RSS feed for The formation of exoplanets

Introduction

The Open University — Wed, 30 Oct 2024 00:00:00 GMT

The idea that our Solar System may not be unique, and that there might be planets orbiting other stars (or exoplanets), has been around for a long time. Important principles that underpin exoplanet research today were foretold by key discoveries in the eighteenth and nineteenth centuries. In 1783, an unseen companion was presented as an explanation of the peculiar periodic dimming observed for the bright star Algol, and in 1844 the observation of a periodic change in position of the bright stars Sirius and Procyon uncovered their two unseen companions. The concept of detecting and exploring an unseen object by studying its influence on a nearby astronomical source has also been applied to exoplanets and their host stars.

However, exoplanets were expected to be extremely hard to observe in practice. Using the orbits and size of planets in our Solar System as a guide, the influence of an exoplanet on its host star was predicted to be very small. Indeed, it took until the late twentieth century for technological advancements to enable the first exoplanet to be identified, and the result was surprising: 51 Pegasi b (named after its Sun-like host star 51 Pegasi in the Pegasus constellation) was unlike anything in our Solar System. 51 Pegasi b is a hot Jupiter: a planet with a similar mass to Jupiter, but with an incredibly high surface temperature as its orbit takes it very close to its host star. The fact that a planet like 51 Pegasi b exists triggered what became a radical overhaul of theories of planet formation and evolution. But the discovery of a hot Jupiter was also encouraging, as it showed that exoplanet detection was perhaps not quite as impossible a challenge as many had assumed. All astronomers needed to do was start looking for something that is different from the planets of the Solar System.

Since the discovery of 51 Pegasi b, thousands more exoplanets have been discovered by a variety of techniques. The main outcome of exoplanet searches and characterisation studies carried out using these techniques is a known exoplanet population with a wide variety of physical and orbital characteristics. This course explores the key astrophysical concepts involved in planet formation and how they can be used to explain the great diversity of configurations observed.

Section 1 focuses on the birthplaces of planets: the protoplanetary discs. Section 2 describes the core-accretion scenario for planet formation. This was initially developed to explain the existence of Jupiter, but has, over the years, become a more general model of planet formation for its ability to account for a large diversity of planetary outcomes, from Earth-sized planetary cores that form first, to ice and gas giants that evolve later. Lastly, Section 3 focuses on the final stages of planet formation, and explores the core-accretion scenario further, as well as an alternative disc-instability scenario, where the formation of massive objects occurs first from the collapse of gas into clumps by self-gravity.

This course material will refer to masses of stars in terms of the mass of the Sun (represented by M_☉ = 1.99 × 10³⁰ kg), and masses of planets in terms of the mass of the Earth (represented by M_⊕ = 5.97 × 10²⁴ kg) and the mass of Jupiter (represented by M_Jup = 1.90 × 10²⁷ kg). Orbital distances from stars will be expressed in terms of the distance of the Earth from the Sun; this is 1 astronomical unit (1 au = 1.496 × 10¹¹ m).

This OpenLearn course is an adapted extract from the Open University course S384 Astrophysics of stars and exoplanets.

Learning outcomes

The Open University — Wed, 30 Oct 2024 00:00:00 GMT

After studying this course, you should be able to:

use mathematical models to calculate properties of protoplanetary discs
understand the core-accretion scenario for the growth of planetary cores from smaller components
understand the disc-instability scenario for the formation of planets from a circumstellar disc
describe how planets migrate and interact after forming
appreciate how planet formation models can explain the observed exoplanet population.

1 Protoplanetary discs

The Open University — Wed, 30 Oct 2024 00:00:00 GMT

The idea that planets form from initially microscopic solid material within protoplanetary discs made predominantly of gas dates back to the Enlightenment, possibly starting with the idea that the planets of the Solar System formed out of a nebula surrounding the Sun, which featured in Kant’s Universal Natural History and Theory of the Heavens. Today, our understanding of protoplanetary discs stems from experimental observations and theoretical models of the behaviour of gases in the gravitational fields of stars.

1.1 Observations of protoplanetary discs

The Open University — Wed, 30 Oct 2024 00:00:00 GMT

The presence of discs around newborn stars is a natural consequence of the collapse of the molecular cloud from which they form, as a mechanism to conserve angular momentum. The first evidence of the existence of protoplanetary discs came thanks to the Hubble Space Telescope (HST) in the mid-1990s, which was more or less at the same time as the first exoplanet discoveries. Figure 1 shows the HST images of the protoplanetary discs around four young stars in the Orion nebula, 1500 light-years from the Sun, compared with the much more detailed view of a protoplanetary disc in the same region obtained with the James Webb Space Telescope (JWST) in 2022. Protoplanetary discs have been observed mainly around young stars (with ages of about one to ten million years) that are close to the final stages of formation. This means that most of the material from the disc has been accreted by the central star, so the mass of the disc (M_disc) is much lower than the mass of the star (M_*).

Figure 1 (a) Some of the first images of protoplanetary discs, in the Orion nebula, taken in 1993 with HST. (b) Protoplanetary disc in Orion, imaged with JWST in 2022. The orbit of Neptune is shown for scale.

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The figure shows the following photos: Part (a) show four images of protoplanetary discs, in the Orion nebula, taken in 1993 with HST. Each image has a central glowing dot of varying size that is surrounded by a dark cloud of varying thickness. Part (b) shows a protoplanetary disc in Orion, imaged with JWST in 2022. A white clove-shaped light is seen with a central horizontal band labelled ‘disc’.

Figure 1 (a) Some of the first images of protoplanetary discs, in the Orion nebula, taken in 1993 with HST. (b) Protoplanetary disc in ...

The presence of planets in protoplanetary discs is strongly supported by observations, which have been supported by the development of ground-based instruments such as SPHERE.

Box 1 SPHERE

Figure 2 The SPHERE instrument (dashed--dot outline) mounted on the side of one of the four telescopes that form the VLT complex in Chile.

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This is a photograph of the SPHERE instrument in an industrial environment.

Figure 2 The SPHERE instrument (dashed--dot outline) mounted on the side of one of the four telescopes that form the VLT complex in Chile.

SPHERE (Spectro-Polarimetric High-contrast Exoplanet REsearch) is an instrument operating at near-infrared and visible wavelengths, installed on one of the four telescopes comprising the European Southern Observatory’s (ESO) Very Large Telescope (VLT) site in Paranal (Chile). SPHERE is one of the first dedicated direct-imaging instruments and its primary science goal is to directly detect and characterise young exoplanets and the discs in which they form.

Recent observations using VLT/SPHERE include the direct detection of two forming planets in the disc around the young T Tauri star PDS 70 (which gets its name from the Pico dos Dias Survey for young stellar objects). One of the directly imaged planets is shown in Figure 3.

Figure 3 VLT/SPHERE image of PDS 70 with a planet in a gap in the disc. The other planet in the system is obscured by the bright region of the disc to the right of the central star.

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An image is shown with scales on the axes. The horizontal axis is labelled ‘Delta R A in arcsec’ and ranges from 0.9 to negative 0.9 in decrements of 0.1 unit. The vertical axis, labelled ‘Delta Dec in arcsec’, ranges from negative 0.9 to 0.9 in increments of 0.1 unit. At the centre of the graph (0.0, 0.0), a central cavity with a star as a tiny bright dot is shown against a deep red background. A planet is shown as a bright light on the lower left of the cavity. The planet’s disc is shown as an elliptical orbit around the cavity, with an axis of 1.4 arcsec in the vertical direction and 1.0 arcsec in the horizontal direction. The disc appears as a brighter shape mostly on the right side of the star.

Figure 3 VLT/SPHERE image of PDS 70 with a planet in a gap in the disc. The other planet in the system is obscured by the bright region ...

1.2 Modelling protoplanetary discs

The Open University — Wed, 30 Oct 2024 00:00:00 GMT

Material in a protoplanetary disc will be in orbit around a central star (or protostar). A first approximation to the motion of the material is that it is in so-called Keplerian motion, that is it obeys Kepler’s laws of planetary motion. In particular Kepler’s third law, which is a consequence of Newton’s law of gravity, states that the square of the orbital period is proportional to the cube of the orbital radius (assuming circular orbits, which is generally the case), i.e. P² ∝ a³. As long as the orbiting particle has a mass that is small compared to that of the star, this may be expressed in the following equation:

where G is the gravitational constant (6.674 × 10^-11 N m² kg^-2) and M_* is the mass of the star. The Keplerian orbital speed is therefore the distance travelled in an orbit (the circumference of the orbit, 2πa) divided by the orbital period, P. This is therefore

Equation label: (Equation 1)

Figure 4 shows a schematic view of a protoplanetary disc, comprised mostly of molecular hydrogen. The rest of this section will aim to derive and solve the differential equations that govern the vertical (out-of-disc plane) gas-density profile as well as the radial dependence of the orbital (in-disc plane) velocity, which turns out to differ from the pure Keplerian motion. The following two subsections therefore will look in turn at how these two properties may be quantified.

Figure 4 Schematic illustration of a protoplanetary disc.

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In the figure, a central yellow sphere labelled ‘M subscript asterisk’ is shown. A horizontal line is drawn across the sphere, such that the sphere lies in the middle of the line. This line is labelled ‘midplane (z equals 0)’. A right arrowhead is drawn at the right end of the line. This arrowhead is labelled ‘r-axis’. An upward arrow labelled ‘z-axis’ is drawn from the sphere. The protoplanetary disc is shown as a semicircular ring around the yellow sphere, lying on the horizontal plane facing away from the observer. The two cross-sections of the ring, lying on either side of the sphere are symmetrical about the z-axis. The height of the disc increases on either side of the horizontal line towards the periphery, forming two triangular cross-sections. A thin green layer covers the upper and lower surfaces of the disc, and the interior portion of the disc is shown as a blue region with several black dots. The black dots lying closest to the horizontal line are bigger, and the size decreases the further the dots are from the horizontal line. A point labelled ‘A’ is taken near the top-right corner of the right disc. A line of length d joins A and the centre of the sphere. This line forms an angle theta with the horizontal line. From A, a vertical line of length z is drawn to the horizontal line.

Figure 4 Schematic illustration of a protoplanetary disc.

1.3 Vertical gas-density profile

The Open University — Wed, 30 Oct 2024 00:00:00 GMT

In the direction perpendicular to the disc plane (vertically, corresponding to the z-axis in Figure 4), the profile of the gas density is such that the vertical pressure gradient and the vertical component of the stellar gravity are in hydrostatic equilibrium. Since, as mentioned earlier, , any disc contribution to the gravitational force can be ignored. Therefore we may write:

Referring to Figure 4, the vertical component of the stellar gravity at a point A located a distance d from the star is , where G is the gravitational constant.

Figure 4 (repeated) Schematic illustration of a protoplanetary disc.

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Figure 4 (repeated) Schematic illustration of a protoplanetary disc.

The angle θ is such that , hence . Now, the distance d is given by but for geometrically thin discs, , so we have simply , and therefore

Note that the Keplerian angular speed ω_K at this same point in the disc is

Equation label: (Equation 2)

so and we may write

This equation can be simplified by recognising that, for an ideal gas, the pressure and density are related by the sound speed c_s such that

Equation label: (Equation 3)

where k_B is the Boltzmann constant, T is the temperature of the disc and is the mean molecular mass. The sound speed may be assumed to be constant for a given disc. Hence, and the expression of hydrostatic equilibrium becomes

Equation label: (Equation 4)

This differential equation has the following solution, which gives the density in terms of the disc scale height H and the density at the midplane ρ₀:

Equation label: (Equation 5)

where

Equation label: (Equation 6)

and

Equation label: (Equation 7)

Here, is the surface density of the disc (i.e. its mass per unit surface area).

What is the gas density at a height of z = H?
The gas density is .

The shape of a disc can be described by its aspect ratio , where is the Keplerian speed at a radius r. Normally, for protoplanetary discs,

Equation label: (Equation 8)

This is because the speed of sound (Equation 3) and the temperature profile of the disc is driven by the stellar irradiation, so that usually . (The reason for this latter dependence is that, from the Stefan-Boltzmann law, the temperature of the disc where the flux received from the star .) Hence, and this result, combined with the fact that (Equation 2), leads to Equation 8.

How does Equation 8 explain the shape of the disc shown in Figure 4?
The aspect ratio of the disc increases with r, so the disc is expected to be thicker on the edge than in the centre, like the one in Figure 4.

1.4 Radial dependence of the orbital velocity

The Open University — Wed, 30 Oct 2024 00:00:00 GMT

Having considered the vertical density profile of the disc, we now turn to the radial dependence of the orbital velocity . In the radial direction, in addition to the gravitational force, there is also a force due to the pressure gradient. Hence, the net centripetal acceleration of a small volume of gas in the disc on an assumed circular orbit at radius r is

Here, since M_* is much greater than the total mass of the disc inside the orbital radius. Therefore, the orbital speed v_orb of the gas in the disc has two components: one due to the Keplerian speed , and one due to this extra pressure gradient. It is given by:

Equation label: (Equation 9)

Usually, in the disc, so the gas will behave as if it was feeling a slightly lower gravitational pull from the star, and its orbital speed will be sub-Keplerian, that is, . The following activity shows how to quantify the magnitude of the deviation between the actual orbital speed of the gas and its Keplerian speed.

Activity 1

Using Equation 9 and approximating the pressure gradient as , where n is a dimensionless constant:

a.Write an approximate expression for as a function of the radius, scale height and Keplerian speed, .
b.Calculate the difference between the orbital and Keplerian speeds, , at a radius of 1 au from a star of the same mass as the Sun, for a disc of constant aspect ratio H/r = 0.05 and n = 3. (You may assume 1 au = 1.496 × 10¹¹ m, 1 M_☉ = 1.99 × 10³⁰ kg, and G = 6.674 × 10^-11 N m² kg^-2.)

Answer

a.From Equation 3, the pressure of the gas P_gas is linked to the sound speed c_s such that . Using this, the expression for the pressure gradient becomes:
then substituting this into Equation 9 gives
Finally, using the fact that
the requested expression is obtained as:
Equation label: (Equation 10)
b.From Equation 10, the difference in velocities is
So the difference is only about 0.4% of the Keplerian velocity.
To evaluate Δv at r = 1 au we need to calculate the Keplerian velocity v_K at 1 au:
which gives . So the difference between the orbital and Keplerian speeds at this radius is about 100 m s^-1.

2 Rising from the dust

The Open University — Wed, 30 Oct 2024 00:00:00 GMT

The basis of the core-accretion scenario is that planets form by accumulation of solids into a core, on which an atmosphere is accreted once a critical value of the core mass is achieved. This section explores the various stages of this process, going from sub-micron-sized dust particles to metre-sized rocks that grow into kilometre-sized planetesimals and finally into Mercury-sized planetary embryos, spanning roughly 12 orders of magnitude in size. The embryos then accumulate further matter, becoming the cores of fully formed planets.

2.1 From dust grains to rocks

The Open University — Wed, 30 Oct 2024 00:00:00 GMT

Consider again the protoplanetary disc from Activity 1. The fact that the velocity of the gas in a protoplanetary disc is usually sub-Keplerian has important consequences for the evolution of solid particles embedded in it. A consequence of Equation 10 is that for geometrically thin discs () the radial pressure gradient makes a negligible contribution to the orbital speed of the gas. However, as seen in Activity 1, the difference in speed can be of the order of Δv ~ 100 m s^-1 at ~1 au from the star and this turns out to be important in determining how the particles in a disc behave. In particular, a finite Δv can cause particles in the disc to slow down and drift inwards towards the star.

One of the most important parameters that determines how a particle of mass m interacts with the gas surrounding it is the stopping time, defined as

Equation label: (Equation 11)

where F_drag is the magnitude of the drag force that acts in the opposite direction to Δv. This stopping time may be related to the Keplerian orbital speed by

Equation label: (Equation 12)

where τ_S is the Stokes number, which characterises how well particles follow fluid streamlines. Large particles will generally have large Stokes numbers (), and small particles will generally have small Stokes numbers ().

Small particles of radius s will be coupled with the gas; that is, they will move at almost the same speed as the gas. Such particles experience a drag force whose magnitude is given by

Equation label: (Equation 13)

where the thermal speed of the gas, v_th, is roughly the same as its sound speed, c_s. For spherical particles, the material density is . So, by combining this with Equations 11 and 13, we have

Equation label: (Equation 14)

We can now define the radial drift speed, v_rad, as the speed with which particles in the disc move radially through it, drifting inwards towards the star. To derive a general expression for the radial drift speed we write the orbital speed as

Equation label: (Equation 15)

where, from Equation 10, we already know that . Remember that n is a dimensionless constant relating the pressure gradient to P_gas/r. After much algebra concerning the equations of motion and making appropriate approximations (the details of which are not important here), a general expression for the radial drift speed is found as:

Equation label: (Equation 16)

The radial drift speed will reach its maximum value when the Stokes number is τ_S = 1. This corresponds to the situation when the stopping time and Keplerian orbital speed are related by . In this case, Equation 16 shows that the radial drift speed is .

Since η will be a small number for geometrically thin discs, make use of the approximation that to obtain a simple relation between Δv and v_K. Hence write the maximum radial drift speed in terms of Δv.
From Equation 10, the difference between the Keplerian speed and the orbital speed is

Therefore, using the approximation for small values of η, we have

and so . Hence, the maximum radial drift speed is

So the maximum radial drift speed is simply the difference between the Keplerian and orbital speeds.

Activity 2

a.For large particles, with s > 1 m, the Stokes number is very large, . Obtain an expression for the radial drift speed in this case, in terms of τ_S and Δv only.
b.For small particles, with s < 1 cm, the Stokes number is very small, . Obtain an expression for the radial drift speed in this case, in terms of τ_S and Δv only.

Discussion

a.For large particles so Equation 16 becomes
Then, substituting for v_K using the approximation , we have
b.For small particles so Equation 16 becomes
Then, substituting for v_K using the approximation , we have

Activity 2 showed that, in the limits of large or small particles, both values of the radial drift speed are independent of η. In each case, the radial drift speed is a small fraction of Δv.

2.2 Assembling the planetesimals

The Open University — Wed, 30 Oct 2024 00:00:00 GMT

Thanks to the coupling with the disc, small (sub-micron-sized) particles will collide with each other gently enough that they will always stick together. Therefore, they will efficiently form millimetre-sized aggregates, in a process referred to as coagulation, that tend to settle on the midplane of the disc.

The simplest assumption for the formation of a planetesimal is that this process continues up to the kilometre-sized scale. As the particles grow, however, so does their speed, and the outcome of a collision no longer necessarily leads to bigger objects, as energetic impacts can be neutral (so the particles will bounce off each other) or even destructive (so the particles fragment and are broken apart once more).

However, there is also another problem that occurs around the metre-sized scale, as illustrated by the following activity. Metre-sized particles, referred to as ‘rocks’ will have τ_S ~ 1, and so move with the maximum radial drift speed.

Activity 3

a.Consider a thin protoplanetary disc with aspect ratio H/r = 0.05 and dimensionless pressure constant n = 3. Calculate the radial drift speed for a particle with τ_S = 1 at 1 au from a star of mass 1 M_☉. Hint: the Keplerian speed at 1 au from a 1 M_☉ star as calculated in Activity 1 is v_K = 29.8 × 10³ m s^-1.
b.Moving at the constant speed from part (a), how long would the particle take to travel a distance of 1 au? Express your answer in terms of the number of orbital periods at 1 au.

Discussion

a.In this case,
so with τ_S = 1, the radial drift speed is
(Note that this is equal to Δv, calculated in Activity 1, which is as expected according to the expression for the maximum radial drift speed derived earlier for the case τ_S = 1.)
b.The time t to travel a distance of 1 au radially at the speed from part (a) is
Since the orbital period at a distance of 1 au from a 1 M_☉ star is 1 y = 3.16 × 10⁷ s, this timescale is only about 40 orbital periods.

Activity 3 showed that the radial drift for metre-sized particles can be very rapid. Radial drift therefore reduces the abundance of rocks in the outer regions of the disc, while potentially increasing it in the regions closer to the star. The fact that rocks move through the disc very rapidly gives rise to the ‘metre-sized barrier problem’ in explaining how planetesimals form: the growth to kilometre-sized planetesimals must happen fast enough to be complete before the medium-sized particles drift toward the centre, but also occur via a mechanism that avoids fragmentation.

Figure 5 shows a schematic view of the current picture of planetesimal formation. Once smaller fragments have formed, they settle vertically into the disc: see Figure 5(a). Next, the fragments drift radially towards the centre, leading to a possible build-up of solids in the inner disc: see Figure 5(b). Over-densities of solid material form through streaming instabilities and then lead to the formation of planetesimals through gravitational collapse: see Figure 5(c).

Figure 5 Schematic illustration of the formation of planetesimals.

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The figure has three parts: In part (a), a cross-section of a protoplanetary disc is shown. The cross-section resembles a triangle, which is broader on the right and tapers towards the left. The leftward arrow at the left end is labelled ‘to star’. A thin green layer is shown bounding the top and bottom surfaces of the disc. The interior of the disc is a blue region having two groups of black dots of varying sizes. From each of the black dots near the top surface, a downward arrow is shown. From each of the black dots near the bottom surface, an upward arrow is shown. This diagram is labelled ‘coagulation and vertical settling’. Part (b) shows the same cross-section of a protoplanetary disc. Now, the blue region is empty. A series of short leftward arrows is shown in a horizontal line passing through the middle of the disc. These arrows are labelled ‘radial drift’. The middle portion of the disc is labelled ‘possible pile-up of solids in inner disc’. A rectangular section of the middle portion of the blue region is magnified. In the magnified image, two thin green bands are shown between thick blue bands. These thin bands are labelled ‘gravitational collapse of streaming instability over-densities’. In part (c), the same cross-section is shown. Now the blue region is filled with black dots. The dots nearer to the green bands at the top and at the bottom are smaller in size, while the dots lying in the middle portion of the blue region are larger in size. The smaller dots are labelled ‘remaining solids’. This diagram is labelled ‘formation of planetesimals with a range of initial masses’.

Figure 5 Schematic illustration of the formation of planetesimals.

2.3 The growth of planetary cores

The Open University — Wed, 30 Oct 2024 00:00:00 GMT

Once kilometre-sized planetesimals have formed with a mass of ~ 10¹² – 10¹³ kg, they are massive enough to interact significantly with their neighbours via gravity and modify their velocity, thus becoming prone to collisions.

In the same way as for the smaller particles, collisions of planetesimals with other planetesimals need to happen at sufficiently low speed to lead to accretion. Under this assumption, the rate at which a planetesimal of mass M_p and radius R_p grows with time t can be written as:

Equation label: (Equation 17)

Here, Σ is the surface density of the disc, v_esc is the planetesimal’s escape velocity, v_rel is the relative velocity between the two impacting planetesimals and F_g is the gravitational focusing, which is a dimensionless parameter that describes how the gravitational attraction between two bodies increases their collision probability.

How does dM_p/dt change with the disc’s surface density Σ?
The growth rate scales linearly with the disc’s surface density, so the growth rate will be higher for discs with a higher mass in planetesimals.

And how does growth rate scale with the distance from the central star?
With everything else being equal, growth is slower at large distances where the Keplerian angular speed, ω_K is smaller.

The following activity shows a quantitative example of the impact of the gravitational focusing on the growth rate, considering a planetesimal in an orbit similar to that of Jupiter.

Activity 4

a.Assuming that F_g is constant, starting from Equation 17 write an expression for the planetesimal’s radius growth rate dR_p/dt as a function of its density ρ_p.
b.Estimate the value of F_g needed for a planetesimal at the same distance as Jupiter with ω_K = 0.16 y^-1 to grow to a radius of R_p = 1000 km in 10⁵ years. Use a surface density of Σ = 100 kg m^-2 and a planetesimal density of ρ_p = 3000 kg m^-3.

Answer

a.Assuming the planetesimal to be spherical, its mass M_p can be expressed in terms of its radius R_p and density ρ_p as:
Equation label: (Equation 18)
then we note that
Since dM_p/dt is given by Equation 17, and (by differentiation of Equation 18), this means
Equation label: (Equation 19)
b.Using the values provided, we obtain:
To reach a radius of 1000 km in 10⁵ years, the desired growth rate must be dR_p/dt= 10⁶ m / 10⁵ = 10 m y^-1.
Hence, the gravitational focusing needs to be approximately

The expression that involves F_g (Equation 17) depends on the relative velocities between planetesimals, the range of which is characterised by the velocity dispersion within the disc. Hence, velocity dispersion plays a crucial role in determining the accretion rates. It turns out that there are two regimes:

In cases where gravitational focusing of planetesimals is initially very strong, runaway growth occurs. This is an accelerated phase in the growth of planetesimals such that the largest bodies get larger at a rapid and increasing rate, proportional to their mass. This phase is thought to be generally quite short, and ends when the velocity dispersion of the resulting planetary embryos increases to the point where gravitational focusing is ineffective.
In cases where the largest planetary embryos grow quickly while the smallest grow slowly, oligarchic growth occurs. This leads to a bimodal mass distribution with a number of embryos comparable to the mass of the Moon, Mercury or Mars (~10²³ kg) embedded in a large population of smaller planetesimals.

2.4 The isolation mass

The Open University — Wed, 30 Oct 2024 00:00:00 GMT

Once the oligarchic growth phase is over, the resulting embryos are relatively isolated and on initially circular orbits. They continue growing into planetary cores, by accreting the nearby leftover planetesimals within a feeding zone which extends a distance Δa either side of the planetary core. We may write the radius of this feeding zone as Δa = CR_Hill, where C is a constant and R_Hill is the Hill radius, defined as:

Equation label: (Equation 20)

Here, M_p is the mass of the planetary core, M_* is the mass of the star and a is the radius of the orbit. The Hill radius is defined as the distance from the planetary core at which its gravitational force dominates over that of the star.

The growth of the cores continues until all the neighbouring planetesimals have been consumed. At this point, the mass of the core reaches the isolation mass M_iso, defined as the total mass of the planetesimals within the feeding zone.

Equation label: (Equation 21)

The origin of Equation 21 is explored in the next activity.

Activity 5

a.Write down an expression for the mass of planetesimals within the feeding zone in terms of the disc surface density Σ, distance to the central star a and feeding zone width Δa. Hence, derive Equation 21.
b.Use Equation 21 to evaluate M_iso in the terrestrial planets region at a_⊕ = 1.0 au and in the Jovian planets region at a_Jup = 5.2 au, for M_* = 1 M_☉, Σ = 100 kg m^-2 and C = 2√3.

Answer

a.The mass of planetesimals within the feeding zone is the area of the annulus with width 2Δa at a radius a, multiplied by the surface density. Hence,
The width of the feeding zone is defined in terms of the Hill radius of the resulting planetary core (Equation 20) as Δa = CR_Hill. Therefore, once this mass is all contained within a single core, its mass is given by
This simplifies to
which may be rearranged to give the requested expression.
b.In the terrestrial planets region (where a = a_⊕ = 1.0 au), this gives
(This is about 0.066 times the mass of the Earth, or around the mass of Mercury.)
Similarly, at the distance of Jovian planets (where a = a_Jup = 5.2 au), Equation 21 gives an isolation mass that is (5.2)³ larger. Hence,
(This is about 9.3 times the mass of the Earth, which is around half the mass of Neptune.)

As shown in Activity 5, the fact that M_iso ∝ a³ means that more massive cores up to several Earth masses can form at larger distances from the central star. However, at a ~ 10 au the speed of the planetesimals is too high for the collisions to lead to accretion, so it becomes increasingly hard to build massive planetary cores.

Planetary cores formed in this way will have sizes from around that of Mercury (with radius a few thousand kilometres) to several times that of the Earth (with radius a few tens of thousand kilometres).

Activity 6

Summarise the typical sizes of objects involved in the various stages of the core-accretion scenario.

Discussion

Initially, the particles are dust grains with a typical size of a micron or less which coagulate into millimetre-sized aggregates. These accumulate into rocks that are around one metre in size, which grow further into kilometre-sized planetesimals. Gravitational focusing helps these grow into planetary embryos with sizes and masses around that of the Moon, Mercury or Mars. These then become planetary cores by accreting leftover planetesimals within their feeding zone to reach a mass and size of a few times that of the Earth.

3 The final stages of planet formation

The Open University — Wed, 30 Oct 2024 00:00:00 GMT

This section explores the possible outcomes of the final stages of planet formation, starting with the formation of giant planets via core accretion, then turning to alternative formation routes and, finally, exploring the effect of migration and planet–planet interaction on the final architectures of planetary systems.

3.1 Forming giant planets via core accretion

The Open University — Wed, 30 Oct 2024 00:00:00 GMT

Once the mass of a planetary core reaches a few Earth masses, it starts to build up a gas envelope. This process can lead to a variety of outcomes because, as seen in Activity 5, the speed at which a core grows and its final mass depend on where in the disc it is found.

Figure 6 Schematic view of possible outcomes of the core-accretion model.

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The figure has three parts. Each part is comprised of two stages: formation and output. In part (a), in the formation stage, two hemispheres with common centres are shown on two sides of a vertical line. The left side of the line is labelled ‘core formation by solid accretion’. Two identical smaller spheres are drawn on the left side of the left hemisphere. An arrow from each of the smaller spheres points towards the centre of the left hemisphere. The right side of the line is labelled ‘gas accretion beyond critical mass’. The hemisphere on the right is bigger in size. A blue ring is shown around the right hemisphere. A set of radial arrows points towards the ring around the right hemisphere. In the output stage, a photo of Jupiter (a gas giant) is shown. In part (b), in the formation stage, a similar diagram to that in part (a) is shown. The left side of the line is labelled ‘core formation by solid accretion’. The right side of the line is labelled ‘slow core accretion’. Here, the blue ring is thinner compared with part (a) and there are fewer radial arrows. In the output stage, a photo of Neptune (an ice giant) is shown. In part (c), in the formation stage, another similar diagram is shown. The left side of the line is labelled ‘core formation by solid accretion’. The right side of the line is labelled ‘growth in the region of the disc with little solids’. Here, both hemispheres are of the same size. The blue ring is far thinner compared with parts (a) and (b), and surrounds both hemispheres. In the output stage, a photo of Earth (a terrestrial planet) is shown.

Figure 6 Schematic view of possible outcomes of the core-accretion model.

Figure 6(a) shows that for Jupiter-like gas giants to form, the core needs to reach a critical mass, high enough that the gas envelope cannot maintain hydrostatic equilibrium and start contracting. (Usually, M_crit is in the approximate range 5 – 20 Earth masses.) Exceeding the critical mass triggers a phase of rapid accretion, which continues until either the gas is dispersed or the planet opens a gap in the disc and the rate of gas accretion slows down. (In the core-accretion scenario, the gas-dispersal timescale is one of the factors that governs the lifespan of the disc and hence the final mass of gas giants.)

Figure 6(b) shows that if the core grows in a region of the disc where accretion is slower than in (a), there will be less gas in the vicinity of the planetary core by the time the critical mass is reached. This will typically occur further out than (a). Therefore, the final mass of the gas envelope will be smaller than it is for gas giants, and the resulting planet will be a core-dominated ice giant, like Uranus and Neptune.

Finally, Figure 6(c) shows that if the timescale for the core growth is much smaller than the gas-dispersal timescale, or if the core grows much closer to the star than gas or ice giants (where very little solid material is available) the planet will only develop a thin hydrogen-dominated atmosphere, like that of the primordial Earth.

Many observed protoplanetary discs show gaps, bright rings, asymmetries, spirals and other structures. Figure 7 provides a stunning example of the variety of configurations that can arise from the interactions of forming planets with the disc.

Figure 7 Gallery of protoplanetary disc images obtained with the Atacama Large Millimeter Array (ALMA).

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In this figure, 21 types of protoplanetary discs are shown, arranged in five rows and four columns. Each disc appears unique with different thicknesses and orientations. Some are circular and some are elliptical, some have ring structures within them.

Figure 7 Gallery of protoplanetary disc images obtained with the Atacama Large Millimeter Array (ALMA).

3.2 Super-Jupiter exoplanets

The Open University — Wed, 30 Oct 2024 00:00:00 GMT

The core-accretion scenario discussed so far has its origin in the search for an explanation for our Solar System which, until the discovery of 51 Pegasi b, was the source of all available observational data used to constrain planet-formation theories. While, in its modern form, the core-accretion scenario succeeds in explaining some of the characteristics of the observed exoplanet population, there are still some aspects of planet formation that the model struggles to describe. One such aspect is connected with the formation of giant planets and, in particular, of directly imaged planets in wide orbits.

As briefly mentioned in Section 3.1, the main problem with the growth of giant planets via core accretion is connected with the lifetime of the discs, which appear to have gas-dissipation timescales that are too short to allow for the formation of many of the observed gas-giant exoplanet systems. This is true for Jupiter-sized exoplanets, but becomes even more crucial when considering the directly imaged super-Jupiters (exoplanets with masses several times that of Jupiter), like the one orbiting the young solar-type star YSES 2 (named after the Young Suns Exoplanet Survey). The YSES 2 planetary system, which is part of the Scorpius–Centaurus association, is shown in Figure 8. The giant exoplanet YSES 2 b, which is visible as a bright dot indicated with the arrow in the figure, has a mass of about 6 Jupiter masses and a semimajor axis of around 110 au, and is one of the few directly imaged planets around a solar-type star.

Figure 8 Near-infrared image of the planet around the young star YSES 2, obtained with the VLT/SPHERE instrument.

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An image is shown with scales on the axes. The horizontal axis, labelled ‘Delta R A in arcsec’, ranges from 1.2 to negative 1.2 in decrements of 0.2 units. The vertical axis, labelled ‘Delta Dec in arcsec’, ranges from negative 1.2 to 1.2 in increments of 0.2 units. At the centre of the graph (0.0, 0.0), a central ring is shown with bright orange streaks and black streaks emanating radially from the ring. The intensity of the orange and black streaks reduce as they move away from the central ring. A planet is shown as an orange sphere on the lower right of the ring close to the periphery of the emanating streaks.

Figure 8 Near-infrared image of the planet around the young star YSES 2, obtained with the VLT/SPHERE instrument.

3.3 The disc-instability scenario

The Open University — Wed, 30 Oct 2024 00:00:00 GMT

There is, however, an alternative to the core-accretion scenario that succeeds in predicting the formation of giant planets, including those with M_p > M_Jup via direct collapse of the gas in the protoplanetary disc. The key idea of this disc-instability scenario is that a sufficiently cold and/or massive disc tends to be gravitationally unstable. Thus, the disc can undergo fragmentation to form gravitationally bound clumps that evolve into giant planets. For fragmentation to occur, the local surface density in the disc needs to be high enough that the self-gravity of the gas and its differential rotation (both drivers of gravitational collapse) are higher than the thermal pressure (which counteracts collapse). These competing effects are nicely summarised by the Toomre criterion, which states that, for a disc to fragment, its Toomre Q parameter must satisfy:

Equation label: (Equation 22)

Here, c_s is the sound speed (Equation 3), ω_K is the Keplerian angular speed (Equation 2) and Σ is the gas surface density.

To understand the significance of the Toomre criterion for exoplanet systems, it is instructive to consider it in the context of a protoplanetary disc similar to the one that formed our Solar System. To that end, it is useful to introduce the concept of the minimum-mass solar nebula. This is a hypothetical protoplanetary disc with a surface density profile Σ_sn(r) defined as the minimum value of the surface density that a disc would need to have (as a function of radius r from a Sun-like star) to form our Solar System. The composition of this model nebula is derived from the observed mass of heavy elements in the Solar System planets, plus enough hydrogen and helium to mimic the solar composition. The distribution of this model nebula is determined by spreading the mass needed for each planet over an annulus extending over the distance between them. The accepted surface density distribution for the minimum-mass solar nebula is:

The following activity shows how to express Q as a function of the stellar mass, disc mass and the disc aspect ratio, and compares the minimum value of Σ required for fragmentation to that of the minimum-mass solar nebula.

Activity 7

a.Starting from the definition of scale height (Equation 6), demonstrate that for a protoplanetary disc with uniform surface density Σ, the Toomre Q parameter can be written as
Equation label: (Equation 23)
where r is the distance from the star, H is the scale height, M_* is the mass of the central star and M_disc is the mass of the disc.
b.Show that for the Toomre criterion to be satisfied, the surface density must obey:
c.Consider a disc with aspect ratio H/r = 0.05 around a solar-type star (M_* = 1 M_☉). Show that the minimum surface density at r = 1 au for the disc to fragment is roughly two orders of magnitude greater than that of the minimum-mass solar nebula at the same radius.

Discussion

a.Using the scale height definition, Equation 22 becomes:
Using (Equation 2), and noting that for a disc with uniform surface density, then Q becomes
as required.
b.Starting from Equation 23 and multiplying by a factor of and the relevant normalised quantities gives
For the Toomre criterion to be satisfied, we need Q < 1, therefore rearranging the equation above:
c.Using the result from part (b), for a solar mass star with a disc whose aspect ratio is 0.05, fragmentation occurs at r = 1 au when Σ > 1.4 × 10⁶ kg m^-2. Comparing this to the surface density of the minimum-mass solar nebula at r = 1 au gives

Activity 7(c) showed that the minimum-mass solar nebula does not meet the Toomre criterion for fragmentation at r = 1 au. In fact, the Toomre criterion would only have been met at distances greater than several thousand astronomical units in the case of the minimum-mass solar nebula. So the Solar System planets probably did not form via disc instability.

Computational simulations show that as a disc becomes unstable, due to Q falling below 1, shock waves are generated within the disc. These shock waves follow a spiral pattern and heat up the disc. Since , the net effect of the shocks is for Q to increase again so the disc stabilises. This effect is known as self-regulation and because of this the disc temperature and surface density tend to reach values for which Q ~ 1. Therefore, an additional condition is necessary for fragmentation: the cooling needs to be fast enough to prevent self-regulation. This is known as the cooling criterion and it is satisfied if the cooling time obeys:

Equation label: (Equation 24)

The fact that both the conditions in Equation 22 and Equation 24 need to be satisfied for fragmentation effectively limits the mass and semimajor axis of the planets that can form via the disc-instability scenario, as shown in Figure 9.

Figure 9 Mass and separation (semimajor axis) of possible planets forming around a solar-type star satisfying both the cooling and Toomre criteria, for a typical disc aspect ratio.

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The figure shows a graph. The horizontal axis is labelled ‘separation in au’ and ranges from 0 to 300 in increments of 50 units. The vertical axis is labelled ‘mass in M_Jup’ and ranges from 0 to 100 in increments of 10 units. In the graph, from approximately (30, 8), two curves are drawn. One curve is increases slowly towards the right, with decreasing gradient, and ends at (300, 46). The region just above this curve is labelled ‘Toomre criterion fulfilled’, and the region below this curve is labelled ‘not fulfilled’. The other curve increases rapidly with increasing gradient, and ends at (80, 100). The region just right of this curve is labelled ‘cooling criterion fulfilled’, and the region left of this curve is labelled ‘not fulfilled’. The region bounded by the two curves and the axes is shaded in red, and the remaining part is shaded in green. The central portion of the green region is labelled ‘allowed range for formation’.

Figure 9 Mass and separation (semimajor axis) of possible planets forming around a solar-type star satisfying both the cooling and Toomre...

3.4 The Jeans mass for fragmentation

The Open University — Wed, 30 Oct 2024 00:00:00 GMT

The typical mass of a planet formed through fragmentation can be estimated starting from the definition of its Jeans mass. The Jeans criterion states that a gas cloud will collapse if the cloud’s kinetic energy is less than the magnitude of its gravitational energy; the minimum mass of a gas cloud for which the Jeans criterion is met is known as the Jeans mass. For the geometry of a protoplanetary disc, the Jeans mass in terms of the surface density is

where T and are the temperature and mean molecular mass of the gas. Noting that the sound speed is given by (Equation 3), so the Jeans mass may be written as

Using Q = 1 to express the surface density Σ for a disc that is just becoming unstable (Equation 22) this becomes:

Then, recognising that (Equation 6) and (Equation 2), we have

Therefore, this gives the final result:

Equation label: (Equation 25)

Estimate the Jeans mass for a typical disc with H/r = 0.05, centred on a Sun-like star.
Inserting values into Equation 25:

(This is about 1.6 Jupiter masses.)

3.5 Migration and planet interaction

The Open University — Wed, 30 Oct 2024 00:00:00 GMT

Regardless of the mechanism involved, once planets have managed to form, they tend to interact with each other and with the remaining gas and planetesimals in the disc. Therefore, planets often end up in a different mass and orbital configuration than the one they had at formation. There are several ways this can happen.

Interaction with remaining gas in the disc

The angular momentum exchange between a planet and the remaining gas in the disc causes the planet to migrate. Migration affects both Earth-sized rocky planets and giant planets, and is one possible mechanism for the formation of hot Jupiters like 51 Pegasi b, which almost certainly formed much further out and migrated inwards.

Interaction with the remaining planetesimals

Giant planets can interact with the leftover planetesimals in the disc. The resulting exchange in angular momentum can cause the planetesimals to be ejected from the system.

Planet–planet interaction

There is no guarantee that newly formed planets will be on stable orbits. Instabilities can cause the planets’ orbits to cross, and the net effect of this is usually the ejection of the smaller-mass body involved in the interaction, leaving the surviving planet on a highly eccentric orbit. This effect could explain the high eccentricities seen in many exoplanetary systems.

Interaction with additional stellar companions

If a distant stellar companion is present and a planet is formed on an orbit that is misaligned with that of the binary companion star, the planet’s eccentricity will change due to the Kozai-Lidov effect. This is a dynamical phenomenon that affects systems where two bodies are orbiting each other in the presence of a third, more distant companion. The presence of the third body causes the position of the inner pair’s orbital periapsis to oscillate, leading to periodic exchanges between the planet’s orbital eccentricity and inclination on a timescale of many orbital periods. This is another possible explanation for the formation of hot Jupiters.

3.6 Comparing theory and observation

The Open University — Wed, 30 Oct 2024 00:00:00 GMT

Figure 10 shows a snapshot of the exoplanet population (as of early 2023). While being a result of the observational biases connected to the various detection techniques, the figure highlights some interesting trends, including the existence of types of planet that are not present in our Solar System. While core accretion successfully explains the bulk of the giant planet population discovered through transit, radial-velocity and microlensing techniques (orange diamonds, red squares and green triangles, respectively, in Figure 10), and disc instability may explain some of the planets discovered by direct imaging (blue triangles in Figure 10), neither scenario is able to easily explain the characteristics of all planets shown here.

Figure 10 Masses of the known exoplanets (in units of Earth mass) plotted against their orbital period in years. The different colours and shapes represent the different detection methods. Solar System planets are plotted as open circles. The dashed-dotted line marks the position of the Earth.

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The figure illustrates the masses of exoplanets plotted against their orbital periods as a graph. The horizontal axis is labelled ‘orbital period (in units of years)’, and ranges from 10 raised to the negative fourth power to 10 raised to the fourth power in increments of 100 units. The vertical axis is labelled ‘mass (in units of Earth Masses)’, and ranges from 10 raised to the negative second power to 10 raised to the fifth power in multiples of 10 units. In the graph, various types of entities are plotted as clusters of points. Solar System planets are plotted as white circles. Exoplanets detected via different methods are also shown. Those from the radial-velocity method are shown as a red squares, the transit method are yellow diamonds, the microlensing method are green triangles, direct imaging are blue triangles and the astrometry method are dark-blue hexagons. A dashed horizontal line is drawn from 10 raised to the zeroth power on the vertical axis. A dashed vertical line is drawn from 10 raised to the zeroth power on the horizontal axis at. At the intersection of these two lines, a white circle labelled ‘Earth’ is shown. The other planets of the Solar System are shown at the following position on the graph: Mercury at approximately (10 raised to the negative first power, 10 raised to the negative 1.2 power); Venus at approximately (10 raised to the negative 0.2 power, 10 raised to the negative 0.9 power); Mars at approximately (10 raised to the 0.3 power, 10 raised to the negative first power); Jupiter at approximately (10 raised to the 1.1 power, 10 raised to the 2.5 power); Saturn at approximately (10 raised to the 1.5 power, 10 squared); Uranus at approximately (10 raised to the 1.8 power, 10 raised to the first power); and Neptune at approximately (10 squared, 10 raised to the first power). A cluster of red squares is shown between approximately (10 raised to the negative second power, 10 raised to the zeroth power) to approximately (10 raised to the negative second power, 10 raised to the fourth power) and approximately (10 squared, 10 raised to the fourth power). A cluster of yellow diamonds is shown to the left of the dashed vertical line. A cluster of green triangles is shown to the right of the dashed vertical line. A cluster of blue triangles is shown between approximately 10 raised to the zeroth power and 10 raised to the fourth power on the horizontal axis. A small cluster of dark-blue hexagons is shown between (10 raised to the zeroth power, 10 raised to the third power) and (10 raised to the second power, and 10 raised to the fourth power).

Figure 10 Masses of the known exoplanets (in units of Earth mass) plotted against their orbital period in years. The different colours ...

In particular, the core-accretion scenario struggles to explain the formation of giant planets at orbital distances larger than a few astronomical units (corresponding to orbital periods longer than a few years), because of the extended time needed to form big enough cores at these distances. On the other hand, discs are unlikely to fragment at small orbital distances from the central star, because the stellar irradiation tends to stabilise the disc by maintaining high temperature (hence high sound speed and high Q), so the Toomre and cooling criteria cannot be satisfied simultaneously. Therefore, unless planets formed by the core-accretion scenario can migrate or be scattered outward to large distances, then the directly imaged giant planets at large orbital distances must have formed on their current orbits via another mechanism, such as the disc-instability scenario. This suggests that giant-planet formation could be bimodal, with different mechanisms dominating depending on the distance from the central star.

Activity 8

A current catalogue of known exoplanets is maintained at the website exoplanet.eu. Visit the website now, and you will see two buttons labelled ‘The catalog’ and ‘The plots’. The first of these allows you to explore the current catalogue of known exoplanets, while the second allows you to plot various planet parameters against each other. Try to produce an updated version of Figure 10 by plotting the masses of known exoplanets against their orbital periods. You can click on the axis labels to change the units in which the quantities are displayed.

4 Quiz

The Open University — Wed, 30 Oct 2024 00:00:00 GMT

Answer the following questions in order to test your understanding of the key ideas that you have been learning about.

Question 1

The disc composed of molecular hydrogen only.

The disc composed of helium only.

The disc composed of molecular hydrogen and helium.

The disc composed of molecular hydrogen, helium and heavier elements.

All four discs will have the same scale height.

The correct answer is a.

Answer

The scale height is given by . The Keplerian angular speed at a given radius will be the same for all four discs since it depends only on the stellar mass and the radius, which are the same in all four cases.

The sound speed is inversely proportional to the mean molecular mass of the gas in the disc. For the disc composed entirely of molecular hydrogen, and for the disc composed entirely of helium, . The disc composed of a mixture of molecular hydrogen and helium will have a mean molecular mass that is somewhere between 2u and 4u, and as noted in the question, the disc composed of molecular hydrogen, helium and other heavier elements has a mean molecular mass of 2.3u.

The largest scale height will be for the disc with the largest sound speed, and this will be for the disc with the smallest mean molecular mass. Therefore the protoplanetary disc composed entirely of molecular hydrogen will have the largest scale height at a given radius. (Conversely, the disc composed entirely of helium will have the smallest scale height.)

Question 2

The disc composed of molecular hydrogen only.

The disc composed of helium only.

The disc composed of molecular hydrogen and helium.

The disc composed of molecular hydrogen, helium and heavier elements.

All four discs will have the same difference in speeds.

The correct answer is b.

Answer

The difference between the orbital and Keplerian speeds at a given radius is given by

As noted in the answer to Question 1, the Keplerian angular speed at a given radius will be the same for all four discs since it depends only on the stellar mass and the radius, which are the same in all four cases. So the difference in speeds will be smallest when the term under the square root is largest. This term will be largest when H/r is smallest. From the information in Question 1, this will be for the disc composed entirely of helium.

Question 3

The disc composed of molecular hydrogen only.

The disc composed of helium only.

The disc composed of molecular hydrogen and helium.

The disc composed of molecular hydrogen, helium and heavier elements.

All four discs will have the same radial drift speed.

The correct answer is a.

Answer

For particles with a Stokes parameter of τ_S = 1, the radial drift speed from Equation 16 is v_rad = -v_Kη/2. As noted in the answer to Question 1, the Keplerian angular speed at a given radius will be the same for all four discs since it depends only on the stellar mass and the radius, which are the same in all four cases. So the radial drift speed will be largest for the disc with the largest value of η. Since this is given by η = n(H/r)², and the disc aspect ratio is largest for the disc with the largest scale height, this corresponds to the disc composed entirely of hydrogen, as revealed in Question 1.

Question 4

The isolation mass increases as the surface density of the protoplanetary disc increases, for a given star at a given orbital radius.

The isolation mass increases with orbital distance from the central star, for a given star and a given disc surface density.

The isolation mass decreases as the mass of the star increases, for a given orbital distance and a given disc surface density.

For the situation described in Activity 5, the isolation mass at 0.1 au is 3.94 × 10²⁰ kg.

For the situation described in Activity 5, an isolation mass of 3.94 × 10²⁶ kg corresponds to a distance of 10 au.

The isolation mass can never be larger than the mass of the Earth.

The correct answers are a, b, c, d and e.

Answer

The isolation mass is given by Equation 21:

Therefore the first three statements are true, since M_iso ∝ Σ, M_iso∝ a³ and M_iso∝ 1/ M_*^1/2. Furthermore, since the isolation mass in Activity 5 at 1.0 au is 3.94 × 10²³ kg, the corresponding masses at distances 10× smaller and 10× larger are 1000× smaller and 1000× larger respectively, therefore the next two statements are also true. Hence all statements are true except the last one.

Question 5

Match the following core-accretion scenarios to the type of planet that results.

Using the following two lists, match each numbered item with the correct letter.

Core formation by solid accretion followed by gas accretion beyond the critical mass.
Core formation by solid accretion followed by slow core accretion.
Core formation by solid accretion followed by growth in the region of the disc with little solids.

a.Gas giant planet
b.Terrestrial planet
c.Ice giant planet

The correct answers are:

1 = a
2 = c
3 = b

Answer

See Figure 6 for details.

Figure 6 (repeated) Schematic view of possible outcomes of the core-accretion model.

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Figure 6 (repeated) Schematic view of possible outcomes of the core-accretion model.

Question 6

The Toomre parameter is less than 1 and so the disc does meet the Toomre criterion.

The Toomre parameter is greater than 1 and so the disc does not meet the Toomre criterion.

The correct answer is b.

Answer

The Toomre criterion for fragmentation is

The sound speed may be written c_s = H ω_K, where H is the scale height, so this becomes

Then we note that the Keplerian angular speed is so this now becomes

So, calculating in this case

Since this is greater than 1, the Toomre condition is not satisfied and the disc will not fragment.

Question 7

H/r > 0.0055

H/r > 0.055

H/r > 0.55

H/r > 5.5

H/r > 55

The correct answer is b.

Answer

The Jeans mass is given by Equation 25 as

So, if the Jeans mass exceeds the mass of Jupiter, we have

In this case

So the disc aspect ratio must be greater than 0.055 (2 s.f.).

Question 8

Hot Jupiter planets at very small orbital distances.

Giant planets at orbital distances larger than a few au.

Terrestrial planets in Earth-like orbits around their stars.

Mini-Neptune planets at orbital periods of a few months.

Super-Earth sized planets at orbital periods of less than a year.

The correct answer is b.

Answer

The core-accretion scenario can explain the formation of most types of exoplanets. However, it struggles to explain the formation of giant planets at orbital distances larger than a few astronomical units (corresponding to orbital periods longer than a few years), because of the extended time needed to form big enough cores at these distances. These planets probably formed by the disc-instability scenario.

5 Conclusion

The Open University — Wed, 30 Oct 2024 00:00:00 GMT

The focus of this course has been on how planets form around stars from the material in protoplanetary discs. These were some of the key learning points:

Protoplanetary discs comprised of gas and solid material are believed to be the birthplaces of planets. In hydrostatic equilibrium, the density profile ρ_gas(z) of the gas in a disc as a function of vertical height z can be expressed as:
Equation label: (Equation 5)
Here, (Equation 6) is the disc scale height, ρ₀ is the density at the midplane (z = 0), (Equation 3) is the sound speed in the gas and (Equation 2) is the Keplerian angular speed for an orbit at a distance r from a star of mass M_*.
In the radial direction, in addition to the gravitational force, there is also a force due to the pressure gradient of the gas dP_gas/dr. Therefore, the orbital speed v_orb(r) of the gas in the disc has two components: one due to the Keplerian speed, (Equation 1), and one due to this extra pressure gradient, given by
Equation label: (Equation 9)
Usually, dP_gas/dr < 0, so the orbital speed is sub-Keplerian, v_orb(r) < v_K(r). The difference between the Keplerian speed and the orbital speed is and is typically ~ 100 m s^-1 at 1 au from a 1 M_☉ star.
The core-accretion scenario predicts that planets form by accumulation of initially sub-micron-sized dust grains to form metre-sized rocks, then kilometre-sized planetesimals and Mercury-sized planetary embryos, and eventually planetary cores up to several times the size of the Earth.
The relation between the orbital speed and Keplerian speed of particles in a protoplanetary disc can be expressed as (Equation 15) where with n a numerical constant. Particles in the disc experience a radial drift inwards with a speed:
where (Equation 12) is called the Stokes number. The Stokes number is related to the stopping time
Equation label: (Equation 14)
where ρ_m is the material density of the particles and s is their radius. The maximum radial drift speed occurs when τ_S = 1 which corresponds to roughly metre-sized rocks. In this case, .
Once planetesimals have formed, their mass M_p grows through collisions with other planetesimals at a rate:
Equation label: (Equation 17)
where R_p is the planetesimal’s radius, v_esc is its escape velocity, v_rel is the relative velocity between the two impacting bodies, Σ is the surface density of the disc and F_g is the gravitational focusing.
Planetary embryos continue growing into planetary cores by accreting leftover planetesimals within a feeding zone that extends a distance Δa either side of the core, such that . Here, C is a constant and R_Hill is the Hill radius that is defined as the distance from the planetary core at which its gravitational force dominates over the gravitational force of the star of mass M_*, which it orbits at a distance a:
Equation label: (Equation 20)
The total mass of material within the feeding zone is called the isolation mass and represents the final mass of the planetary core:
Equation label: (Equation 21)
Once the mass of the core reaches a few Earth masses, it starts to build up a gas envelope. This can lead to the formation of gas giant planets, ice giant planets or terrestrial planets, depending on the amount of gas accreted by the time the critical mass for hydrostatic equilibrium is reached. Many observed protoplanetary discs show gaps, bright rings, asymmetries, spirals and other structures where planets are forming within them.
The disc-instability scenario provides an alternative way to form gas giants. In this model, a cold and/or massive disc fragments into clumps due to gravitational instabilities, and these clumps eventually evolve into gas giants. Two conditions need to be satisfied for disc fragmentation: the Toomre criterion:
Equation label: (Equation 22)
where c_s is the speed of sound, ω_K is the Keplerian angular speed and Σ is the gas surface density; and the cooling criterion:
Equation label: (Equation 24)
The fact that both conditions need to be satisfied for fragmentation effectively limits the mass and semimajor axis values of the planets forming via the disc-instability scenario.
The typical mass of a planet formed via fragmentation can be estimated from the Jeans mass, which may be expressed as
Equation label: (Equation 25)
The Jeans mass is of order 1–2 times the mass of Jupiter for typical discs.
Once formed, the planets interact with the disc and with each other, undergoing migration in some cases, until the system reaches its final configuration. Factors influencing the final composition and orbital configuration of planets can include interactions with the remaining gas in the disc, interactions with remaining planetesimals, planet–planet interactions and interactions with additional stellar companions.
Neither the core-accretion nor disc-instability scenarios can explain all of the observed exoplanet population. Therefore, it is plausible that both scenarios play a role in planet formation, where different mechanisms are at play at different distances from the parent star.

Acknowledgements

The Open University — Wed, 30 Oct 2024 00:00:00 GMT

This free course was written by Andrew Norton and Mariangela Bonavita.

Except for third party materials and otherwise stated (see terms and conditions), this content is made available under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 Licence.

The material acknowledged below is Proprietary and used under licence (not subject to Creative Commons Licence). Grateful acknowledgement is made to the following sources for permission to reproduce material in this free course:

Images

Course image: NASA/JPL-Caltech

Figure 1a: Mark McCaughrean (Max Planck Institute for Astronomy), C. Robert O’Dell (Rice University), and NASA/ESA, https://esahubble.org/images/opo9545b/, released under the Creative Commons Attribution 4.0 International license, https://creativecommons.org/licenses/by/4.0/

Figure 1b: NASA/ESA/CSA; Data reduction and analysis: PDRs4All ERS Team; Graphical processing: Fuenmayor, S. and Berne, O.

Figure 2: ESO/J. Girard (djulik.com), https://www.eso.org/sci/facilities/paranal/instruments/sphere.html. Licensed under a Creative Commons Attribution 4.0 International License, https://creativecommons.org/licenses/by/4.0/

Figure 3: Muller, A. et al. (2018) ‘Orbital and atmospheric characterization of the planet within the gap of the PDS 70 transition disk’, Astronomy & Astrophysics, vol. 617, pp. 11, EDP Sciences

Figure 4: Cleeves, L. (2015) ‘Molecular signposts of the physics and chemistry of planet formation’, PhD Thesis, University of Michigan

Figure 5: Armitage, P.J. (2017) ‘Lecture notes on the formation and early evolution of planetary systems’, V6, based on lectures given at the University of Colorado

Figure 6: Venturini, J., Ronco, M.P., Guilera, O.M. (2020) ‘Setting the stage: planet formation and volatile delivery’, Space Science Reviews, vol. 216 (5), article 86, Springer Science + Business Media

Figure 7: ALMA (ESO/NAOJ/NRAO), Andrews, S. et al., (NRAO/AUI/NSF), Dagnello, S. Images are licensed for use under the Creative Commons Attribution 3.0 Unported license, https://public.nrao.edu/news/2018-alma-survey-disks/, https://creativecommons.org/licenses/by/3.0/

Figure 8: Bohn, A.J. et al. (2021) ‘Discovery of a directly imaged planet to the young solar analog YSES 2’, Astronomy & Astrophysics, vol. 648, pp. 15, EDP Sciences

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Glossary

The Open University — Wed, 30 Oct 2024 00:00:00 GMT

angular momentum: The momentum associated with the rotational motion of a body.
aspect ratio: The ratio of the height to the radius for a two-dimensional structure such as a protoplanetary disc or an accretion disc. Typically where is the sound speed and is the Keplerian speed.
coagulation: The process by which small (micron-sized) particles in a protoplanetary disc collide with each other gently enough that they stick together to form millimetre-sized aggregates.
cooling criterion: The condition necessary for a protoplanetary disc to undergo self-regulation when forming planets via the disc-instability scenario. It is satisfied if the cooling time obeys where is the Keplerian angular speed.
cooling time: The characteristic timescale for a system to reduce its temperature to some previous level.
core-accretion scenario: A model for planet formation in which planets form by accumulation of solids into a core, on which an atmosphere is accreted once a critical value of the core mass is achieved. Initially, micron-sized dust grains in a protoplanetary disc coagulate to form metre-sized rocks, then kilometre-sized planetesimals, Mercury-sized planetary embryos and eventually planetary cores. Contrast with disc-instability scenario.
critical mass: In relation to planet formation, the limiting mass of a planetary core above which the gas surrounding it cannot maintain hydrostatic equilibrium and starts contracting. Exceeding the critical mass triggers a phase of rapid accretion onto the core until the gas in the protoplanetary disc is dispersed.
disc-instability scenario: A model for planet formation in which planets form directly from gravitational instabilities within a protoplanetary disc. It may be responsible for the formation of massive planets that lie at large distances from their star. Contrast with core-accretion scenario.
disc scale height: The scale height of an accretion disc or protoplanetary disc. It is generally given by where is the sound speed and is the Keplerian angular speed.
escape velocity: A quantity that gives the minimum speed required for an object to escape the gravitational influence of a massive body. In Newtonian gravity, the magnitude of the escape velocity is given by where is the universal gravitational constant, is the mass of the gravitating body and is the initial distance from its centre.
exoplanet: A planet orbiting a star other than the Sun. According to the International Astronomical Union (IAU), an exoplanet has a mass that is below the limiting mass for nuclear fusion of deuterium (currently calculated to be 13 times the mass of Jupiter for objects with the same isotopic abundance as the Sun) and orbits a star or stellar remnant. This definition takes no account of how the object formed, so it is possible that the definition may include objects that would otherwise be classified as brown dwarfs.
feeding zone: The distance either side of the core from within which further planetesimals are accreted during the growth of planetary cores in a protoplanetary disc. Typically where is a small constant and is the Hill radius.
fragmentation: The process by which a contracting interstellar cloud breaks up into a number of separate cloudlets as energy is radiated from the cloud and the Jeans mass decreases.
gravitational focusing: A dimensionless parameter that describes how the gravitational attraction between two bodies increases their collision probability. It is expressed as where is the escape velocity and is the relative velocity between the two impacting bodies.
Hill radius: The radius of the Hill sphere defined by where is the semimajor axis of the planet’s orbit around a star, is the mass of the planet and is the mass of the star.
hot Jupiter: A giant exoplanet in an extremely close orbit around a star.
hydrostatic equilibrium: A situation in which the forces acting on a fluid (normally gravitational forces) are balanced by the internal pressure of the fluid (including thermal, degeneracy and radiation pressure), so that the fluid neither collapses nor expands.
isolation mass: During the growth of a planetary core, this is the total mass of planetesimals within the feeding zone.
Jeans mass: In a disc geometry (such as a protoplanetary disc undergoing planet formation via the disc-instability scenario), the Jeans mass is where is the surface density of the disc.
Kepler’s first law: One of three laws of planetary motion stated by Johannes Kepler. The first law states that planets orbit stars in elliptical orbits with the star at one focus of the ellipse.
Kepler’s laws: Three laws summarising the nature of planetary motion.
Kepler’s second law: One of three laws of planetary motion stated by Johannes Kepler. The second law states that a line joining a planet and its star sweeps out equal areas in equal times. The consequence of this is that planets move fastest when they are closest to their star.
Kepler’s third law: One of three laws of planetary motion stated by Johannes Kepler. The third law states that the square of a planet’s orbital period is proportional to the cube of the semimajor axis of its orbit . More generally: where is the total mass of the star and planet.
Keplerian: A term used to denote quantities that relate to properties of a (circular) Keplerian orbit, e.g. Keplerian speed, Keplerian angular speed.
Keplerian angular speed: The angular speed of a body in a Keplerian orbit, i.e. where is the mass of the central body and is the orbital radius.
Keplerian orbit: The orbit a point mass executes if it is subject only to the gravitational force from another point-like mass. Quite often this term is used in a stricter sense to denote a circular orbit with constant angular speed that obeys Kepler’s third law.
Keplerian orbital speed: The tangential speed of a body in a Keplerian orbit, i.e. where is the mass of the central body and is the orbital radius.
Keplerian speed: The tangential speed of a body in a Keplerian orbit, i.e. where is the mass of the central body and is the orbital radius. Contrast with Keplerian angular speed.
Kozai-Lidov effect: Synchronised changes in the eccentricity and inclination of an orbit such that one increases while the other decreases, in a cyclic manner, caused by the presence of a third, more distant companion.
migration: The process by which protoplanets move away from their place of formation in a protoplanetary disc.
minimum-mass solar nebula: A hypothetical protoplanetary disc with a surface density profile defined as the minimum value of the surface density that a protoplanetary disc would need to have to form our Solar System.
molecular cloud: A cloud of dense cold gas containing molecules, principally molecular hydrogen (), together with dust. Molecular clouds are generally detected through emission lines of molecular species at radio frequencies; important species include , and . Because molecular clouds are cold and dense, they are important sites for star formation.
oligarchic growth: In planetary formation, this describes the situation where the largest planetary embryos grow quickly while the smallest grow slowly.
planetary core: A solid body resulting from a planetary embryo that will accumulate further material to form the core of a planet.
planetary embryo: An object that will likely grow into a planet. Planetary embryos comprise roughly Mercury-sized bodies formed from planetesimals and may grow into planetary cores.
planetesimal: Solid, roughly kilometre-sized bodies that are intermediate in size between rocks and planetary embryos during the growth of planets in protoplanetary discs.
protoplanet: A planet growing by a process of accretion in the protoplanetary disc of a young star or protostar. Small inhomogeneities in the disc are thought to lead to the growth of protoplanets.
protoplanetary disc: A protoplanetary disc consists of cold gas and dust, and is left over from the material that formed the central protostar. Small inhomogeneities in the disc are thought to lead to the growth of protoplanets. Radiation pressure and the solar wind compete against the gravity of the protoplanets and eventually drive off the remaining material of the protoplanetary disc.
radial drift speed: The speed with which particles in a disc move radially through it. It depends on the Stokes number typically according to where is the Keplerian speed and where is a dimensionless constant and is the aspect ratio of the disc.
runaway growth: An accelerated phase in the growth of planetesimals.
self-regulation: In relation to the disc-instability scenario for planet formation, the situation where, as a protoplanetary disc becomes unstable (due to the Toomre Q parameter falling below ), shock waves are generated in the disc. These heat up the disc, so increasing 𝑄, and the disc stabilises. A disc will undergo self-regulation if the cooling criterion is met.
sound speed: The speed at which the wavefronts of a sound wave propagate. In an ideal gas, the sound speed is given by where is the gas pressure and is its density, or equivalently by where is the temperature, is the Boltzmann constant and is the mean mass of the particles involved.
Stokes number: A dimensionless parameter which characterises how well particles embedded in a fluid flow follow streamlines. It is given by where is the stopping time and is the Keplerian angular speed. Large particles will generally have large Stokes numbers () and will detach from the flow when it changes velocity abruptly. Small particles will generally have small Stokes numbers () and will closely follow fluid streamlines at all times.
stopping time: A characteristic timescale that describes how a particle of mass interacts with gas surrounding it. It is defined as where is the magnitude of the drag force that acts in the opposite direction to , which is the speed of the particle with respect to the gas.
streaming instabilities: A mechanism for the formation of planetesimals in which the drag felt by solid particles orbiting in a gas disk leads to their spontaneous concentration into clumps which can gravitationally collapse.
surface density: The density, in units of mass per unit area, of an (essentially) two-dimensional structure such as a protoplanetary disc or accretion disc.
Toomre criterion: The necessary condition that must be satisfied for a protoplanetary disc to undergo planet formation via the disc-instability scenario. For fragmentation to occur the local surface density of the disc needs to be high enough that the self-gravity of the gas and its differential rotation are higher than the thermal pressure.
Toomre Q parameter: For a protoplanetary disc to fragment, and for planets to form via the disc-instability scenario, the disc must satisfy the Toomre criterion. For this to happen, the Toomre parameter must satisfy where where is the Keplerian angular speed, is the sound speed, and is the disc surface density.
velocity dispersion: The spread of velocities present in a given population of objects.