##  Chapter 0 Giant branch planetary systems: Dynamical and radiative
evolution

Alexander J. Mustill  Lund University, Department of Physics, Box 118, SE-22
100 Lund, Sweden

XXX Chapter Article tagline: update of previous edition,, reprint..

###### Abstract

[Abstract] In seven billion years, the Sun will be dead. As stars like the Sun
pass from their present state to that of a dead white dwarf star, they undergo
two phases of extremely high luminosity and radius—the red giant branch and
the asymptotic giant branch—during which they will lose half or more of their
mass. These changes to the star have a significant impact on orbiting planets,
asteroids and comets. The large stellar radius (beyond the current orbit of
the Earth) leads to the engulfment of bodies entering the stellar envelope, a
process enhanced by strong tidal interactions. The high luminosity affects
bodies’ orbits and physical properties, while mass loss can later trigger the
destabilisation of bodies around white dwarfs. It is necessary to understand
these processes to understand both the future of our Solar System, and to
interpret growing observations of planetary systems around evolved stars.

[

Glossary]

Adiabatic [Dynamics] Slow in relation to some characteristic timescale.
(usually the orbital timescale in this context).

Asymptotic Giant Branch Phase in a star’s life following exhausting of helium
fuel in the core, and preceding the WD.

Main Sequence Stage of a star’s life where it is fusing hydrogen in its core.

Planetary Nebula The expelled stellar envelope after the AGB, surrounding and
illuminated by the star as it transitions to a WD. Nothing to do with planets.

Planetesimal Generic term for a small body, asteroid or comet.

Radiation/radiative forces Forces acting on orbiting bodies through
absorption, emission and scattering of radiation, including radiation
pressure, Poynting–Robertson drag, the Yarkovsky effect, and the YORP effect.

Red Giant Branch Evolutionary state after the Main Sequence, when hydrogen
fuel in the core is exhausted but prior to the ignition of helium fuel.

Tidal Related to a _differential_ gravitational potential felt across an
extended body such as a star or planet.

White Dwarf Final state of stellar evolution, composed of electron-degenerate
matter with no nuclear burning.

[

Abbreviations]  AGB Asymptotic Giant Branch MMR Mean motion resonance MS Main
Sequence PN Planetary Nebula PR Poynting–Robertson RGB Red Giant Branch WD
White Dwarf YORP Yarkovsky–O’Keefe–Radzievskii–Paddack

###  1 Introduction

Key points

  * •

Stars have a finite life, limited by their nuclear fuel. When hydrogen is
exhausted in the core, the Sun swells to a large size and luminosity as a _red
giant branch (RGB)_ star. It will later begin fusing helium in its core, and
when that is exhausted it will again swell to an even greater size as an
_asymptotic giant branch (AGB)_ star. Following that, it will expel its
envelope and become a “dead” _white dwarf,_ not undergoing any further nuclear
reactions.

  * •

The most significant changes happen on the AGB, where the Sun will grow to
become larger than Earth’s orbit, increase in brightness to several thousand
times today’s value, and lose half of its mass.

  * •

The hugely increased stellar radius can engulf and destroy bodies such as
planets, either directly, or by dragging them in through enhanced _tidal_
forces. The result of this should be a large cleared region round a white
dwarf, an expectation increasingly at odds with observations.

  * •

The extreme luminosities can change the orbits of small dust grains and
asteroids up to several tens of km through _radiation forces_ such as
_radiation pressure, Poynting–Robertson drag,_ and the _Yarkovsky effect_.
These bodies are also potentially subject to sublimation or loss of their
volatiles. Planets that evade the reach of the star’s strong tidal forces may
have their atmospheres stripped or chemically altered by the extreme
irradiation.

  * •

The loss of mass from the star causes the orbits of any surviving bodies to
expand. When more than one body is present in a system, forces between them
become relatively stronger after the star has lost mass. This can result in
orbital instabilities which may be responsible for explaining planetary
phenomena close to white dwarfs, such as transiting planets and planetesimals,
discs of dusty debris and vaporised rock, and rocky material deposited into
white dwarf atmospheres.

While stars like the Sun live for a long time, they do not live forever.
Eventually they exhaust their source of nuclear fuel, and ultimately, unless
sufficiently massive, end their lives as slowly cooling white dwarfs. In this
process, however, they undergo spectacular changes, growing in radius and
luminosity as giant stars, while casting off much of their mass which becomes
briefly visible as a planetary nebula. These changes—large stellar radius,
high luminosity, and mass loss—have significant effects on any planets,
asteroids, comets, and other bodies orbiting the star, which are described in
this chapter.

Understanding the changes wrought by giant stars on their planetary systems is
important for a number of reasons. On a cultural level, we want to know what
the future of our own planet will be, and that of any distant descendants we
may leave behind. Scientifically, the study of giant stars offers a way to
probe the population of planets around stars slightly more massive than the
Sun, which are not readily amenable to radial-velocity planet surveys when
they are on the main sequence; but the population of planets orbiting such
giant stars has evolved somewhat from what it was on the main sequence. And
perhaps most significantly, there is now a growing body of evidence for
planetary systems orbiting white dwarf stars. This evidence comes in the form
of:

  1. 1.

a handful of detections of planets and dusty structures such as disintegrating
asteroids;

  2. 2.

many discs of dust and gas surrounding white dwarfs which are thought to be
formed from the remnants of asteroids or other bodies that passed too close to
the star;

  3. 3.

the very high fraction of white dwarfs that are observed to have rock-forming
elements in their atmospheres, again thought to originate from such disrupted
bodies.

Study of these can tell us about the bulk abundances of extra-Solar asteroids;
but we must always be mindful that bodies orbiting white dwarfs may have been
altered by the significant changes that the stars undergo as giants.

This Chapter begins by summarising the relevant parts of stellar evolution
theory (Section 2) and celestial mechanics (Section 3) needed to understand
the effects on planetary systems of stellar RGB and AGB evolution. I then
describe the strong tidal forces induced by stellar radius expansion (Section
4), and how these compete with stellar mass loss which tends to cause orbits
to expand (Section 1). After that, I discuss the effects of mass loss in
destabilising systems, which can explain the source of planetary material
later detected close to or accreted onto white dwarfs (Section 2). I finally
discuss the effects the increase stellar irradiation has both on bodies’
orbits and on their physical properties (Section 6).

###  2 Overview of relevant stellar evolution

Figure 1: Evolution of a $1\mathrm{\,M}_{\odot}$ star with the same chemical
composition as the Sun, from MIST stellar evolutionary tracks (Choi et al.,
2016). Top row: stellar luminosity (A), radius (B) and mass (C) for the final
$\sim 500$ Myr before the star becomes a white dwarf, encompassing both the
RGB and the AGB. Note the large increases in luminosity and radius during the
AGB and the RGB, as well as the strong mass loss especially during the AGB.
Bottom row: the same, for 1 Myr around the AGB tip, showing the strong
variations in luminosity (D) and radius (E), as well as the mass loss (F),
during the thermal pulses.

We begin by revising some basic features of stellar evolution for low- and
intermediate-mass stars. A more detailed treatment can be found in textbooks
on stellar evolution, such as that by Prialnik (2009). We shall be concerned
exclusively with the progenitors of WDs, meaning stars of mass
$M_{\star}\lesssim 8\mathrm{\,M_{\odot}}$, and more particularly with the
range $1-3\mathrm{\,M_{\odot}}$. The progenitor host stars of most evolved
planetary systems would have lain in this mass range, whether we are
considering planet-hosting giant stars, or WDs observed with photospheric
pollution or other signs of remnant planetary systems. Stars more than
$3\mathrm{\,M_{\odot}}$ are comparatively rare owing to the steepness of the
stellar initial mass function, and some evidence suggests that planets may
become less common around the more massive stars (Reffert et al., 2015). At
the lower mass end, few stars $<1\mathrm{\,M_{\odot}}$ have had the chance to
evolve to become WDs within the age of the Universe, particularly when
excluding metal-poor stars that have a lower likelihood of hosting planets.
Finally, we also confine our attention to stars that are “effectively single”
for the purposes of their evolution; i.e., they either have no stellar
companion, or they have only a wide-orbit companion that does not affect the
evolution of the star. Thus we exclude from consideration the future evolution
of circumbinary planetary systems, which has received comparatively little
attention.

A star spends most of its life as a _main-sequence (MS)_ object, fusing
hydrogen into helium in its core, and most known planetary systems orbit such
MS stars. The MS ends when hydrogen is exhausted in the core; at this point,
the core collapses while hydrogen burning continues in a shell around the
core. The combination of core collapse plus shell hydrogen burning drives the
star up the giant branch, which can be partially understood as a consequence
of the virial theorem (Prialnik, 2009): as the core shrinks and heats up, the
envelope expands and cools, while the luminosity rises significantly (see
Figure 1A and B). The high luminosity, combined with the high opacity in the
cool envelope, leads to the envelope becoming unstable to convective gas
motions. We now refer to the star as a _red giant branch (RGB)_ star, and the
large radius of the convective envelope plays a crucial role in strengthening
tidal interactions, described later.

Eventually, RGB evolution ends when helium ignites in the core. There is a
qualitative difference in RGB evolution at around $2\mathrm{\,M_{\odot}}$:
above this limit, core helium burning begins to occur gradually in a non-
degenerate core before the star has ascended far up the RGB. Below this limit,
core temperature and pressure are too low for He burning to begin before the
core becomes degenerate; when it starts, a runaway _helium flash_ occurs which
abruptly terminates the RGB after a much larger radius ($\sim 1$ au) has been
attained. After a relatively brief time, core helium is exhausted, and a
second phase of core collapse and shell burning drives star back up the giant
branch as an _asymptotic giant branch (AGB)_ star, attaining still larger
radii and luminosities than on the RGB. Again, the envelopes of these objects
are deeply convective. Towards the tip of the AGB, _thermal pulses_ occur
where nuclear reactions in the hydrogen- and helium-burning shells
alternatively switch on and off, leading to large oscillations at intervals of
of $\sim 10^{5}$ yr in luminosity and radius imposed on the overall growth
trend (Figure 1D and E).

The combination of the large stellar radius, the high luminosity, and the high
opacity on the RGB and particularly the AGB means that the envelopes of these
stars are rather weakly bound. While the Sun is currently losing mass at a
rate of only $10^{-14}\mathrm{\,M_{\odot}\,yr}^{-1}$, mass loss rates around
giant stars can reach $10^{-7}\mathrm{\,M_{\odot}\,yr}^{-1}$ at the RGB tip
and $\sim 10^{-4}\mathrm{\,M_{\odot}\,yr}^{-1}$ at the AGB tip in the
_superwind_ phase. While these phases of strong winds are relatively brief,
they dominate the overall mass loss over a star’s life: the Sun will lose only
$\sim 0.1\%$ of its mass over its 10 Gyr MS lifetime, but will overall lose
around half its mass in becoming a WD, mostly at the AGB tip (Figure 1C and
F). Modern stellar evolution models have much of the mass loss taking place
towards the AGB tip (e.g., MESA/MIST Choi et al., 2016), but the reader should
bear in mind that some literature on modelling planetary systems around
evolving stars makes use of older stellar evolution models (such as SSE) which
had more of a balance between the total mass lost on the RGB and the AGB for
low-mass stars. Models of intermediate-mass stars which do not undergo a core
helium flash always result in most mass loss occurring towards the AGB tip.

After the AGB tip and the final stages of mass loss, a young WD is left behind
surrounded by the expelled envelope which is illuminated to become briefly
visible as a _planetary nebula_.

We now provide some rough rules of thumb to aid the reader’s intuition in
quantifying these stellar evolutionary processes. A $1\mathrm{\,M_{\odot}}$
star has a MS lifetime of order $10^{10}$ yr. The MS luminosity is a steep
function of mass owing to the sensitive temperature dependence of nuclear
fusion reactions; hence, the MS lifetime decreases significantly with
increasing stellar mass, and a $2\mathrm{\,M_{\odot}}$ star will spend only
$\sim 10^{9}$ yr on the MS. As an RGB star, a $1\mathrm{\,M_{\odot}}$ star
will attain a radius of around 1 au, with a $\gtrsim 2\mathrm{\,M_{\odot}}$
star above the mass for nondegenerate He burning attaining a smaller radius.
At the AGB tip, the maximum radius increases with stellar mass, with a
$1\mathrm{\,M_{\odot}}$ AGB tip star attaining a radius of over 1 au and a
$3\mathrm{\,M_{\odot}}$ AGB tip star attaining a radius of around 3 au. The
fraction of mass lost also grows with stellar mass (although more massive
stars still give rise to slightly more massive WDs), with a
$1\mathrm{\,M_{\odot}}$ star losing roughly half its mass and a
$3\mathrm{\,M_{\odot}}$ star losing roughly three-quarters. These values also
depend somewhat on the stellar metallicity, but rather weakly over the
metallicity range important for planetary systems.

The MS lifetime is a steep function of stellar mass. Many “old” or “evolved”
planetary systems orbiting giants or WDs are therefore actually younger than
the Solar System’s 4.6 Gyr! They can be younger in dynamical terms (planetary
orbits completed) than the Solar System, while any potentially habitable
planets may not have had a chance to develop complex life.

###  3 Overview of orbital dynamics

####  1 Two-body Keplerian orbits

Figure 2: Keplerian orbital elements inclination $I$, longitude of ascending
node $\Omega$, argument of periapsis $\omega$, and true anomaly $f$, for a
bound elliptical orbit.

It is a happy coincidence that the gravitational potential outside a sphere of
matter is the same as that generated by a point mass of equivalent mass
located at its barycentre: the _Kepler potential_

$\Phi(r)=-\frac{\mathcal{G}M_{\star}}{r},$ (1)

where $\mathcal{G}$ is the gravitational constant and $M_{\star}$ the stellar
mass. Thus the motion of two spherical bodies is equivalent to the motion of a
body orbiting a point mass. In this case, as in any static potential, a body’s
position and velocity at a given time uniquely determine its future orbit, an
orbit thus defined by six independent parameters. While position and velocity
may specify an orbit, they are constantly changing with time. It is therefore
considerably more convenient to use the _Keplerian orbital elements,_ which
relate more intuitively to the geometry of the orbit which, for a bound orbit,
is elliptical. These orbital elements are:

  * •

the _semimajor axis_ $a$ (half the length of the ellipse);

  * •

the _eccentricity_ $e$ (deviation from circularity of the orbit);

  * •

the _longitude or argument of periapsis_ 111Also called the _periapse,
pericentre,_ or, for orbits around specific bodies, _periastron, perihelion,
perigee_ etc. $\varpi$ or $\omega$ (the orientation of the ellipse, relative
to a fixed reference axis or to the intersection of the orbital and reference
planes);

  * •

the _inclination_ $I$ and _longitude of ascending node_ $\Omega$ (two angles
that together specify the 3-dimensional orientation of the orbital plane);

  * •

one angle to parametrise the orbital phase, common choices being the _true
anomaly_ $f$ or _true longitude_ $\theta$ (the physical angle round the orbit,
measured from the pericentre or a fixed reference axis respectively), or the
_mean anomaly_ $\mathcal{M}$ or _mean longitude_ $\lambda$ (corresponding
angles which increase linearly in time).

Several of these are illustrated in Figure 2.

The utility of the Keplerian orbital elements is that all, save the mean
anomaly (or equivalent parametrisation of orbital phase), are constant in the
Kepler problem. This arises because of symmetries of the potential. We recall
_Noether’s Theorem_ from classical mechanics, which informally states that for
every continuous symmetry of a system there is a corresponding conserved
quantity. Conservation of energy arises from time invariance of the potential,
and for the Kepler problem corresponds to a constant semimajor axis:

$E=-\frac{\mathcal{G}M_{\star}m_{\mathrm{pl}}}{2a},$ (2)

where $m_{\mathrm{pl}}$ is the planetary mass. Conservation of the angular
momentum vector arises from spherical symmetry, and corresponds in the Kepler
problem to fixed orbital eccentricity, inclination, and longitude of ascending
node; the magnitude of the angular momentum vector can be expressed as

$L=m_{\mathrm{pl}}\sqrt{\mathcal{G}M_{\star}a\left(1-e^{2}\right)}.$ (3)

Any spherically-symmetric potential will place these four independent
constraints on particles’ orbits, confining each orbit to its own plane. A
near-unique property of the Kepler potential is the existence of a further
“hidden” mathematical symmetry, which gives rise to invariance of the
longitude of periapsis. These symmetries then place five constraints on the
Keplerian orbit, confining it to a one-dimensional manifold: as it happens, a
conic section. The one degree of freedom permitted is then the orbital phase.

Evolution of these orbital elements arises from symmetry-breaking; one example
highly relevant for this chapter is the change to the potential caused by
stellar mass loss, which breaks time invariance and hence energy conservation.
Additional effects that can break the symmetries of the Kepler potential and
induce orbital evolution include relativistic effects, deviations from
sphericity of one or both bodies (such as tidal deformation), and radiative
forces; and a major class of such effects is the impact of additional massive
bodies in the system such as planets or stars, and we deal with these in the
following subsection. Usually these extra effects are weak compared to the
dominant Keplerian motion: a body follows an approximately Keplerian orbit on
an orbital timescale, and we study slow, long-term changes to the orbital
elements. Sometimes, however, elements can change near-impulsively, such as
when two planets experience a close encounter, or when mass loss is extremely
rapid (as happens in binary systems where one component undergoes a
supernova).

####  2 Systems of $N>2$ bodies

The introduction of an additional massive body into a two-body system
complicates the dynamics considerably, and general analytical solutions do not
exist. For analytical work, we must make use of various approximations with
different ranges of applicability. Alternatively, we may simulate orbital
evolution numerically on a computer; here, the analytical approximations
remain useful to aid interpretation and understanding. Textbook treatments of
this area include those by Murray and Dermott (1999), covering analytical
aspects, and more recently by Tremaine (2023), who additionally covers
numerical integrations.

Systems of multiple bodies may broadly be stable or unstable. By “stable” we
usually mean simply that the system maintains the same qualitative behaviour
for some long but not infinite time of interest (such as the lifetime of a
star, or of the Universe), while we speak informally of a system “becoming
unstable” or “undergoing an instability” when a significant change happens to
a body or the system as a whole, such as the excitement of large
eccentricities leading to orbits intersecting222This differs from the formal
mathematical notion of instability as motion away from a fixed point or limit
cycle: all informally unstable systems are mathematically unstable, but so
also may be many informally stable systems.. The outcome of orbital
instability may be the ejection of one or more bodies on an unbound orbit, or
the close approach of two bodies, which itself may lead to a physical
collision, orbital change through strong tidal interactions, or the disruption
of a body by a strong tidal force.

##### Strong interactions: close encounters and scattering

A simple but important case of three-body dynamics occurs when two bodies
undergo a _close encounter_ or gravitational _scattering_ event, often as a
result of the other dynamical processes to be described below. For example,
two planets can scatter each other, or a planet can scatter an asteroid or
comet. The region around a planet where this process is significant can be
described by the _Hill sphere,_ whose radius (the _Hill radius_) approximates
the volume where the planetary gravitational potential dominates over the
stellar and centrifugal potentials. The Hill radius is given by

$r_{\mathrm{H}}=a\left(\frac{m_{\mathrm{pl}}}{3M_{\star}}\right)^{1/3}.$ (4)

Roughly speaking, outside the Hill sphere a body’s orbital motion is dominated
by the Keplerian potential of the star, while inside the Hill sphere it is
dominated by the Keplerian potential of the planet. An orbit inside the Hill
sphere may be a bound circle or ellipse, as in the case of a satellite, or a
hyperbola in the case of a scattering event333A more sophisticated treatment,
_Hill’s approximation,_ also takes into account the Keplerian motion of both
bodies around the star.. The outcome of such a scattering event depends on the
relative masses and velocities of all bodies, as well as the impact parameter
of the event, but broadly the outcomes are either a deflection of the orbit,
or collision of the two bodies444Readers coming from other sub-fields, such as
the dynamics of stellar clusters or galaxies, should note that in planetary
dynamics the term “collision” refers specifically to a physical impact, rather
than to any gravitational scattering as in a “collisional” stellar system.;
tidal disruption of the less dense body may also be possible. A planet has a
larger cross section for scattering versus collision the larger its Safronov
number is, where the Safronov number is given by

$\Theta=\frac{2m_{\mathrm{pl}}}{M_{\mathrm{\star}}}\frac{a}{R_{\mathrm{pl}}}=\left(\frac{v_{\mathrm{esc}}}{v_{\mathrm{Kep}}}\right)^{2}$
(5)

which is the square of the ratio of the planet’s surface escape velocity to
its Keplerian orbital velocity (Tremaine, 2023). In Equation 5,
$R_{\mathrm{pl}}$ is the planet’s physical radius. A higher $\Theta$ means a
planet is more prone to scattering as a result of a close encounter, while a
small $\Theta$ means it is more prone to collision. A more massive planet can
induce a larger velocity change during an encounter without experiencing a
physical collision. To provide some examples, a low-mass planet on a
relatively close-in orbit, such as Earth, has a small Safronov number
($v_{\mathrm{Kep}}=30\mathrm{\,km\,s}^{-1}$,
$v_{\mathrm{esc}}=11\mathrm{\,km\,s}^{-1}$) and is relatively more efficient
at accreting bodies, and less efficient at scattering them, than a more
massive planet on a wider orbit such as Jupiter
($v_{\mathrm{Kep}}=13\mathrm{\,km\,s}^{-1}$,
$v_{\mathrm{esc}}=60\mathrm{\,km\,s}^{-1}$).

The single-planet Hill radius defined above is suitable for use when one body
is of negligible mass; for two planets of comparable mass, we can define
similarly the _mutual Hill radius_

$r_{\mathrm{H,mut}}=\frac{a_{1}+a_{2}}{2}\left(\frac{m_{1}+m_{2}}{3M_{\star}}\right)^{1/3}.$
(6)

Using the mutual Hill radius, there is a particularly simple expression for
the stability of two planets initially on circular orbits: their orbits may
not intersect if

$\Delta>3.5R_{\mathrm{H}}$ (7)

(Gladman, 1993). Note that this criterion for _Hill stability_ does not
preclude that one planet escape to infinity, or collide with the star.

##### Long-term evolution

In many systems, gravitational scattering forms only a part of the dynamics,
or may not occur at all. Most of the evolution occurs under weaker, slower-
acting interactions, with orbital elements changing gradually, which may
subsequently lead to the close encounters described above. In such cases, we
often make use of _osculating_ orbital elements, which are those
instantaneously derived from a body’s position and velocity, and which may be
taken as fixed on an orbital timescale. To calculate the changes on long
timescales, we are uninterested in short-period fluctuations owing to
planetary conjunctions, which typically average out in the long run, and so we
seek expressions for the averaged perturbing potential a body feels from
another planet or star, as a function of orbital elements rather than
Cartesian position. This perturbing potential can be expressed in several
ways. One approach, common for near-coplanar systems such as the Solar System,
is to utilise Fourier series where the angles are combinations of orbital
longitudes (mean, apsidal, and nodal) and the coefficients are functions of
semimajor axes, eccentricities and inclinations. Here the potential takes the
form (see, for example, Murray and Dermott, 1999, Chapter 6)

$R=\sum_{\phi}S(a_{1},a_{2},e_{1},e_{2},I_{1},I_{2})\cos\phi,$ (8)

for coefficients $S$ and angles $\phi$ where

$\phi=j_{1}\lambda_{1}+j_{2}\lambda_{2}+j_{3}\varpi_{1}+j_{4}\varpi_{2}+j_{5}\Omega_{1}+j_{6}\Omega_{2},$
(9)

where $j_{i}$ are integers summing to zero555This ensures invariance of the
angle $\phi$ to the choice of reference axis.. The coefficients $S$ are
expressible as power series in the small parameters of eccentricity and
inclination, and this is therefore unsuitable for high inclinations and
eccentricities. As a suitable alternative for this situation, a multipole
expansion may be performed, valid for arbitrary eccentricity and inclination
but now with the semimajor axis ratio as a small parameter; this is useful for
scenarios such as planets in binary systems experiencing the Kozai effect.
These series are averaged over the mean longitudes, and only significant terms
retained in order to study long-term evolution.

Under orbit-averaging, most of the Fourier terms involving mean longitudes
vanish, as they vary rapidly over the course of an orbit. The exception comes
if two bodies’ periods are close to an integer commensurability, such as 2:1.
In this case, the terms involving the angles
$\phi_{1}=\lambda_{1}-2\lambda_{2}+\varpi_{1}$ and
$\phi_{2}=\lambda_{1}-2\lambda_{2}+\varpi_{2}$ would not average out on an
orbital timescale, as the two angles are slowly varying. Such a situation is
known as a _mean motion resonance (MMR)_. In this configuration, conjunctions
between bodies occur close to the same locations in inertial space, which
means configurations of close or even intersecting orbits can remain stable,
as in the case of Neptune and Pluto. This also means that the kicks bodies
receive from each other at conjunction add coherently, leading to significant
dynamical interactions on relatively short timescales: the semimajor axis,
eccentricity and resonant angle typically librate around a fixed point. On
long timescales, mean motion resonances can be either stable or unstable,
depending on the details of the resonance and how it interacts with the global
properties of the planetary system: witness the stable resonances of Trans-
Neptunian Objects such as Pluto with Neptune, versus the unstable _Kirkwood
gaps_ sculpted by Jupiter’s mean motion resonances in the asteroid belt. MMRs
are stronger the smaller the difference between integers, so first order MMRs
such as 2:1, 3:2 etc are usually more significant than high-order resonances
such as 11:4. MMRs have a finite width in semimajor axis, which depends on
eccentricity and on the planet:star mass ratio

$\Delta
a_{\mathrm{res}}\propto\left(\frac{m_{\mathrm{pl}}}{M_{\star}}\right)^{1/2}$
(10)

for moderate eccentricities. Instability of MMRs is linked to the overlap of
resonances, and/or to secondary resonances with other dynamical frequencies in
the planetary system (Lecar et al., 2001). The overlap of neighbouring mean-
motion resonances leads to chaotic behaviour (usually implying orbital
instability) in a _chaotic zone_ of width

$\Delta
a_{\mathrm{chaos}}=1.4\left(\frac{m_{\mathrm{pl}}}{M_{\star}}\right)^{2/7}$
(11)

surrounding the planet’s orbit.

The interplay of different mean motion resonances determines the stability of
systems of multiple planets. When there are three or more planets in a system,
additional three-planet resonances play a role, where the resonant angles are
given by $\phi=j_{1}\lambda_{1}+j_{2}\lambda_{2}+j_{3}\lambda_{3}$; a familiar
example from our Solar System is the 4:2:1 _Laplace resonance_ between Io,
Europa and Ganymede. These three-planet resonances form a “web” in the
parameter space of bodies’ semimajor axes, through which the system slowly
diffuses in chaotic motion (Petit et al., 2020). Instability can then occur
when the system reaches a stronger two-planet MMR. The typical lifetime for
this to happen is given by

$t\propto\left(\frac{m_{\mathrm{pl}}}{M_{\star}}\right)^{1/4},$ (12)

which has a slightly weaker mass scaling than the mutual Hill radius often
historically used to characterise the dynamical spacing of multi-planet
systems.

Finally, we turn to perturbations independent of bodies’ mean longitudes,
dependent only on the longitudes of periapsis and node. These, known as
_secular_ perturbations, are always present to some level in any multi-planet
system. To lowest order, a secular system can be regarded as a system of
coupled harmonic oscillators where the planets’ eccentricities and
inclinations are oscillating. The eigenfrequencies of this system play a
significant role in its dynamics: they set the frequencies for oscillation of
the planets’ eccentricities and inclinations, as well as the corresponding
precession rates of their apsides and nodes. Strong instability can occur when
one of these eigenfrequencies matches another frequency in the system. This
occurs at _secular resonances,_ where the precession frequency of one body
forced by the system’s secular perturbations matches one of the secular
eigenfrequencies. It also occurs at _secondary resonances,_ where a secular
precession frequency matches the libration frequency of a MMR (Lecar et al.,
2001). Secular and secondary resonances are responsible for carving the
structure of the asteroid belt, its edges and Kirkwood gaps.

In systems of very high eccentricity or inclination, a multipole expansion of
the perturbing potential is appropriate. In such a expansion, the leading-
order term is a quadrupole666Unlike the case of electrostatics, there can be
no negative mass to give rise to a gravitational dipole.. The effects of this
quadrupolar potential are known as the _Kozai effect_ 777Having been
(re-)discovered by several authors, it also also known as the Lidov–Kozai, or
von Zeipel–Lidov–Kozai, effect.. The Kozai effect leads to a change of the
inner body’s eccentricity and inclination while conserving the $z-$component
of the orbital angular momentum. Starting from a circular orbit, high
eccentricities can be attained if the initial mutual inclination is greater
than $\approx 39^{\circ}$. Inclusion of octupolar terms can lead to the
excitation of yet higher eccentricities (Naoz, 2016). This effect can be
significant in planetary systems with wide binary companions, as well as
planetary systems that attain large mutual inclinations during or after a
phase of scattering.

Owing to the complexity of these effects—which often operate in series or in
parallel within the same system—the planetary $N$-body problem is often
studied numerically. The planetary $N-$body problem, compared to galactic or
cosmological simulations, involves comparatively few bodies but often a huge
duration in terms of dynamical timescales, which has motivated the development
of specialised integrators and software packages. Common integration
algorithms include _symplectic integrators_ (constructed so as to guarantee
conservation of a quantity close to the system’s true energy), and high-
accuracy non-symplectic integrators such as RADAU (often in practice more
accurate than the symplectic integrators). The effects of stellar evolution
have been incorporated into commonly-used $N-$body packages such as Mercury
(Veras et al., 2013; Mustill et al., 2018) and Reboundx (Baronett et al.,
2022). In so doing, to ensure good convergence, care must be taken to ensure
that the changing stellar parameters are interpolated correctly within the
integration timesteps, rather than merely being periodically updated (Veras et
al., 2013; Baronett et al., 2022).

###  4 Tidal orbital decay

Hitherto we have dealt with point or spherical masses generating and moving
within each other’s Keplerian potentials. However, any extended body in a
gravitational potential feels a differential gravitational force across its
volume; this is known as the tidal force. Here we consider the case of an
extended giant star distorted by an orbiting planet, but the same physics
applies to such systems as the planet distorted by the star, planets and their
satellites, and stellar clusters orbiting the Galaxy. The tidal mechanism
relevant for giant stars is the _equilibrium tide,_ where the star is
distorted to align with equipotential contours generated by the combined
gravity of the planet and the star in the reference frame rotating with the
planet’s orbital mean motion. An alternative class of tidal interaction, the
_dynamical tide,_ involves the excitation and dissipation of oscillation modes
in one or both bodies; this is not significant for giant stars.

Figure 3: Cartoon illustrating the principle of the equilibrium tide. In a
reference frame corotating with the planet’s orbital motion, the undisturbed
star (yellow) is distorted and attempts to fill an equipotential (grey dashed
ellipse). As the star here is rotating more slowly than the planet orbits, the
tidal bulge lags the planet’s position, and the star instead aligns with the
black ellipse (the lag angle here is greatly exaggerated for clarity). The
planet then feels a stronger force from the nearer bulge ($F_{1}$) than from
the further bulge ($F_{2}$), resulting in a net force opposing orbital motion.
The outcome of this is the loss of orbital angular momentum and the inspiral
of the planet towards the star.

The equilibrium tide mechanism is illustrated in Fig 3. The star being a fluid
body, its surface attempts to fill an equipotential contour in the rotating
reference frame, giving rise to a deformation that can be described as two
bulges: one pointing towards the planet, and a second pointing away from the
planet at the antipodal point. In general, the fluid of the stellar envelope
has a finite response time to the changing potential, known as the _lag time_.
This means that the tidal bulges will not align exactly with the axis directed
towards the planet, unless the stellar rotation period matches the planetary
orbital period _(tidal synchronisation)._ Instead, the tidal bulges lag or
lead this axis, depending on whether the stellar rotation is slower or faster
than the planet’s orbital motion. The lag time or angle depends on the nature
of the energy dissipation in the star’s envelope and in general may depend
(often non-trivially) on the frequency of the forcing from the orbital motion.
This misalignment of the tidal bulges results in a tangential force (and
therefore torque) between the planet and the deformed star, which changes both
the spin of the star and the orbit of the planet.

Giant stars have large radii and their envelopes have large moments of
inertia, and so they typically spin very slowly. Therefore for planets
orbiting giant stars that are subject to significant tidal forces, the
planetary orbit is more rapid than the stellar spin and the tidal bulge lags
the planet’s orbital location. This removes angular momentum from the planet’s
orbit, causing the planet to slowly spiral inwards towards the star until it
is eventually engulfed in the envelope.

The exact mathematical derivation of the tidal torque is complex, but it is
easy to appreciate that the torque must be an extremely strong function of the
ratio of the star’s radius $R$ to the planet’s orbital semimajor axis $a$. The
star feels a tidal potential which is a quadrupole, proportional to
$(R/a)^{3}$, and it attempts to fill these equipotential surfaces. This small
quadrupolar distortion of the star then generates its own potential that
differs from the Kepler potential by a further $(r/R)^{3}$ quadrupole at a
radius $r$ exterior to the star; at the orbital radius of the planet, these
combine to give a perturbing potential felt by the planet proportional to
$(R/a)^{6}$. The exact scaling with radius depends on the details of the
dissipation mechanism; for giant stars, the energy of the tidal distortion is
dissipated by convective motions in the stellar envelope, and the resulting
rate of orbital decay is given by

$\left(\frac{\dot{a}}{a}\right)=\frac{f}{\tau}\frac{M_{\mathrm{env}}}{M_{\star}}q(1+q)\left(\frac{R_{\star}}{a}\right)^{8},$
(13)

where $q=m_{\mathrm{pl}}/M_{\star}$, $\tau$ the convective turnover timescale,
and $f$ a frequency-dependent term accounting for the efficiency with which
the tidal motions couple to the convective motions. (Zahn, 1989; Villaver and
Livio, 2007). Owing to the extremely strong dependence of the tidal torque on
$R/a$, as $R_{\star}$ grows from the MS to the RGB or AGB tip by a factor of
100, tides go from utterly insignificant to of overriding importance for
planets on orbits out to several au. The inwards orbital decay induced by
tidal forces competes with orbital expansion induced by stellar mass loss, as
discussed in the next section.

It is common to parametrise the strength of tidal dissipation within a body
with a parameter $Q$, the _tidal quality factor,_ quantifying the relative
amount of the energy stored in the distortion that is dissipated in a single
cycle. A constant $Q$ is rarely a suitable model for long-term evolution, as
the quality factor depends on forcing frequency often in a very non-trivial
manner. However, an effective instantaneous $Q$ can be calculated and is
instructive to compare the relative strengths of tides in different systems.
For planets orbiting giant branch stars just prior to engulfment of a planet,
the effective $Q$ is $\sim 10^{2}$ (Nordhaus et al., 2010), meaning the
strength of the tide in this situation is orders of magnitude stronger than
the strength of a tide raised by a planet on a MS star ($Q\gtrsim 10^{7}$,
e.g., Collier Cameron and Jardine 2018).

###  5 Stellar mass loss and its effects on orbital dynamics

####  1 Mass loss in two-body systems: orbit expansion and competition with
tidal inspiral

We now consider the effects of stellar mass loss on orbiting bodies, beginning
with the star–planet system covered comprehensively by Veras et al. (2011). In
most cases stellar mass loss does not create a significant perturbative force
(though there are weak aerodynamic drag forces on very small bodies); instead,
mass loss from the star causes a change to its gravitational potential, which
then changes bodies’ orbital elements. Recalling Noether’s Theorem from
Section 1, we remember that conservation of energy in a system arises from a
time symmetry. By allowing mass loss from the star, we now have a time-
dependent potential, meaning that bodies’ orbital energy is no longer
conserved: in general, orbits expand as the star loses mass. Some constraints
can, however, still be placed on the orbital evolution. Mass loss from RGB and
AGB stars is unlikely to be strongly directional, meaning that rotational
invariance of the potential is not broken and therefore that orbital angular
momentum is conserved as the orbit expands888Although anisotropic effects may
matter for bodies on very wide orbits, such as Oort cloud comets close to
apoapsis.. Isotropic mass loss also means that the orbital inclination and
longitude of ascending node do not change. In the important case of _adiabatic
mass loss,_ where the time-scale for mass loss is significantly longer than
the orbital timescale, the orbital eccentricity is also an _adiabatic
invariant_ and does not change as mass loss proceeds; nor does the argument of
periapsis. The orbit therefore expands while maintaining the same shape, and
from conservation of angular momentum (Equation 3) we have the following
relation between the initial and final semimajor axes:

$\frac{a_{\mathrm{final}}}{a_{\mathrm{inital}}}=\frac{M_{\star\mathrm{,initial}}}{M_{\star\mathrm{,final}}}.$
(14)

Thus, planets orbiting a $1\mathrm{\,M}_{\odot}$ star, which loses around half
of its mass, will roughly double their orbital semimajor axes; those orbiting
a $3\mathrm{\,M}_{\odot}$ star, which loses around three-quarters of its mass,
will roughly quadruple theirs.

While mass loss is typically slow and adiabatic for planets on orbits of a few
to tens of au, it can be significantly non-adiabatic for bodies with wider
orbits and longer orbital periods, such as wide binary stellar companions and
Oort cloud comets. Under non-adiabatic mass loss, orbital angular momentum is
still conserved, but the eccentricity can change; the orbital semimajor axis
and periapsis distance still increase, typically faster than in the adiabatic
case (Veras et al., 2011). For sufficiently rapid mass loss, bodies can become
unbound: such is for example the case for _impulsive_ (effectively
instantaneous) mass loss such as that resulting from a supernova explosion,
where the loss of over half of the star’s mass will cause a body initially on
a circular orbit to possess positive energy and hence to be on a hyperbolic
orbit after the explosion. For the low- and intermediate-mass stars with which
we are concerned, non-adiabatic mass loss leads to the unbinding of their Oort
Cloud comets and their release into the Galaxy as interstellar objects (ISOs)
such as 1I/‘Oumuamua and 2I/Borisov.

##### Mass loss vs tides

Figure 4: Orbital evolution of $1\mathrm{\,M_{J}}$ planets orbiting
$1.5\mathrm{\,M}_{\odot}$ stars. A suite of one-planet systems is evolved
through the thermally-pulsing AGB, each system having its planet placed on a
circular orbit at a different initial semimajor axis. The planets’ orbits
evolve under the combined effects of stellar mass loss and tidal orbital
decay, and are removed if they enter the stellar envelope (blue line). Planets
beginning interior to $\sim 3.2$ au are engulfed (red lines), while those on
wider orbits survive (black lines). Tidal models from Mustill and Villaver
(2012) with stellar evolution models from Vassiliadis and Wood (1993).

We see now that tidal forces and mass loss are in competition for planets
initially orbiting at a few au: tides drag planets in towards the star, while
mass loss causes their orbits to spiral outwards (Figure 4). The balance
between engulfment and survival has been probed in many studies under
different assumptions about stellar evolution, mass loss and the strength of
tidal forces (e.g., Villaver and Livio, 2007; Nordhaus et al., 2010; Mustill
and Villaver, 2012; Sanderson et al., 2022). Restricting attention to planets
on circular orbits, when considering a planet of a given mass, the tidal force
is a monotonically decreasing function of initial orbital radius; therefore,
within a certain radius planets of that mass will be engulfed, while exterior
to it they will survive. This critical radius depends on the mass of both the
star and the planet, and is in the region of a few au (Mustill and Villaver,
2012; Nordhaus and Spiegel, 2013). As the tidal forces decay very rapidly with
orbital semimajor axis, most planets beyond the destruction radius will simply
expand their orbits according to Equation 14. Over a relatively narrow range
of intermediate separations, planets’ orbits expand less than anticipated from
mass loss alone (lowest few black lines in Figure 4), or may even in rare
cases shrink. However, most young WDs should be surrounded by a cleared volume
of at least several au in radius, where no planets are found. The extent of
this volume depends strongly on stellar progenitor mass: it is larger for the
more massive WDs with more massive progenitors, which attain greater AGB radii
and lose a larger fraction of their mass. It also depends on the modelled
strength of stellar mass loss and tidal effects. In principle these models
could be tested with a large population of planets detected at suitable
distances (_i.e.,_ a few to ten au) around WDs. While JWST has recently imaged
some giant planet candidates around WDs, the best way to detect large numbers
of these planets may be astrometrically in future data releases of the _Gaia_
spacecraft. However, although _Gaia_ is expected to yield detections of
planets around WDs they will likely not be numerous enough to meaningfully
constrain the tidal models (Sanderson et al., 2022).

Once engulfed, planets’ orbits shrink rapidly owing to gravitational friction
with the stellar envelope, taking $<100$ yr to reach $1\mathrm{\,R}_{\odot}$
(O’Connor et al., 2023): this is essentially a form of _common-envelope (CE)
evolution_ with an extremely low-mass companion. While a stellar companion can
survive such an event, transferring its orbital energy to the stellar envelope
and expelling it, and ending up as a short-period _post-common envelope
binary,_ the prospects for planets are much bleaker. They lack enough energy
to eject the stellar envelope, meaning that their inspiral is not terminated
by ejection of the envelope and continues until the planet is destroyed. This
happens either by evaporation within the denser regions of the stellar
envelope, or by tidal disruption at the Roche limit at $\sim
1\mathrm{\,R}_{\odot}$. The lower mass limit for survival of bodies inside the
stellar envelope is thought to be of the order of $10\mathrm{\,M_{J}}$
depending on factors such as the efficiency of energy transfer in the CE
phase, and is coincidentally around the planet/brown dwarf mass boundary
(Villaver and Livio, 2007; Nordhaus and Spiegel, 2013; O’Connor et al., 2023).

The fact that planets and other material are in fact seen on tight orbits
close to WDs shows that many systems undergo significant dynamical evolution
throughout the WD’s lifetime. This can be triggered by the dynamical changes
in multi-planet systems during the AGB, as discussed next.

####  2 Mass loss in multi-body systems: changes to dynamics and stability

Planets in systems of multiple planets, as those in single-planet systems,
experience adiabatic or non-adiabatic orbit expansion in accordance with their
semimajor axes. In systems spanning a large range of semimajor axes, it is
possible for closer bodies to be in the adiabatic regime and more distant ones
in the non-adiabatic regime, leading to an increase in semimajor axis ratios.
For planetary systems located within a few tens of au, however, orbit
expansion will be adiabatic for all bodies and the system will expand in a
self-similar manner.

In multi-planet systems, the planet–star mass ratio increases following mass
loss, by a factor of 2–4 for 1–3 $\mathrm{M}_{\odot}$ progenitor stars. This
has significant effects on the system’s dynamics, since most of the dynamical
interactions described in Section 2 have a strength that depends on some power
of this ratio. Speaking generally, following mass loss the planets are heavier
relative to the star than they were before; this strengthens planet–planet and
planet–asteroid interactions and leads to the destabilisation of bodies, or
indeed entire systems, that survived their MS lifetime (Debes and Sigurdsson,
2002). This is now thought to be responsible for the delivery of material
close to the WD that is observed as photometric transits, dust and gas discs,
and ultimately photospheric pollution. In this paradigm, asteroids, planets,
or moons are destabilised from their original orbits and have their
eccentricities excited to near unity. Their orbital periapsis then lies
interior to the _Roche limit_ or _tidal disruption limit_

$R_{\mathrm{Roche}}=\left(\frac{3\rho_{\star}}{\rho_{\mathrm{pl}}}\right)^{1/3}R_{\star}$
(15)

around the WD, where $\rho_{\star}$ and $\rho_{\mathrm{pl}}$ are the average
stellar and planetary densities, resulting in their disruption as tidal forces
overcome internal binding forces (Murray and Dermott, 1999).

There are now numerous studies of the details of how different dynamical
mechanisms are affected by stellar mass loss, of which there is only space to
mention a few here. In systems of a single planet and a disc of
comets/asteroids, unstable MMRs broaden during stellar mass loss [width
$\propto\left(m_{\mathrm{pl}}/M_{\star}\right)^{1/2}$; Equation 10], and MMRs
interior to a planet’s orbit can be an efficient deliverer of material to the
WD, although the extent of a disc affected is rather small (Debes et al.,
2012). Where the planet is close enough to the star to be affected by tidal
forces, MMRs migrate along with the planet, sweeping through the system and
resulting in more complex dynamics that can enhance or reduce the
survivability of interior bodies (Ronco et al., 2020). The planet’s chaotic
zone, with its width scaling as $\left(m_{\mathrm{pl}}/M_{\star}\right)^{2/7}$
(Equation 11), also broadens with mass loss (Bonsor et al., 2011), although an
external disc configuration (such as Neptune and the Kuiper Belt) is
inefficient at delivering material close to the star and requires the presence
of extra planets to deliver destabilised bodies to the WD (Bonsor et al.,
2011).

In systems of multiple planets, the stability scalings in two-planet systems
[scaling as the Hill radius
$\propto\left(m_{\mathrm{pl}}/M_{\star}\right)^{1/3}$; Equation 6, 7] and in
three-planet systems [scaling with
$\left(m_{\mathrm{pl}}/M_{\star}\right)^{1/4}$; Equation 12] are affected by
mass loss, leading to the destabilisation of entire systems that survived MS
evolution (e.g., Debes and Sigurdsson, 2002; Veras et al., 2013; Mustill et
al., 2018; Maldonado et al., 2021). In terms of both the efficiency and
longevity of the delivery of material to a WD, destabilisation of multi-planet
systems can be more effective than the moderate growth of chaotic regions in
single-planet systems. For planets on wide orbits around WDs, the Safronov
number (Equation 5) is high, favouring strong gravitational scattering and
excitation of planetary eccentricities over collisions between planets once
instability has occurred. The high-eccentricity planets can then interact with
and destabilise entire planetesimal discs, rather than the relatively narrow
regions affected by the growth of chaotic zones and Kirkwood gaps (Mustill et
al., 2018). Instability in systems of giant planets resolves rather quickly,
and most bodies are ejected rather than being forced down to the WD. However,
in systems of low-mass (e.g., Earth- or Neptune-mass) planets the planets are
inefficient at ejecting either one another or asteroids. Instead, the
gravitational scattering can continue for many Gyr as the planetesimal disc is
slowly depleted through a combination of ejection from the system and
scattering down to the WD’s Roche limit. Despite the relatively weak scaling
of the stability timescale with planet–star mass ratio, high-multiplicity
systems of low-mass planets may be sufficiently closely spaced that the
majority of them can be destabilised by mass loss (Maldonado et al., 2021),
and they are therefore a good candidate to explain WD planetary phenomena on a
population level.

Slightly different is the case of secular dynamics in planetary systems, where
adiabatic mass loss _per se_ doesn’t change the locations or widths of secular
resonances (the changing planet–star mass ratio manifests instead in globally
rescaling the secular frequencies). However, tidal decay or engulfment of the
inner planets does change the location of secular resonances (Mustill and
Villaver, 2012; Smallwood et al., 2021), which can be highly efficient at
exciting the near-unity eccentricities needed for delivery of bodies to the
WD. A changed system architecture due to tidal migration can therefore place
secular resonances in previously stable and well-populated regions of a
planetesimal disc. Furthermore, radiative forces on the AGB can redistribute
minor bodies through a planetary system (see Section 6), repopulating
resonances that were destabilised on the MS. Finally, mass loss and tidal
engulfment can change the nature of Lidov–Kozai interactions, for example by
removing a planet interior to a disc that was suppressing Lidov–Kozai
oscillations. Similar to the case of scattering by low-mass planets, such
secular mechanisms can provide efficient delivery of asteroids to the WD over
a long timescale (Petrovich and Muñoz, 2017).

A major challenge in modelling these processes is our significant lack of
knowledge about the relevant regions of planetary systems that survive RGB and
AGB evolution, beyond a few au around stars more massive than the Sun. This
state of ignorance applies to both the WD systems themselves and their MS and
RGB/AGB progenitors. Our knowledge here is limited to a reasonable census of
Jupiter-analogue planets on orbits of a few au, a handful of wide-orbit young
massive planets detected through direct imaging, and a growing number of
planet candidates detected indirectly from structures observed in
protoplanetary discs, together with relatively good statistics on the broad
occurrence rates of minor body populations in the form of debris disc studies
around MS stars. The prevalence and multiplicity of lower-mass planets (Earth-
mass, super-Earths and Neptunes), known to be extremely common on close-in
orbits, is entirely unconstrained. In light of the weak observational data,
theoretical studies can take three approaches to creating planetary systems on
wider orbits. Most studies make synthetic systems based on simple
prescriptions such as setting up 3 planets separated by a number of Hill radii
(e.g., Debes and Sigurdsson, 2002; Veras et al., 2013; Mustill et al., 2018).
This has the advantage of control over system parameters, but the systems may
not be representative of the real planetary population. Maldonado et al.
(2021) took a different approach, taking the well-studied systems close to the
star (many being multi-planet systems discovered by the _Kepler_ spacecraft)
and using them as templates, scaling them up to wider orbits. This implicitly
assumes that processes of planet formation and migration are self-similar at
sub-au distances around Solar-mass stars and at $\sim 10$ au around
intermediate-mass stars. Finally, in principle one could consistently model
the formation of a planetary system, including its minor body populations, and
then study its long-term evolution under mass loss. However, planet formation
models are not well constrained for planets on wide orbits around stars more
massive than the Sun, and this has not been explored in practice.

###  6 Radiative effects

####  1 Radiative forces and their effects on orbits

The motion of most bodies around stars is dominated by gravitational forces,
both from the star and from other bodies in the system. However, _radiative_
or _radiation forces_ —forces on orbiting bodies arising from the emission,
absorption or scattering of electromagnetic radiation—may induce slow changes
to Keplerian orbits, in a similar manner as planet–planet interactions in
multi-planet systems. The strength of radiative forces is proportional to the
cross-sectional area of an orbiting body, while that of the gravitational
force is proportional to its mass (thus, volume); therefore, radiative forces
are most significant for the smallest bodies such as dust grains, and are
unimportant for planetary bodies. These radiative forces become stronger as
the stellar luminosity increases on the RGB and the AGB, and bodies for which
these forces were previously insignificant begin to be affected by them.

##### Radiation pressure

_Radiation pressure_ is the radial force arising from momentum transfer from
stellar photons to a body (Burns et al., 1979). Its strength is given by

$F_{\mathrm{rad}}=\frac{L_{\star}AQ_{\mathrm{PR}}}{4\pi a^{2}c},$ (16)

where $c$ is the speed of light, $A$ is the body’s cross-sectional area and
$Q_{\mathrm{PR}}$ its radiation pressure efficiency which includes absorption
and scattering (Burns et al., 1979). As the radiation pressure is a radial
inverse square force, it can be considered as an effective reduction in
stellar mass, with the parameter $\beta=F_{\mathrm{rad}}/F_{\mathrm{grav}}$,
the ratio of radiative to gravitational force, being constant round the orbit.
Objects on circular orbits that suddenly feel a strong radiation pressure (for
example, dust grains released from an asteroid) will suddenly change their
orbital eccentricity and semimajor axis, while the orbit will be unbound if
$\beta\geq 0.5$ (this is the same mass reduction factor as leads to unbinding
after impulsive mass loss described in Section 1). This condition is met at
the _blow-out size,_ which is roughly $1\,\mu\mathrm{m}$ for dust orbiting the
Sun. $F_{\mathrm{rad}}$ increases with increasing stellar luminosity, and the
blow-out size increases to $\sim$mm sizes at the AGB tip. Particles below this
limit are removed on an orbital timescale, as they follow hyperbolic
trajectories, although it is worth noting that as the AGB tip luminosities are
not maintained for long, such bodies may not actually be able to escape before
the end of the AGB.

##### Poynting–Robertson drag

Let us now consider bound small particles, above the blow-out size. Such a
small particle will absorb radiation from the star, be heated uniformly, and
re-radiate the energy isotropically in its rest frame. When switching back to
the inertial frame of the system, the emission therefore beamed in direction
of orbital motion, resulting in a retarding force opposed to the orbital
motion, known as _Poynting–Robertson drag_ or the _Poynting–Robertson effect_.
This creates a negative torque on the orbit leading to orbital in-spiral and
eccentricity decay, with the rate of decay of semimajor axis being given by

$\dot{a}=-\frac{L_{\star}Q_{\mathrm{PR}}A}{4\pi
amc^{2}}\frac{2+3e^{2}}{\left(1-e^{2}\right)^{3/2}},$ (17)

where $m$ is the body’s mass (Burns et al., 1979). In terms of the $\beta$
parameter introduced above, this leads to a lifetime of

$\frac{t_{\mathrm{PR}}}{\mathrm{yr}}=\frac{400}{\beta}\left(\frac{a}{\mathrm{au}}\right)^{2},$
(18)

and so as $\beta$ increases significantly on the RGB and AGB orbital inspiral
can become very rapid.

##### Yarkovsky effect

Larger bodies in the km size range, instead of being uniformly heated and
radiating isotropically, are large enough to have a significant temperature
gradient across their volume, which can have seasonal and diurnal components
(Bottke et al., 2006). The result of this is an anisotropic absorption and
emission of radiation, giving rise to a force acting to change the orbit,
known as the _Yarkovsky effect_ or _Yarkovsky force_. The orbit can expand or
contract depending on the spin direction of the body. The Yarkovsky effect is
responsible in the Solar System for delivering asteroids to unstable Kirkwood
gaps in the Asteroid Belt. As radiation forces become significant on the AGB,
rapid migration of bodies becomes possible.

####  2 Physical alteration and destruction of bodies by radiation forces

So far we have considered the effects of stellar evolution on bodies’ orbits,
which are always affected at least in their response to stellar mass loss.
Stellar evolution can, in some cases, also have an effect on the bodies
themselves, primarily through the star’s increased luminosity.

A simple estimate for the surface temperature of a body in orbit around a star
is the _equilibrium temperature_ $T_{\mathrm{eq}}$, calculated by balancing
the total incoming radiation from the illumination of one hemisphere by the
star, and the total outgoing radiation assuming the body redistributes this
energy effectively and reradiates it isotropically from its entire surface:

$T_{\mathrm{eq}}=\left[\frac{\left(1-A_{\mathrm{b}}\right)L_{\star}}{16\pi\epsilon\sigma
a^{2}}\right]^{1/4},$ (19)

where $A_{\mathrm{b}}$ is the Bond albedo of the body, $\epsilon$ its
emissivity, and $\sigma$ the Stefan–Boltzmann constant. Despite the relatively
weak $L_{\star}^{1/4}$ scaling, the strong increase of stellar luminosity
entails a significant increase in bodies’ equilibrium temperatures on the RGB
and AGB.

##### The YORP effect

Related to the Yarkovsky effect is the _Yarkovsky–O’Keefe–Radzievskii–Paddack
(YORP)_ effect, where anisotropic reflection, absorption and emission of
starlight causes a torque on the body itself and affects its rotation rate and
spin axis (Bottke et al., 2006). The effect of this is seen in the Solar
System, where small asteroids are spun up to the break-up limit. As do all
radiation effects, while present on the MS it strengthens on RGB and AGB. Then
bodies up to $\sim 10$ km can be rapidly spun up to break-up, leading to the
release of their component parts (rocks, pebbles, etc) as debris (Veras et
al., 2014). Being much smaller than the parent body, these debris particles
are then susceptible to rapid orbital evolution under the other radiative
forces described above, and can migrate through the system to repopulate
regions destabilised during a system’s MS evolution and so provide a source
for WD pollution.

##### Sublimation and devolatilisation of bodies by strong irradiation

Strong heating can result in physical alteration of bodies by sublimation of
their more volatile components. Most easily lost are volatile species such as
water with low sublimation points:for $1\mathrm{\,M}_{\odot}$ progenitors,
smaller asteroids (few km) are totally dessicated out to 30–40 au, while 100km
asteroids are capable of retaining almost half of their water even at 5 au
(Malamud and Perets, 2016). This process has implications for understanding
extrasolar planetesimal compositions from WD pollution, as these have been
devolatilised by RGB and AGB irradiation before we can observe them, and so
may not reflect the composition of their progenitor bodies on the MS.

##### Planetary atmospheric loss

Planetary atmospheres are heated thermally as the star ascends RGB and AGB,
when the stellar irradiation is intense but the photons of relatively low
energy, since stars cool as they ascend giant branch. The heating of the
planetary atmosphere, coupled with the accretion of stellar wind material for
giant planets, can have effects on atmospheric chemistry (Spiegel and
Madhusudhan, 2012).

As the star and its expelled envelope transitions into the planetary nebula
phase, the star core is briefly highly luminous in X-rays owing to its
temperature. Planetary exospheres can then be heated by the XUV irradiation,
leading to blow-off conditions and the loss of significant fractions of the
planets’ envelope. It is expected that the innermost giant planets that can
survive stellar radius expansion and tidal interactions, within $\sim 3$ au,
will lose most or all of their envelopes in this phase (Villaver and Livio,
2007).

##### Changes to planetary habitability

The significant increase in stellar luminosity as the star ascends the RGB is
ultimately lethal to Earth-like life on any planets in the liquid water
habitable zone during the main sequence, as the planetary equilibrium
temperature increases. Planets with surface water such as Earth will see a
stronger increase in their surface temperature then suggested by Equation 19
owing to the greenhouse effect, and Earth will likely become uninhabitable to
complex life in roughly 1 Gyr and to all life in around 3 Gyr (O’Malley-James
et al., 2013), long before the Sun leaves the MS. As the Sun’s luminosity
continues to increase, Mars and the satellites of the giant planets will
briefly enter the liquid water habitable zone. Whether life would have time to
develop on analogous bodies in extra-Solar systems is an open question.

To end this section on a speculative note, a technological civilisation may
attempt to construct a _starshade,_ a structure interposed between the planet
and the star to counter the increasing stellar luminosity and maintain a
habitable climate (Gaidos, 2017). Even considering the relatively modest
luminosity increase as the star progresses along the MS, these structures can
be large enough to yield a transit signal detectable with current
instrumentation, should they exist and cross our line of sight to the star.

###  7 Summary and Conclusions

We have seen that both stars and their planetary systems undergo significant
changes after the star leaves the Main Sequence, progresses through RGB and
AGB evolution, and becomes a white dwarf. Stellar mass loss causes orbits to
expand, while the increased stellar radius enhances tidal forces which means
that close-in planets out to a few au are engulfed into the envelope, where
they likely do not survive unless they are around the brown dwarf mass limit
($\gtrsim 10\mathrm{\,M_{J}}$). This means that a young white dwarf should be
surrounded by a cleared volume, several au in radius, purged of all planetary
material.

The fact that we see many white dwarfs surrounded by dusty debris discs within
$\sim 1\mathrm{\,R}_{\odot}$, or with rocky material in their atmospheres,
together with a handful of transits of planets, or structures likely arising
from disrupting asteroids, means that this picture cannot be complete. It is
likely that stellar mass loss, in addition to causing orbits to expand,
triggers orbital instability in a significant fraction of planetary systems
orbiting WDs whose progenitors were stable on the main sequence. While in
principle this is easy to understand, since after stellar mass moss the
planets are relatively heavier compared to the star than they were before, the
exact mechanism or mechanisms at play are not yet known, with changes to
orbital resonances, changes to dynamics in stellar binary systems, and the
outright destabilisation of entire planetary systems all likely playing some
role in unknown proportions. Our understanding of the orbital dynamics at work
here is hampered by our lack of knowledge of the relevant populations of
planets: likely low-mass (sub-Saturn), on orbits of a few au or beyond
(ensuring survival beyond the AGB), and orbiting stars of around
$2\mathrm{\,M}_{\odot}$. These are all extremely challenging regimes to probe
observationally.

Finally, the large stellar luminosity on the AGB also affects orbiting bodies.
This can be through the enhancement of radiative forces such as radiation
pressure, Poynting–Robertson drag, and the Yarkovsky and YORP effects, which
can cause a significant orbital migration of dust and small asteroids; this
may provide an additional source for delivery to white dwarfs. The large
luminosity can also physically alter bodies, stripping planets of their
atmospheres and asteroids and comets of their volatile content. These effects
must also be taken into account when attempting to link planetary systems
around white dwarfs to their progenitors around giant branch and main sequence
stars, in order to get a full picture of the whole lifespans of planetary
systems.

[Acknowledgments]

AJM acknowledges support from the Swedish Research Council (Project Grant
2022-04043) and the Swedish National Space Agency (Career Grant 2023-00146).
He would like to thank the Section Editor Dimitri Veras, and Paula Ronco, for
useful comments on the manuscript.

_see also_ An excellent and comprehensive review on the effects of stellar
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time of its publication, can be found in Veras (2016). Thorough treatments of
celestial mechanics are to be found in Murray and Dermott (1999) and Tremaine
(2023), while an introduction to the required stellar evolution theory is to
be found in Prialnik (2009). 37

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