The secrets of other worlds as revealed by white dwarfsAlexander J. Mustill1, Eva Villaver2, Dimitri Veras3, Boris T. Gänsicke3, Amy Bonsor41 Lund Observatory, Department of Astronomy & Theoretical Physics, Lund University, Box 43, SE-221 00, Lund, Sweden. Email: alex@astro.lu.se2 Universidad Autónoma de Madrid, Departamento de Física Teórica, E-28049 Madrid, Spain3 Department of Physics, University of Warwick, Coventry, CV4 7AL, UK4 Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK

Surtr riðr fyrst ok fyrir honum ok eftir eldi brennandi.Snorri Sturluson, Gylfaginning 50

Þar liggr hann í böndum til ragnaröSnorri Sturluson, Gylfaginning 50

Ok brenna allan heim með eldi.Snorri Sturluson, Gylfaginning 4

Jörð skal rifna ok upphiminn.Skarpåkersstenen

Er þá nökkur jörð eða himinn?Snorri Sturluson, Gylfaginning 53

1. What planets can survive a star’s AGB evolution?

2. Are multi-planet systems safe?3. A system of super-Earths destabilised by stellar mass loss provides

long-term delivery of asteroids to pollute the white dwarf

Stellar radius

Surviving 1M⊕ planets

Engulfed 1M⊕ planets

Orbits expand as star loses mass at AGB tip

The evolution of a planet’s orbit at the end of a star’s life is affected by a battle between two opposing effects:• The loss of mass from the star; for typical white dwarf progenitors

of 2–3 Solar masses, this occurs primarily at the AGB tip. Planets’ orbits expand to conserve specific orbital angular momentum, so that afinal/ainitial = M★,initial/M★,final. This results in a factor 3–4 expansion for planets orbiting typical white dwarf progenitors, in the absence of other forces.

• Orbital decay induced by the greatly enhanced tidal forces as the stellar radius expands to 1 au or more. Jupiter-mass planets experience strong tidal decay, but for Earth-mass planets the tidal decay is weaker.

The figure above (Mustill & Villaver 2012) shows the orbits of 1M⊕ planets orbiting a 2.5M⊙ AGB star. Planets within ~2.3 au will experience modest orbital decay due to tides before being engulfed into the stellar envelope. Those on wider orbits outrun the expanding envelope as stellar mass loss accelerates towards the AGB tip. In the N-body studies of Mustill et al (2018) we focus on wide-orbit (>10 au) planets, for which tidal forces can be neglected.

3 M⊙

The loss of mass from the star as it becomes a white dwarf means that planets become relatively more massive and affect each others’ orbits more. In particular, the mutual Hill radius

rH,mut = 0.5(a1+a2)[(M1+M2)/3M★]1/3

increases. Planetary systems which remained stable throughout their star’s main sequence lifetime can therefore be destabilised by stellar mass loss. The range of planetary separations vulnerable to being destabilised can be estimated analytically as a function of stellar mass, planet mass, and planetary orbital radius (see figure above; Mustill et al 2014). Numerical integrations of three planet systems confirms that there exists a range of planetary separations for which systems remain stable for the star’s main sequence lifetime, but are then destabilised by AGB mass loss, although the boundaries of this region are fuzzy (figure below; Mustill et al 2018).

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In Mustill et al (2018), we have conducted the first full-lifetime simulations of asteroid belts in multi-planet systems that are destabilised by stellar mass loss, showing that systems of super-Earth planets can cause ongoing delivery of asteroids to the white dwarf over several Gyr. The key ingredients of the work are:

• Low-mass, eccentric planets can drive long-term delivery (Frewen & Hansen 2014).

• Instabilities in systems of low-mass planets can take Gyr to resolve, during which duration the planets spend long periods of time on eccentric orbits (Veras et al 2016).

• The observation that at least for close-in planets orbiting Sun-like stars, the planet occurrence rate rises steeply towards smaller planets (e.g., Fressin et al 2013).

• The modification of the RADAU N-body integrator in the MERCURY package (Chambers 1999), permitting planetary orbital evolution to be followed accurately for Gyr. Our modified integrator reads from a stellar evolution file generated by SSE (Hurley et al 2000), adjusting the stellar mass and radius at every timestep in the N-body integration.

The figure above (Mustill et al 2018) shows the evolution of orbits in one simulation. 200 asteroids (red) are placed in a belt interior to three planets (black, masses 1.3 to 31M⊕) in orbit of a 3M⊙ star. The evolution has three phases:• Through the star’s main sequence lifetime, and as it

approaches the AGB tip at ~477Myr, some asteroids are lost due to the inner planet’s chaotic zone and Kirkwood-type gaps. These asteroids are typically ejected from the system.

• Stellar mass loss at the AGB tip causes the entire planetary system to expand outwards, and some further erosion of the belt occurs early in the white dwarf’s lifetime. As before, these particles are ejected; they do not accrete onto the white dwarf.

• The stellar mass loss triggers a delayed instability in the planetary system: here, planetary orbit-crossing begins at around 530 Myr. This depletes the belt on a timescale of Gyrs. In contrast to the previous erosion, significant numbers of asteroids are delivered to the WD. In this simulation, 23% of the original asteroids are accreted onto the white dwarf.

(MS) (AGB) (WD)

Belt erosion prior to AGB tip

Orbital expansion due to

AGB mass loss

Planetary orbit-crossing begins

Delivery of material to WD

begins

Particles remaining in simulation

Particles collided with WD

4. The observed age dependence of accretion rates is reproduced

The amount and time-dependence of the delivery of asteroids to white dwarfs is a function of the masses of the planets in the system.• Systems of Jovian planets deliver only 1–2% of the asteroids to the

white dwarf, and this is concentrated into a short window of a few 10s of Myr after orbit crossing begins.

• Systems of super-Earth planets can deliver ~20% of the asteroids to the white dwarf. The rate is initially high once orbit-crossing begins, and decays exponentially on a timescale of ~1Gyr (figure above, Mustill et al 2018), matching the age dependence of accretion rates observed by Hollands et al (2018).

• For asteroid belts of the same mass, material is delivered from belts interior to the planets 2–3 times more efficiently than from outer belts (the latter are more likely to lose their asteroids to ejections). The observed composition of material accreted into white dwarf atmospheres is refractory, favouring interior belts.

• While the above statistics refer to asteroids that collide with the white dwarf, over twice as many cross its Roche limit where they will be tidally disrupted. In the N-body simulations presented here, many of these bodies are ejected by ongoing scattering before they hit the star. In reality, these particles will form debris trails which can interact with each other before ejection, enhancing accretion onto the white dwarf.

5. Asteroids delivered during planetary scattering reach the white dwarf with random inclinations, creating a dynamic environment close to the star

The asteroids that cross the Roche limit or directly collide with the white dwarf arrive with an isotropic inclination distribution. The figure left (Mustill et al 2018) shows the mutual inclination between the orbits of successive asteroids accreted in the same simulation, together with the time between accretion events. We also show the timescales for circularisation of debris trails by radiation forces (Veras et al 2015) as well as one estimate of the lifetime of dust discs within the Roche limit (Rafikov 2011).• Even with just 200 asteroids per belt, the time between

successive deliveries to the white dwarf is short compared to the lifetime of debris trails, and can be shorter than the lifetime of circular close-in dust discs.

• Asteroids whose pericentres approach or cross the Roche limit will encounter, at high inclination, debris trails from previously scattered asteroids, or close-in circumstellar dust or gas discs within the Roche limit.

• Owing to the high orbital velocities at pericentre, scattered asteroids and their debris will experience significant collisional processing, fragmentation and vapourisation, as proposed by Jura (2008).

References

Chambers, 1999, MNRAS, 304, 793Debes & Sigurdsson, 2002, ApJ, 572, 556Farihi, 2016, New Astron. Rev., 71, 9Fressin et al, 2013, ApJ, 766, 81Frewen & Hansen, 2014, MNRAS, 439, 2442Hollands, Gänsicke & Koester, 2018, MNRAS, 477, 93Hurley, Pols & Tout, 2000, MNRAS, 315, 543Jura, 2008, AJ, 135, 1785Mustill & Villaver, 2012, ApJ, 761, 121Mustill, Veras & Villaver, 2014, MNRAS, 437, 1404Mustill, Villaver, Veras, Gänsicke & Bonsor, 2018, MNRAS, 476, 3939Rafikov, 2011, MNRAS, 416, L55Skarpåkersstenen, c1000Sturluson, c1220, GylfaginningVanderburg et al., 2015, Nature, 526, 546Veras, Leinhardt, Eggl & Gänsicke, 2015, MNRAS, 451, 3453Veras, 2016, RSOS, 3, 150571Veras, Mustill, Gänsicke, Redfield, Georgakarakos, Bowler & Lloyd, 2016, MNRAS, 458, 3942

Acknowledgements

AJM acknowledges support from the Swedish Research Council starting grant 2017-04945 “A unified picture of white dwarf planetary systems”, the project grant “IMPACT” from the Knut & Alice Wallenberg Foundation, and hardware and travel funds from the Walter Gyllenbergs fund of the Royal Fysiographic Society in Lund.

Motivation Summary White dwarfs offer a surprising insight into the compositions of extra-Solar planets and asteroids. Over 25% are found to contain elements heavier than helium in their atmospheres, signs of ongoing accretion of planetary or asteroidal material. The bulk elemental abundances of the bodies accreted can be found spectroscopically, and compared to Solar System planets and asteroids. Evidence for planetary systems orbiting white dwarfs comes from these atmospheric signatures, from IR excess and circumstellar emission features, and from the direct detection of transits (Vanderburg et al 2015). These lines of evidence are summarised in the cartoon to the right, and reviewed in Farihi (2016). Debes & Sigurdsson (2002) proposed that the destabilisation of planetary systems by stellar mass loss could deliver material to close to the white dwarf. However, the exact dynamical pathways involved are not yet clear (see Veras 2016 for a review). Here, we present the first full-lifetime simulations of systems of planets and asteroids experiencing stellar evolution, and show that the destabilisation of asteroids in super-Earth systems reproduces observed trends.

• Spectroscopic signatures of metals accreted into ~40% of WD atmospheres

• Dust discs detected through IR excesses

• Gas discs detected through Keplerian emission features

• Transits of disintegrating asteroids

time

flux

λflu

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Roche limit

• More massive planets are less efficient: they deliver a smaller fraction of asteroids to the white dwarf, and the delivery does not extend over the Gyr observed.

• Our simulations therefore support the existence of super-Earths on wide (>10 au) orbits around a significant fraction (>25%) of white dwarfs.

• Asteroids scattered to the white dwarf have an isotropic inclination distribution. Significant collisional interactions between asteroids, debris streams and dust and gas discs will occur.

• We modify the RADAU algorithm in the MERCURY package to incorporate changes to stellar mass and radius as the star evolves from the main sequence, through the red giant and asymptotic giant branch stages, and on to become a white dwarf.

• We set up systems of three planets, with asteroid belts interior to or exterior to the planets’ orbits.

• We show, in self-consistent N-body integrations over the star’s lifetime, that planet–planet scattering in the presence of asteroid belts can provide ongoing accretion to the white dwarf for several Gyr.

• Super-Earth and Neptune-mass planets are highly efficient at delivering material, and the delivery can be maintained for several Gyr, as the low-mass planets are very inefficient at ejecting each other or the asteroids and so they persist on eccentric orbits.

• The rate at which the delivery of material decays in these simulations matches the observed dependence of accretion rate on white dwarf cooling age.

For full details, see Mustill et al, 2018, MNRAS, 476, 3939

Observations:

Simulations:{

Planetary systems orbiting white dwarfs: