manner we have the Stokes regime where s > 94λ for larger pebbles. Weidenschilling comes up with stopping times for both regimes:

ts = (ρ ss ρcs if s < 9 4λ (Epstein) 2ρss2 9ζ if s > 9 4λ (Stokes)

Where ρs is the density of the particle and ζ the viscosity of the gas. These regimes

overlap when s = 94λ. At this point the pebbles feel the biggest effect of the drag. A consequence of this drag is a loss of angular momentum due to torque. Or, in other words, a measurable radial drift inwards.

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Figure 2: Sketch of the radial drift of a pebble in both the Stellar and Keplerian frame. We know the total azimuthal velocity in the Stellar frame is vg+ w. By changing frames

and using equation 6 we know the azimuthal velocity becomes vg+ w − vk= w + vhw.

2.5 Pebbles and their radial drift

We can describe a radial force balance by saying the pebble moves inwards at radial velocity u and azimuthal velocity w with respect to the gas. This makes the pebble feel a total headwind of v =√u2+ w2. This makes the drag force in radial direction equal to

FDuv. Also the pebble will be moving at velocity vg+ w in the azimuthal direction in the

stellar frame. If we write a radial force balance we find d2r dt2 = FD m u v + (vg+ w)2 r − vk2 r (20)

Where the first term is drag, the second centrifugal acceleration and lastly we have stellar gravity. We can equate this to zero since the radial drift is a constant velocity. Rewriting this using vg = (1 + η)vk, ∆g =

v2 g−v2k

r and after neglecting small terms like w2 r we receive FD m u v + ∆g + 2vkw r ' 0 (21)

We can calculate the torque by either multiplying the azimuthal force component with the radial distance, or we can take the time derivative of the angular momentum

dL dt = −rFD w v = d dt[mr(vg+ w)] −rFD m w v = (vg+ w) dr dt + r ∂ ∂r(vg+ w) dr dt (22)

From here we can use the fact that drdt = −u combined with ∂vg

∂r ' −(1 + η) vk 2r to receive FD m w v = uvk 2r + u  ηvk 2r + w r + ∂w ∂r  (23) The last term on the right we can neglect since it’s of quadratic order whereby we receive

FD m w v − uvk 2r ' 0 (24)

Lastly we can use the definitions for ts, Ω0 and vhw to combine equations 21 and 24 and

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of the gas. u = − 2tsΩ0 t2 sΩ20+ 1 ηvk= − 2τs τ2 s + 1 vhw (25) w = τs 2u = − τ2 s τ2 s + 1 vhw (26)

Where τs is the dimensionless Stokes number

τs≡ tsΩ0

For our research however we want to evaluate the velocities in the Keplerian frame (moving with vk in the azimuthal direction). We therefore have to change equation 26 as seen in

figure 2 to receive the total azimuthal drift w + vhw=

1 1 + τ2

s

vhw (27)

3

Pebble accretion, in the co-moving frame

In order to make sense of these past calculations and to see the physical significance, we can use computational methods to simulate the accretion process of the planetesimals. We will have to make some assumptions and simplifications, for instance we will be moving along in the co-moving frame of the planetesimal at radial distance r from the star while rotating at a frequency Ω0 as seen in figure 3

Figure 3: Sketch of the three-body problem in the heliocentric frame (centred on the star). The large body with mass M rotates with Keplerian frequency Ω0, the gas with (1 + η)Ω

and the pebble m with Ω, around the z-axis.

3.1 Equation of motion of pebble

First we need to come up with an equation of motion for the pebbles falling into the planetesimal. We can begin by completing this quest in two dimensions. All the known forces acting on the pebble can be summed up as

dv

dt = acor+ agrav+ acent+ astar+ adrag (28) Where acor is the Coriolis force because we’re in the co-moving frame moving along with

the planetesimal at frequency Ω0.

acor= −2Ω0× v = 2Ω0

 vy

−vx 

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The agrav is simply the gravitational pull of the planetesimal agrav = − GM (x2+ y2)3/2 x y  (30) Where M is the mass of the planetesimal. We can combine the centrifugal force acent

and the stellar gravity astar doing a first order Taylor expansion remembering x  r to

simplify to acent+ astar = Ω20rex− Ω2rex = GMstar r3 − GMstar (r + x)3  rex ' 3x r GMstar r3  rex '3xΩ 2 0 0  (31) Lastly, we know from the definition of tsthat we can write the acceleration due to drag as

adrag = −

vpebble− vgas

ts

(32) For the gas velocity in the rotating frame of reference we know it only has an azimuthal component depending on the x-coordinate. We can use the same technique we did before by doing a Taylor approximation to receive

vgas= ((1 + η)Ωr − Ω0r)ey '  (1 + η)  Ω0− 3x 2rΩ0  r − Ω0r  ey '  vhw− 3 2Ω0x  ey (33)

Which will result in a total drag acceleration adrag= − 1 ts  vx vy− vhw+32Ω0x  (34) We can combine equations 29-34 until we ultimately receive the final equation of motion for the pebble

dv dt = 2Ω0vy−(x2GM x+y2)3/2 + 3xΩ 2 0−vtxs −2Ω0vx− GM y (x2+y2)3/2− 1 ts vy− vhw+ 3 2Ω0x  ! (35)

3.2 Effect of the co-moving frame

We saw in section 2.3 that vhwis independent of r (or x in our frame). However, we cannot

take the gas velocity to be a constant value of vhw in the negative y-direction. When we

moved our frame of reference from the central star to a Cartesian frame co-moving with the planetesimal at frequency Ω0, we neglected the curvature we should see since x, y  r, r0.

This change from a curved to a Cartesian frame needs a correction for the velocity of the gas. This correction is called the Keplerian shear and is given by 32Ω0xey as seen in

equations 34 and 35.

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of reference. We find this region by simply solving for vg = 0. When doing so we find a

vertical line, we call this the co-rotation line xco

xco≡

2vhw

3Ω0

Where xco is negative since vhw is also negative.

Figure 4: Basic sketch of the vector field of the velocity of the surrounding gas. The effect of the Keplerian shear is shown by the correction of −32Ω0x in the y-direction. The dashed

red line shows the co-rotation line xco where vg = 0

In equation 31 we see the combined forces of the centrifugal and stellar gravitational force also known as tidal force atidal. When we equate this expression with the gravitational

force of the planetesimal (taking y = 0) and solve for x atidal= agrav

3xΩ20= GM

x2 (36)

We eventually obtain the so-called Hill-radius which sets the, although somewhat arbitrary, boundary of gravitational domination.

rH≡ 3

r M 3Mstar

3.3 Computational method

In order to simulate this process of debris falling into the planetesimals due to Pebble Accretion, the Runge-Kutta-Fehlberg integration method is used (Es-hagh, 2005). This is an integration method similar to the ordinary Runge-Kutta approach but with a variable step method making sure simulation speed is improved, while not compromising precision. A more detailed explanation of the integration method can be found in appendix A. Every simulation is started from a certain starting point (x0, y0), where y0 is chosen as

max(50rplanet, 102

q

GM ts

vhw ). This is to ensure the gravity has negligible influence on the

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azimuthal velocities as seen in equations 25 and 26, only with the azimuthal velocity corrected for the co-moving frame (27) (R. G. Visser & Ormel, 2016). Note that vhw is,

as always, negative. vx,start = 2τsvhw 1 + τ2 s (37) vy,start= vhw 1 + τ2 s −3 2Ω0x (38)

Where τs = 0.1, vhw = −32 ms−1 and Ω0 = 8.20 · 10−10 s−1. A hit has occurred when

the position of the pebble is within one planetesimal radius of the center of mass of the planetesimal (px2+ y2 < r

planet)

3.4 Different pebble trajectories

In figure 5 we see different trajectories that pebbles are taking coming near a planetesimal. The radii of the planetesimals are ranging from 200 km to 3000 km. The general path of the pebbles is clear, at first they fall onto the planetesimal from up above where they either collide with, or drift alongside the planetesimal to continue their path inwards to the central star. There are some extra noteworthy trajectories to elaborate on though. With the smaller bodies (figure 5(a) & (b)) there are many near misses that cause the pebbles to take some turbulent paths before ultimately follow the path alongside the gas again.

In figure 5(b) the co-rotation line xco is first seen. Figures 5(c), (d) and (f) actually show

vividly what happens after a pebble crosses this line. As we know the pebbles begin in the upper right corner falling in a left downwards direction. When crossing xco the pebbles

are drifted upwards as a consequence of the gas velocity we have seen before in figure 4. At a certain point planetesimals become large enough xco is within the Hill-radius of the

massive body. At that point, the effect of the Hill-sphere is clearly seen. In 5(e) there is a more detailed view of two pebbles that were released at nearly the same x-coordinate. One of them is pulled inside the Hill-sphere just before the gas makes it travel in an upwards direction, while the other escapes the gravitational pull of the planetesimal and continues its journey towards the star. We must not forget however that the Hill-sphere is mostly an arbitrary border which can be seen from 5(d). The pebble released from the furthest x-coordinate travels distinctly well inside the Hill-radius of the planetesimal, while clearly escaping the gravitational pull. This can mainly be attributed to the fact that rH is calculated by looking at what massive body dominates solely on the x-axis.

The distance between the most outer x-positions where a hit still occurs is called the impact parameter. Before a certain threshold of planetesimal size is met, the impact parameter is of order rplanet as we can see for instance in 5(a), (b) & (c). When planetesimals

get massive enough (they may even be considered planets at this point), their impact parameters have become considerably larger. Even of the order of Hill radii. This is clearly seen for example in 5(f) where there are pebbles which in the first instance escape the Hill-sphere before crossing xco. After which they reappear into the Hill-sphere in their

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(a) (b)

(c) (d)

(e) (f)

Figure 5: Pebble trajectories for different planetesimal radii in the co-moving frame. For these simulations τs= 0.1 and vhw = −32 ms−1

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4

Computing the binary system, the result

Figure 6: Two planetesimals of masses M1and M2moving around their combined center of

mass at distance r1and r2, respectively. A phase of ϕ = 0 corresponds to the planetesimals

sitting on the x-axis at −r2 and r1.

4.1 Dynamics of two planetesimals

First, we need to describe the system of two planetary bodies orbiting their center of mass. Let us give the planetesimals velocities v1 and v2 and masses M1 and M2 as described in

figure 6. We want the center of mass to be located at the origin of the co-moving frame. Therefore the combined momentum of both bodies must add up to zero. To keep it simple, the fiducial model will be a circular orbit. If we work out their radial force balance we get their orbiting velocity

vK,1= s GM2 2 (M1+ M2)(r1+ r2) (39) vK,2= s GM12 (M1+ M2)(r1+ r2) (40) Where r1 and r2 is the distance between the two objects and the origin, respectively. Or,

when we use two bodies of same mass M as in our fiducial model vK,1= vK,2=

r GM 4r0

(41) Where r0is the distance from either planetesimal and the origin. Adding another

planetes-imal will change the equation of motion (35) of falling pebbles slightly. The gravitational part agrav as portrayed in equation 30 will change to

agrav= − GM1 |rpebble− r1|3/2 (rpebble− r1) − GM2 |rpebble− r2|3/2 (rpebble− r2) (42)

Where rpebble is the position of the pebble, and r1 & r2 the position of the two

planetesi-mals, respectively.

As described in section 3.3, the starting point y0 is set quite arbitrarily just to make sure

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simulation. Since the two planetesimals will begin moving around their combined center of mass at the start of the simulation this also means their phase ϕ at the time of passing is also quite arbitrary. In order then to say something of meaning, we need to make sure every angle between 0 & 2π is covered in the simulations. More on that in section 4.3. The path these planetesimals will be following is predetermined by the Keplerian velocities as described in equations 39 & 40. We’ve chosen not to let these paths be calculated by the simulation since the deviations of these paths will be negligible since Mpebble  mplanet

4.2 Fiducial model and its reasoning

For this research we’ve chosen one particular situation similar to the binary system of Pluto and Charon. The corresponding values for every notable parameter are listed in table 1. Charon is chosen since it can still be considered as a planetesimal (nearly being classified as a protoplanet). Parameter Value r 39 AU vhw −32 ms−1 τs 0.1 r1 = r2 19.600 km M1= M2 1.6 · 1021 kg (MCharon) rplanet,1 = rplanet,2 606 km Mstar 2 · 1030 kg (M )

Table 1: Characteristic values of the fiducial model

4.3 Graphing the model, making sense of it all

Figure 7: A hit map of different xstart positions spanning the accretion cross-section where

every coordinate corresponds with different starting angles of both planetesimals ranging from 0 to 2π. The two different colours correspond with hits on the different planetesimals, respectively.

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As said in section 4.1, for every x-position we test we need to cover all possible phases ϕstart between 0 & 2π in order to say something quantitative. On top of that we need to

test all possible x-starting positions spanning the complete accretion cross-section. This to make sure we cover all possible events in the accretion process. We can then make the case for three different outcomes: A flyby of the pebble where after it continues its radial drift towards the central star, or a collision with one of the two planetesimals.

Figure 7 shows the complete hit map of our fiducial model. Let us first appreciate the beauty of this graph. Every single pixel translates to a complete simulation and clearly there are some regions here where we can give definite descriptions of what is actually physically happening. Since our planetesimals are exactly equal in mass and are moving in mirror image of each other we can safely make the assumption there should be a symmetry differing with a starting phase of π. Lucky for us we can see this happening exactly. With the exception of a few pixels, the pattern is repeating itself vertically with a period of π. As told in section 3.3 the starting y-position y0 is set quite arbitrarily, only

making sure that at the moment of release gravity is of negligible importance. Thus the meaning of the starting angle is also quite arbitrary. If we had taken y0 to be anything

higher or lower figure 7 would be shifted up or down but the main pattern of the graph would remain.

(a) (b)

Figure 8: Trajectory of a pebble starting at a distance of x0 = 0.052rH. The starting

phase ϕ0 differs with π/2

As said before, every pixel in this hit map corresponds to a complete trajectory a single pebble has taken. Therefore we can look at some of these points in the graph to make sense of the physical processes happening here. Let us first evaluate the clearest ”sweeping” process of the planetesimals at x0 = 0.052rH. In figure 8 we see two different starting

angles of ϕstart = 0 and ϕstart = π2. From figure 7 we see that starting phase 0 at this

position should be a clear miss while π2 should give a clear hit with a planetesimal. From the trajectory seen in 8(a) we can conclude this large region of misses at xstart ≈ 0.05rH

corresponds with a pass right through the middle of both planetary bodies. All the while we can see from 8(b) the large region of accretion corresponds to one of the planetesimals moving towards the direction the pebble is coming from.

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(a) (b)

Figure 9: Trajectory of a pebble starting at a distance of x0 = 0.044rH. The starting

phase ϕ0 differs with π/2

In the hit map we see some sort of tail forming on the left side of the big sweeping regions at x0 ≈ 0.052rH. These tails show a very clear, albeit small, region of phases where

accretion occurs. In figure 9(a) and (b) we see two trajectories in these tails released at x0 = 0.044rH. Here we see paths similar to what we saw in figure 8, albeit with a clear

and straight path when moving in between the two planetesimals not like 8(a) where the pebble makes a sharp turn while moving through the planetesimals. This would explain why a larger portion of the angles results in a flyby instead of a collision in these narrow tails.

Figure 10: Trajectory of fiducial model at x0= 0.06rH

From figure 7 we see that at xstart ≈ 0.06rHthere is a lot of chaos when determining which

planetesimal the pebble will eventually land on. There seems to be no clear distinction into which of the two planetesimals will be the end result or if the pebble will fly by without a collision. Figure 10 shows one of the chaotic paths where it seems like it has a lot of similarities with the highly chaotic three-body problem.

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(a) (b)

Figure 11: Trajectory of a pebble starting at a distance of x0 = 0.03rH. The starting

phase ϕ0 differs with π/6

At a release point of x0 ≈ 0.03rH there is a clear sweeping planetesimal at a phase of

ϕstart ≈ π/6 while it seems chaotic at a phase of ϕstart ≈ 0. Figures 11(a) and (b) show

the importance of the phase the planetesimals are in. In this instance the slight change of phase results in a near miss before being ejected into a chaotic path similar to the one illustrated in figure 10.

5

Changing parameters, the effects on the model

We can say a lot about figure 7, but we can make more elaborate statements when altering some properties of the system, and looking at the resulting scenario’s. In the next few sections, we will keep changing one parameter at a time and looking at the consequences.

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5.1 Pro- & retrograde rotation

Figure 12: A hit map of our model with the two planetesimals moving in retrograde as opposed to the prograde rotation with which they were moving beforehand.

Figure 12 shows the exact same situation as seen in the hit map of figure 7, except for the fact the two planetesimals are turning in retrograde direction. Planetary binaries mostly rotate in prograde direction. But since they are thought to happen in 20% of planetary binaries (Nesvorny et al., 2019), it is still worth looking at the alternative.

Firstly we see the boundaries of accretion are shifted to the right compared to the fiducial model. Good to note is the fact that the exact same x-positions are used as with the fiducial model. We can attribute this shift to the rotation of the planetesimals as described in figure 6. When looking at the prograde scenario the left boundary hitting pebbles are travelling along with one of the two planetesimals. This results in a longer time where the gravitational pull of said planetesimal pulls on the pebble. The same process happens in the retrograde situation, except it will be on the right side of both planetesimals. Therefore the complete accretion parameter is shifted to the right.

Other than the shift of the accretion parameter there are other interesting things we notice. In the fiducial model of figure 7 we see clear accretion bands slightly on the right side of the middle. On the left side however there also seemingly clear bands with chaotic clutter in between them. This also happens in the retrograde situation albeit in a mirror image. This of course is not really surprising as the rotation of the planetesimals is mirrored. What is interesting however is that the clear bands of the retrograde case appear to be much smaller than the ones in the fiducial model, while the chaotic bands appear thickened. An explanation for this could be the fact that the gravitational pull of the planetesimals isn’t the only factor in the equation of motion of the pebbles. The drag forces will push the pebbles in a certain, mostly lower left, direction. This creates different clear sweeping scenario’s as compared to the fiducial model.

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When we look at all the simulations done in the retrograde scenario, 25.1% of them result in a hit (on either planetesimal). This is slightly lower compared to the fiducial model which has a hit ratio of 27.1%. This is however not a large enough difference to be considered noteworthy.

5.2 Stokesnumber

Figure 13: A hit map of our model with the Stokesnumber set at τs= 0.01. Corresponding

to smaller pebbles experiencing a larger effect of the gas drag.

Figure 13 shows the hit map of our fiducial model with a lowered Stokesnumber of τs= 0.01

instead of τs = 0.1. This can be imagined as smaller pebbles and therefore the impact of

the surrounding forces are larger compared to pebbles where τs = 0.1.

One noteworthy effect is the size of the accretion parameter. In our fiducial model the complete accretion distance is approximately 0.06rH, while with τsit is around 0.16rH. On

top of the increased accretion parameter the percentage of pebbles hitting either one of the planetesimal also increased substantially (42.8% compared to 27.1%). This is an arbitrary number however since the boundaries of x-positions tested are chosen to be just wider than the accretion boundaries, somewhat indiscriminately. Also the incoming flux of pebbles due to the starting velocities is neglected. However the large difference in percentage does make it noteworthy. The reason for this difference can be found in the earlier stated fact that since the stopping time ts is lower, the impact of the planetesimals gravity is higher

on these pebbles.

Other than the number of pebbles actually hitting the planetesimals another big differ-ence with respect to the fiducial model is the mirroring of the clear accreting bands at approximately 0.13rH. Those mirrored bands would probably be a trajectory similar to

the one seen in figure 8(b) where the pebble meets a planetesimal head on but will miss it just on the right side. In our fiducial model this would result in a flyby, but since these are smaller pebbles it would turn a bit further and subsequently land on the other planetesimal rotating towards the incoming pebble.

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5.3 Planetesimal masses and their ratio

Figure 14: A hit map of our model where the masses of planetesimals are set at M2 =

2M1= 2MCharon

The beauty that was so apparent in the structures of figures 7, 12 and 13, is lacking in figure 14. There is also no symmetry with starting angles differing π. This is not a surprise since a phase shift of π swaps places of the planetesimals as seen in figure 6, and the planetesimals are not of equal mass.

The chaotic band on the right seems considerably larger than the one in the fiducial model. Like the trajectory seen in figure 15, this chaotic band corresponds to a large range of xstart-positions where the pebble flies along side of both planetesimals to then

spiral around them for a while to eventually either falling in to one or being ejected out of the system.

(a) (b)

Figure 15: Trajectory of a pebble starting at a distance of x0 = 0.08rH with a starting

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There is a clear difference between the amount of pebbles being accreted by the planetes-imal with mass 2MCharon in comparison to the other. The ratio found here is M˙2/ ˙M1 ≈

1.29. However, as with most numerical results in this section, this is again quite arbitrary. Mostly due to the different flux of pebbles on different xstart-positions not being accounted

for. If we want to say something quantitative, a good thing to start would be to look at the normalised accretion ratio between the two planetesimals:

˙ M1 M1 ? ∝ M˙2 M2 (43) If we know this relation between these two planetesimals gaining mass we could maybe say something about the effect they inflict on each other. It could either be a snowball effect where the larger mass keeps on gaining mass at a faster rate compared to the smaller planetesimal. On the other hand we could see a stabilising effect where the larger mass ”helps” the smaller one gaining mass. In the latter case we could even get a binary system somewhat similar to Pluto-Charon where the masses are eerily close to each other. For now the value of ˙M2/ ˙M1≈ 1.29 looks eerily close to the cube root of two, which might

suggest a growth relation corresponding to ˙ M1 3 √ M1 ? = √3M˙2 M2 (44) However we can’t say anything conclusive since the only mass ratio scenario tested is M2 = 2M1

5.4 Distance between planetesimals

Figure 16: A hit map of our model were the two planetesimals are a distance 4r1 (78.400

km) apart from each other instead of 2r1.

Figure 16 shows the case where the two planetesimals are a distance of 4r1 a part, in

contrast to the 2r1 distance in the fiducial model. What is clear is that the same structure

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slightly larger (±8rH in stead of ±6rH), while the distance between the two planetesimals

doubled. One would think the left en right boundaries of xstart-positions that result in

accretion would also be twice the distance a part but this seems not to be the case. That could suggest the accretion parameter is more attributed to the total mass of the planetesimals (which is the same as with the fiducial model), than it is to the distance between those masses.

6

Discussion and errors

Since this thesis is merely a simplified version of the actually planet accreting scenario, there are multiple aspects of this research we have to take in consideration before making conclusions. Firstly one obvious matter is the fact every simulation here is done in two dimensions. An actual protoplanetary disk obviously exists in a three dimensional space. In three dimensions we would probably see some differing results, while hopefully the general lines will remain.

On top of a lacking dimension, we’ve also simplified the path of both planetesimals. These paths are predetermined to make the simulations faster. In addition we have chosen these paths to be circular so there is eccentricity in their orbits. Likewise we have disregarded inclination of the binary system. Something that would be impossible anyway since the simulations are two-dimensional.

The impact of gravity of the pebbles acting on planetesimals and their momentum transfer has been labelled negligible. Also their mass remains constant while accreting the mass of the incoming pebbles. Since this is a accreting scenario by definition the masses should change. However we are looking at one particular moment in time.

Since every simulation here is of single pebbles, the mechanics of colliding pebbles (G¨uttler, Blum, Zsom, Ormel, & Dullemond, 2010; Blum & Wurm, 2008) have been ignored. This is an extremely interesting part of planetary creation which might have a considerable effect on this research. The interested reader is referred to the review by Blum & Wurm (2008).

Also the gas surrounding the planetesimals will behave differently when these masses become large enough because of the gravitational pull creating the beginnings of atmo-spheres. This however only becomes a significant problem when the planetesimal size is approximately one order of magnitude larger than in our simulations (Ormel, Kuiper, & Shi, 2014).

7

Summary and further research

This thesis mostly aims to describe the physical processes of planetary binaries in a pebble accreting scenario. There is no definite conclusion on how the known binaries such as Pluto-Charon or 67P/C-G might have gotten their characteristics. It did however generate an interesting new way to look for an explanation on how the parameters of these kind of binaries evolve.

In general accretion maps as illustrated in figures 7, 12-14 & 16 are distinguished by chaotic and orderly bands where accretion occurs with missing trajectories in between. These maps also give insight into what percentage of surrounding debris falls into the planetesimals whereby we can subsequently make statements about how different parameters changes the growth timescale of these planetary binaries.

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When looking at two planetesimals of differing mass (2M1 = M2) as seen in section 5.3,

the accretion ratio found is ˙M2/ ˙M1 ≈ 1.29. This suggests a sublinear growth between the

two, in other words, the mass-ratio will decay over time. However in what matter it will sparks some new questions. Equation 44 might give an answer, it even seems more likely given that it directly correlates with the radius of planetesimals. Nevertheless further research must be done in order to say something conclusive.

References

Blum, J., & Wurm, G. (2008, September). The growth mechanisms of macroscopic bodies in protoplanetary disks. , 46 , 21-56. doi: 10.1146/annurev.astro.46.060407.145152 Boss, A. P. (1997). Giant planet formation by gravitational instability.

Sci-ence, 276 (5320), 1836–1839. Retrieved from https://science.sciencemag.org/ content/276/5320/1836 doi: 10.1126/science.276.5320.1836

Churyumov, K. (2004). Discovery, observations and investigations of comet 67p/churyumov-gerasimenko in kyiv. In L. Colangeli, E. M. Epifani, & P. Palumbo (Eds.), The new rosetta targets: Observations, simulations and instrument perfor-mances (pp. 1–13). Dordrecht: Springer Netherlands. Retrieved from https:// doi.org/10.1007/978-1-4020-2573-0 1 doi: 10.1007/978-1-4020-2573-0 1 Es-hagh, M. (2005, 01). Step-variable numerical orbit integration of a low earth orbiting

satellite. Journal of the Earth Space Physics, 31 , 1-12.

G¨uttler, C., Blum, J., Zsom, A., Ormel, C. W., & Dullemond, C. P. (2010, Apr). The out-come of protoplanetary dust growth: pebbles, boulders, or planetesimals? Astron-omy and Astrophysics, 513 , A56. Retrieved from http://dx.doi.org/10.1051/ 0004-6361/200912852 doi: 10.1051/0004-6361/200912852

Halliday, A. (2000, 02). Terrestrial accretion rates and the origin of the moon. Earth and Planetary Science Letters - EARTH PLANET SCI LETT , 176 , 17-30. doi: 10.1016/S0012-821X(99)00317-9

Hayashi, C., Nakazawa, K., & Nakagawa, Y. (1985, January). Formation of the solar system. In D. C. Black & M. S. Matthews (Eds.), Protostars and planets ii (p. 1100-1153).

Nesvorny, D., Li, R., Youdin, A. N., Simon, J. B., & Grundy, W. M. (2019). Trans-neptunian binaries as evidence for planetesimal formation by the streaming instabil-ity.

Ormel, C. W., Kuiper, R., & Shi, J.-M. (2014, 11). Hydrodynamics of embedded planets’ first atmospheres – I. A centrifugal growth barrier for 2D flows. Monthly Notices of the Royal Astronomical Society, 446 (1), 1026-1040. Retrieved from https:// doi.org/10.1093/mnras/stu2101 doi: 10.1093/mnras/stu2101

Visser, R., Ormel, C., Dominik, C., & Ida, S. (2020, Jan). Spinning up planetary bodies by pebble accretion. Icarus, 335 , 113380. Retrieved from http://dx.doi.org/ 10.1016/j.icarus.2019.07.014 doi: 10.1016/j.icarus.2019.07.014

Visser, R. G., & Ormel, C. W. (2016, Jan). On the growth of pebble-accreting planetes-imals. Astronomy Astrophysics, 586 , A66. Retrieved from http://dx.doi.org/ 10.1051/0004-6361/201527361 doi: 10.1051/0004-6361/201527361

Weidenschilling, S. (1977a, 06). Aerodynamics of solid bodies in solar nebula. Monthly Notices of the Royal Astronomical Society, 180 , 57-70. doi: 10.1093/mnras/180.2.57 Weidenschilling, S. (1977b, Sep). Distribution of mass in the planetary system and solar

nebulae. Astrophys. Space Sci.; (Netherlands), 51:1 . doi: 10.1007/BF00642464 Whipple, F. L. (1972, January). On certain aerodynamic processes for asteroids and

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Appendices

A

Integration method

For the simulations in this thesis a step variable numerical integration method is used based on the Runge-Kutta-Fehlberg method as described by Es-hagh (2005). This method makes sure accuracy remains high while simulation speed is maximized. It does this by varying the step size h based on the differences in calculations of different orders. It makes sure the desired accuracy ε is held during the whole simulation.

We begin with the starting arrays X0 for position and V0 for velocity

X0 = [x0, y0]

V0 = [vx0, vy0]

Which we combine in a single array

R0 = [x0, y0, vx0, vy0]

Where we define the time derivative f (R) such that f (R) = ˙R = [vx, vy, ax, ay]

From here we calculate certain values kn

k1= hf (Rn) k2= hf (Rn+ 1 4k1) k3= hf (Rn+ 3 32k1+ 9 32k2) k4= hf (Rn+ 1932 2197k1− 7200 2197k2+ 7296 2197k3) k5= hf (Rn+ 439 216k1− 8k2+ 3680 513k3− 845 4104k4) k6= hf (Rn− 8 27k1+ 2k2− 3544 2565k3+ 1859 4104k4− 11 40k5) After which we calculate the next steps in the simulation

Rn+1= Rn+ 25 216k1+ 1408 2565k3+ 2197 4104k4− 1 5k5 ˜ Rn+1= Rn+ 16 135k1+ 6656 12825k3+ 28561 56430k4− 9 50k5+ 2 55k6

Where Rn+1 is of fifth order and ˜Rn+1 of sixth. To check if step size h is acceptable we

look at δ = 0.84  ε |rn+1| 1/4 where rn+1 = ˜ Rn+1− Rn+1 h

if δ ≤ 0.1 then h is replaced by 0.1h, if δ ≥ 4 then h is replaced by 4h and if 1 < δ < 4 then h is replaced by δh. If not h remains.

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B

Table of parameters used

Table 2: Table of (some of) the parameters used Parameter Description Dimension G Gravitational constant m3kg−1s−2

R Gas constant m2kg s−2K−1mol−1 r Radial distance from star m

s Radius of pebble m

Mstar Mass of star kg

M Mass of planetesimal kg

m Mass of pebble kg

¯

m Mean molecular mass of gas particle g mol−1 T Temperature of the gas K ρs Density of pebble kg m−3

ρ = ρg Density of surrounding gas kg m−3

g = GMstar

r2 Radial acceleration of planetesimal m s

−2 vk= q GMstar r Kepler velocity m s −1

vg Orbiting velocity of gas m s−1

∆g = v2g−v2k r = 1 ρg dP dr Residual gravity m s −2 η = vg−vk vk =  ∆g 2g 

Deviation in velocity (gas-Kepler) Dimensionless ¯

v =q3RTµ Mean thermal velocity of gas m s−1 ts= mvF

D Stopping time s

tsEpstein= ρρ¯svs Stopping time Epstein regime s

tsStokes = 2ρss

2

9ζ Stopping time Stokes regime s

Ω0 = vrk =

q

GMstar

r3 Kepler frequency s−1

τs = tsΩ0 Stokes number Dimensionless

λ = µ

4√2πl2

mρg Mean free path of a gas particle m

ζ = µ¯v

4√2πl2 m

Kinetic viscosity of gas kg m−1s−1 lm Radius of spherical gas particle m

cs=

q

RT

µ Speed of sound in a gas m s −1

H = cs

Ω0 Scale height of planetary disk m

rH= r

q

M

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