Introduction
Violent variability, such as flaring activity, of planet-hosting stars is known to strongly affect prebiotic processes on their planets; therefore, its role in such processes is being extensively studied nowadays (see Airapetian et al., Reference Airapetian, Glocer, Gronoff, Hébrard and Danchi2016; Garcia-Sage et al., Reference Garcia-Sage, Farrish, Airapetian, Alexander, Cohen, Domagal-Goldman, Dong, Gronoff, Halford, Lazio, Luhmann, Schwieterman, Sciola, Segura, Toffoletto, Vievering, Ahmed, Bali and Rau2023; Namekata et al., Reference Namekata, Airapetian, Petit, Maehara, Ikuta, Inoue, Notsu, Paudel, Arzoumanian, Avramova-Boncheva, Gendreau, Jeffers, Marsden, Morin, Neiner, Vidotto and Shibata2024, and references therein). On the contrary, the role of regular variability, such as variability of pulsating stars, has not yet been explored.
Planet-hosting pulsating stars are indeed known to exist, although it is technically difficult to discover and observe planets belonging to variable stars. The set of such stars include WASP-33, WASP-167 (KELT-13, TOI-748), KIC 7917485, HAT-P-2 (HD 147506), HD 31221, HAT-P-57, Kepler-976; they are either δ Scuti or γ Doradus type stars (Kálmán et al., Reference Kálmán, Derekas, Csizmadia, Pál, Szabó, Smith, Nagy, Hegedüs, Mitnyan, Szigeti and Szabó2024; Ahlers et al., Reference Ahlers, Barnes and Myers2019; Christian et al., Reference Christian, Pollacco, Skillen, Street, Keenan, Clarkson, Collier Cameron, Kane, Lister, West, Enoch, Evans, Fitzsimmons, Haswell, Hellier, Hodgkin, Horne, Irwin, Norton, Osborne, Ryans, Wheatley and Wilson2006; Murphy et al., Reference Murphy, Bedding and Shibahashi2016; Collier Cameron et al., Reference Collier Cameron, Guenther, Smalley, McDonald, Hebb, Andersen, Augusteijn, Barros, Brown, Cochran, Endl, Fossey, Hartmann, Maxted, Pollacco, Skillen, Telting, Waldmann and West2010; Wong et al., Reference Wong, Shporer, Kitzmann, Morris, Heng, Hoeijmakers, Demory, Ahlers, Mansfield, Bean, Daylan, Fetherolf, Rodriguez, Benneke, Ricker, Latham, Vanderspek, Seager, Winn, Jenkins, Burke, Christiansen, Essack, Rose, Smith, Tenenbaum and Yahalomi2020; Herrero et al., Reference Herrero, Morales, Ribas and Naves2011; Bakos et al., Reference Bakos, Kovács, Torres, Fischer, Latham, Noyes, Sasselov, Mazeh, Shporer, Butler, Stefanik, Fernández, Sozzetti, Pál, Johnson, Marcy, Winn, Sipöcz, Lázár, Papp and Sári2007; de Wit et al., Reference de Wit, Lewis, Knutson, Fuller, Antoci, Fulton, Laughlin, Deming, Shporer, Batygin, Cowan, Agol, Burrows, Fortney, Langton and Showman2017; Kálmán et al., Reference Kálmán, Derekas, Csizmadia, Szabó, Hegedüs, Smith, Kovács, Ziegler, Pál, Szabó, Parviainen and Murgas2023; Hartman et al., Reference Hartman, Bakos, Buchhave, Torres, Latham, Kovács, Bhatti, Csubry, de Val-Borro, Penev, Huang, Béky, Bieryla, Quinn, Howard, Marcy, Johnson, Isaacson, Fischer, Noyes, Falco, Esquerdo, Knox, Hinz, Lázár, Papp and Sári2015; de Beurs et al., Reference de Beurs, de Wit, Venner, Berardo, Bryan, Winn, Fulton and Howard2023; Kálmán et al., Reference Kálmán, Bókon, Derekas, Szabó, Hegedüs and Nagy2022).
Here, we address the question – which effect of regular variability of host stars can be expected for prebiotic chemistry? Today, broad classes of physical non-equilibria are discussed as directly relevant to prebiotic nucleic acid chemistry (see Lathe, Reference Lathe2004, Reference Lathe2005; Ianeselli et al., Reference Ianeselli, Salditt, Mast, Ercolano, Kufner, Scheu and Braun2023, and references therein). Short-period (on a timescale of hours) variations of environmental temperature, water salinity, ultraviolet irradiation, and other factors could control processes of prebiotic replication (Lathe, Reference Lathe2004, Reference Lathe2005; Ianeselli et al., Reference Ianeselli, Salditt, Mast, Ercolano, Kufner, Scheu and Braun2023). To provide the melting of extremely stable double-stranded RNA (i.e., to overcome the “product inhibition problem”), so that the RNA replication could be continuously iterated, several forms of environmental cycling were suggested: cycling of water salinity (Lathe, Reference Lathe2004, Reference Lathe2005; Ianeselli et al., Reference Ianeselli, Mast and Braun2019), pH of water (Mariani et al., Reference Mariani, Bonfio, Johnson and Sutherland2018), temperature (Kreysing et al., Reference Kreysing, Keil, Lanzmich and Braun2015; He et al., Reference He, Gállego, Laughlin, Grover and Hud2017; Edeleva et al., Reference Edeleva, Salditt, Stamp, Schwintek, Boekhoven and Braun2019); for a review, see Tupper and Higgs (Reference Tupper and Higgs2021).
Planetary rotation has long been considered an important factor in bioevolution, as it causes the periodic environmental change for biochemical processes (Blum, Reference Blum and Rudnick1957; Reference Blum1962; Lathe, Reference Lathe2004, Reference Lathe2005): the incoming radiation flux from the host star is modulated by planetary rotation, for a typical axial tilt.
Here, we consider an astrophysical factor that provides a similar, but perhaps even more suitable, periodic environmental change: the regular high-amplitude short-period variability of the host star; in this case, the incoming radiation flux is intrinsically modulated. In this way, we explore how the variability of pulsating stars may affect the prebiotic evolution on their eventual planets. We show that the RR Lyr type variables are particularly relevant, because their lightcurves possess amplitudes, periods, and time profiles that are potentially most suitable to drive prebiotic reactions. In this context, we introduce a “photospin resonance” notion. (As defined, the 1/1 photospin resonance means synchrony of stellar pulsation with planetary spin, and the 0/1 one corresponds to the planetary tidal lock, which results in effective suppression of diurnal forcing).
Prebiotic reactions on planetary surfaces
In Lathe (Reference Lathe2004, Reference Lathe2005, Reference Lathe2006), a theory of abiogenesis on Earth was proposed, based on a tidal “boosting” effect for the biomolecules production in near-shore lakes and ponds. The repeatedly drying-wetting tidal pools can be a possible place of origin of self-replicating biopolymers, as such pools provide favorable conditions for concentrating organic molecules. In Lathe’s tidal chain reaction (TCR), water salinity is periodically driven (on the early Earth affected by the Lunar tides), leading to exponential amplification of nucleic acids (Lathe, Reference Lathe2004, Reference Lathe2005). At low-salinity phases, the association is promoted and complementary strands are synthesized, and, at high-salinity ones, the duplex strands dissociate. However, the reaction product is unstable and rapidly decays; the Lunar tide cycling period might have been too long to allow for maintaining the TCR. This constitutes a major problem for the TCR scenario for the early Earth abiogenesis (Lathe, Reference Lathe2004, Reference Lathe2005).
The difference between TCR and authentic polymerase chain reaction (PCR) is that, in PCR, exponential amplification of nucleic acids is driven by periodic cycling of temperature (not water salinity), with an amplitude of
$\sim $
50 degrees: at low temperature, association is promoted and complementary strands are synthesized (the molecule number doubles), whereas at high temperature, the duplex strands dissociate (Mullis et al., Reference Mullis, Faloona, Scharf, Saiki, Horn and Erlich1986). (These two steps of the cycle are called annealing/elongation and denaturation ones, respectively). When, in the next cycle, temperature is again decreased, the number of molecules again doubles, and so on.
Apart from driving abiogenesis on Earth, Lathe’s TCR could be potentially active on circumbinary planets of main-sequence stars, due to the photo-tidal synchronization inherent for such planets (Shevchenko, Reference Shevchenko2017). The related Binary Habitability Mechanism was earlier suggested in Mason et al. (Reference Mason, Zuluaga, Clark and Cuartas-Restrepo2013) and Mason et al. (Reference Mason, Zuluaga, Cuartas-Restrepo and Clark2015); in this mechanism, circumbinary planets experience periodic variations in flux across the spectrum as the host stars alternately approach and recede.
In Shevchenko (Reference Shevchenko2020), we suggested that eventual planets of contact-binary stars of W UMa type and pulsating stars of SX Phe type are suitable for maintaining biopolymer chain reactions (BCR), once the planet is in the host star’s zone of habitability. Indeed, on surfaces of such planets, the basic conditions for BCR are automatically satisfied: suitable amplitudes and periods of environmental temperature periodic variations. In the present article, the scenario proposed in Shevchenko (Reference Shevchenko2020) is essentially generalized, considered in application to hypothetical planetary worlds of RR Lyr variables.
Among various known classes of pulsating stars, we concentrate on RR Lyr variables because, as we will see further on, the amplitudes, periods, and, moreover, time profiles of their luminosity variations are suitable for driving BCR on their eventual planets. Other known variable star classes, with exception of W UMa and SX Phe stars (considered already in Shevchenko, Reference Shevchenko2020), do not seem to provide such favorable conditions, as they have either unsuitable variation periods (too long, as in the case of Cepheids), or unsuitable amplitudes (too small, in many classes), or unsuitable time profiles. However, short-period Cepheids might also deserve analysis, especially in view that, as discussed below, any small reproduction factor of chain reaction can be compensated by a large enough number of pulses.
Generally, any pulsating star classes with longer stellar cycles (such as Cepheids) look much less appropriate candidates for driving the BCRs, because the biopolymers being synthesized degrade, during each cycle, exponentially with time (until a stable product is achieved after many cycles); indeed, BCR cycles with duration of a day or greater seem to be totally ineffective (see Lathe, Reference Lathe2004, Reference Lathe2005).
The RR Lyr class of variables has two main subclasses:
$\sim $
90% of its observed representatives are the so-called RRab variables, whereas the remaining minority is formed by the so-called RRc variables (Smith, Reference Smith2004). We are mostly interested in the RRab variables, which display steep rises in brightness and thus have particularly suitable lightcurve time profiles. Variations of RRc stars are more sinusoid-like.
On the Hertzsprung–Russell diagram, the RR Lyr variables populate the so-called “Schwarzschild gap,” located at the “instability strip — horizontal branch” intersection. (When building the HR diagram, any variable stars are usually ignored, hence the term “gap”). The stars evolving on the horizontal branch eventually enter this intersection area and start to radially pulsate. According to Catelan and Smith (Reference Catelan and Smith2015), Ch. 6, their brightness and temperature periodically vary with amplitude in visual magnitude within the range 0.2–1.6, and period within 0.2–0.9 d.
Note that before a star enters the horizontal branch, it suffers the red-giant stage and helium flash; therefore, when considering possible planets of RR Lyr stars, one should have in mind that planets of the first (preceding) generation could lose their gas envelopes or could even be totally destroyed or lost. If this happens, a new, second, generation of planets could form. Thus, the star’s planetary system could be radically transformed, with potentially fatal consequences for any life that could have been already present. However, in this article, we discuss opportunities for abiogenesis, not opportunities for life survival.
Therefore, let us consider how BCR can be naturally maintained on planets in habitability zones (HZ) of RR Lyr stars. Let
${F_{{\rm{min}}}}$
and
${F_{{\rm{max}}}}$
be the minimum and maximum stellar fluxes at the planetary surface during the stellar pulsation cycle. The corresponding stellar magnitude variation is
${\rm{\Delta }}m = 2.5{\rm{lo}}{{\rm{g}}_{10}}\left( {{F_{{\rm{max}}}}/{F_{{\rm{min}}}}} \right)$
. In a simplest model setting, the temperature variation on the planetary surface is determined by the Stefan–Boltzmann law, which gives
$${{\Delta T} \over {{T_{{\rm{min}}}}}} = {\left( {{{{F_{{\rm{max}}}}} \over {{F_{{\rm{min}}}}}}} \right)^{1/4}} - 1,$$
where
${\rm{\Delta }}T = {T_{{\rm{max}}}} - {T_{{\rm{min}}}}$
.
The calculation of planetary surface temperature variation is based here on a direct radiative equilibrium model, as realized in formula (1); it should be acknowledged that this approach is an idealized one, as it ignores that a planet’s atmosphere, oceans, and cloud cover introduce thermal inertia. During each stellar pulsation cycle, these factors would moderate the swing of planetary surface temperature and introduce a time lag between variations of stellar flux and the surface temperature. Taking these factors into account represents an interesting but a complicated (requiring massive modeling with many parameters being varied) challenge, which is beyond the scope of our study.
Let
${\rm{\Delta }}m = 0.6$
, which is an average typical for the RR Lyr variables (see lightcurves in Udalski, Reference Udalski2022) and
${T_{{\rm{min}}}} \approx 320$
K (i.e., for a planet in the HZ). Then, one has
${F_{{\rm{max}}}}/{F_{{\rm{min}}}} \approx 1.74$
; therefore,
${\rm{\Delta }}T/{T_{{\rm{min}}}} \approx 0.15$
, and
${\rm{\Delta }}T \approx 50$
K. Such
${\rm{\Delta }}T$
allows for maintaining the authentic PCR; see descriptions of PCR in Mullis et al. (Reference Mullis, Faloona, Scharf, Saiki, Horn and Erlich1986); Lathe (Reference Lathe2004, Reference Lathe2006).
In Figure 1, a scheme of the typical light curve (one cycle time profile) of the RR Lyr prototype variable is shown, along with the corresponding time profile of the radiation flux variation at the surface of a planet in the star’s HZ, and the corresponding time profile of the effective temperature variation on the planetary surface. The latter is calculated by means of Eq. (1).
Figure 1. Left panel: a scheme of the typical lightcurve (one cycle time profile) of the RR Lyr prototype variable.
$V$
is stellar
$V$
magnitude, for its amplitude, the typical value
${\rm{\Delta }}V = 0.6$
is adopted. Time
$t$
is in units of the pulse cycle duration; the latter is set to unity. Middle panel: the corresponding time profile of the radiation flux (in relative units) variation at the planetary surface. Right panel: the corresponding time profile of the effective temperature variation at the planetary surface.
Let us show that, in providing massive outcomes of “biomaterial,” the prolonged temperature cycle duration (in comparison with the standard PCR cycle) is not an obstacle for the BCR performance: the prolonged cycle duration can be compensated by enormous numbers of cycles.
The number of molecules produced in
$n$
cycles of the BCR is given by
$$N\left( {\alpha, n} \right) = {(1 + \alpha )^n},$$
where
$\alpha $
is the reproduction factor, characterizing the reaction efficiency; it can take values from zero to one, corresponding to null and 100% efficiencies, respectively. To calculate
$N\left( {\alpha, n} \right),$
it is convenient to employ an approximation: at
$\alpha \ll 1$
and
$n \gg 1$
,
$$N\left( {\alpha, n} \right) \approx e \cdot {(1 + \alpha )^{n - {1 \over \alpha }}},$$
where
$e = 2.718 \ldots $
.
Using formula (3), it is straightforward to find that, notwithstanding the generic decay, any smallness of
$\alpha $
can be compensated and overcome by setting a large enough value of
$n$
. For example, if
$\alpha = {10^{ - 5}}$
and
$n = {10^6}$
, then, one has
$N\left( {\alpha, n} \right) \approx 22000$
; if
$\alpha = {10^{ - 4}}$
and
$n = {10^6}$
, then
$N\left( {\alpha, n} \right) \approx 2.7 \cdot {10^{43}}$
. If
$\alpha = {10^{ - 6}}$
and
$n = {10^8}$
, then, again,
$N\left( {\alpha, n} \right) \approx 2.7 \cdot {10^{43}}$
; but if
$\alpha $
is a little bit greater,
$\alpha = {10^{ - 5.8}}$
, then, at the same
$n = {10^8}$
, one has
$N\left( {\alpha, n} \right) \approx 6.8 \cdot {10^{68}}$
. We see that a very small variation of
$\alpha $
at a fixed
$n$
may result in colossal changes in the final number of produced biomolecules.
As shown in Shevchenko (Reference Shevchenko2020), the BCR-induced potential inflation of biomass from one biopolymer molecule mass to a World Ocean mass, if
$\alpha = 1$
and there are enough precursors, is completed in
$\sim $
100 stellar eclipse cycles (for W UMa variables) or
$\sim $
100 pulsation cycles (for SX Phe variables); that is, instantaneously in comparison with the stellar evolution timescales. For any RR Lyr star, 100 pulses would take
$\sim $
1000 hr, that is, about one or two months, which is negligible in comparison with the time needed for the star to traverse the instability strip,
${10^7}$
–
${10^8}$
yr typically, see Catelan and Smith (Reference Catelan and Smith2015).
The same biomass can be obtained with a much smaller reproduction factor, but on longer time intervals. The number
$n$
of pulsation cycles performed by any RR Lyr variable, while it traverses the instability strip, is indeed huge. Since the strip is passed in about
${10^7}$
–
${10^8}$
years, the number
$n\sim \!{10^{10}}$
–
${10^{11}}$
. Let us set
$n = {10^{10}}$
. Using formula (3), we find that the reproduction factor
$\alpha $
as small as
$\sim \!{10^{ - 8}}$
is enough to produce the mentioned mass (that of
${10^{37}}$
molecules). This demonstrates that the prolonged duration of the BCR cycle is indeed not an obstacle for the BCR operability. Moreover, the longer BCR period would promote pre-Darwinian selection for more stable biopolymer products and allow for copying longer molecules.
Concluding, we find that in the hypothetical planetary worlds of RR Lyr stars, the amplitude/frequency conditions for the BCR are expected to be indeed provided. What is more, the lightcurves of the RR Lyr variables possess suitable (for maintaining the BCR) saw-like time profiles (Figure 1) of the cycles, with sharp jumps and slow decays.
Photospin resonances
Any cycled prebiotic reaction should be maintained continuous and uniform in time, which is provided if the stellar flux periodic variation is synchronized with the planetary rotation. Any tidally-locked planet of a pulsating star automatically resides in such a suitable synchronism, because one and the same side of the planet is constantly exposed to the variable stellar flux.
Recent studies in tidal interactions of giant planets with pulsating host stars (Bryan et al., Reference Bryan, de Wit, Sun, de Beurs and Townsend2024) demonstrate that time sequences of various resonant locks of planetary orbital and rotational periods with stellar pulsations may naturally arise.
Another kind of suitable synchronism, which we call a “photospin resonance 1/1,” is also possible. Indeed, among the RR Lyr variables, the typical pulsation periods range from 0.2 to 0.9 d (Catelan and Smith, Reference Catelan and Smith2015; Neilson et al., Reference Neilson, Percy and Smith2016); therefore, if a planet of an RR Lyr star has, like Jupiter or Saturn, the rotation period
$ \approx $
10 hr, it may well occur to reside in, or close to, this resonance.
This opportunity is illustrated in Figure 2, where we present the Bailey (period–amplitude) diagram for the RR Lyr stars in the globular cluster M15 (NGC 7078). The known rotation periods of the Solar system giant planets, along with the variability amplitude equal to 0.6 (characteristic for BCR), are superimposed as vertical lines and a horizontal one, respectively. One may see that the two major dot clusters (corresponding to RRc, on the left of the diagram, and to RRab, on the right) concentrate to the crossings of the vertical and horizontal lines. Therefore, any RR Lyr star planets, rotating as outer ones in the Solar system, may indeed reside near the 1/1 photospin resonance.
Figure 2. The Bailey diagram (period–amplitude graph) for the observed RR Lyr stars in the globular cluster M15. Green dots: the RR Lyr stars, according to the data in Table A1 in Hoffman et al. (Reference Hoffman, Murakami, Zheng, Stahl and Filippenko2021). Black vertical solid line: Jupiter’s rotation period. Orange vertical dashed line: Saturn’s rotation period. Royal color vertical dotted line: Uranus’s rotation period. Blue vertical dash-dotted line: Neptune’s rotation period. Gray short-dashed line: the mean rotation period of the main-belt asteroids. Cyan horizontal solid line: amplitude
, as adopted in Figure 1.
We suppose that eventual giant planets around RR Lyr stars rotate with angular velocities similar to those of the Solar system giant planets, just because inside our system they are rather similar. We concentrate here on giant planets for several reasons. First, in the Solar system, they have much (about two times) shorter rotation periods than Earth or Mars, and thus they are more likely to reside in the 1/1 photospin resonance, as we have just discussed in connection with Figure 2. Second, giant planets seem to be automatically placed into their host star’s HZ when the star is within the instability strip, as discussed below in connection with Figure 3. Third, any eventual inner (presumably rocky) planets can be destroyed during the red-giant phase of the host star evolution, while any existing giant planets may lose their vast atmospheres during this phase, only rocky nuclei being left whose surfaces may harbor subsequent reactions.
Figure 3. Habitability zones: that of the modern Sun (left panel) and those of the prototype star RR Lyr at its minimum (middle panel) and its maximum (right panel) luminosity. Dark green: basic HZs; light green: extended HZs. Note that the scale in the first panel is ten times different from that adopted in the second and third panels. The graphs were built using the TWAM package; see text.
Note that, in eventual planetesimal belts of RR Lyr stars, photospin synchronisms may also emerge. Indeed, in our Solar system, “asteroids larger than a few tens of kilometres in diameter spin with a mean rotation period around 10 hours, with some minor variation with size” (Harris and Pravec, Reference Harris, Pravec, Lazzaro, Ferraz-Mello and Fernández2006). Therefore, while the variability period of any host star of RR Lyr kind slowly changes, the planetesimals in its planetesimal belts may eventually enter and traverse the photospin resonance 1/1.
On the timescale of ten or hundred megayears (for an RR Lyr variable to stay within the instability strip), the number of pulses of the host star is indeed huge (
$\sim \!{10^{10}}$
–
${10^{11}}$
). On the other hand, the pulsation period of any RR Lyr star is subject, as mentioned above, to slow evolution, with a rate
$\omega$
of up to
$\sim $
0.1 d per million years. Therefore, it is quite probable that the system may stay for some time close to the 1/1 photospin resonance. However, any mechanism of capture in such a resonance is currently not known to exist, and thus, the close-to-resonance state can be only transient. Whether this transient state, in the absence of resonance lock, could be appropriate for maintaining the BCR, requires a detailed modelling, taking into account the changing of conditions on the surface of the planet while it rotates; such modelling can be a subject of a separate study, but is beyond the scope of the present one.
A more efficient way appears if we consider a tilted planet (i.e., a planet with rotation axis non-orthogonal to its orbital plane; recall that, in the Solar system, four planets are moderately tilted, with tilts from 23 to 28 arc degrees, and one (Uranus) is extremely tilted, with tilt about 98 arc degrees). Consider, for example, a Uranus-kind tilted planet with orbital period
$\sim \!100$
yr. Then, its “Northern” and “Southern” polar regions would be alternately exposed (for
$\sim \!50$
yr time intervals) for continuous stellar illumination. Thus, each exposure would include
$\sim \!{10^5}$
stellar pulses. Of course, this would require a rather high value of the reproduction factor
$\alpha $
for the BCR to work out, but, while the star is within the instability strip (say, for
$\sim \!{10^7}$
yr), there would be
$\sim \!{10^5}$
such continuous exposures. Therefore, the polar regions of planets seem to be more favorable, than equatorial ones, for BCR to work out.
Another promising mechanism is the following. Any planet, if given enough time for tidal evolution, can be tidally despun to the 0/1 photospin resonance state, when one side of the planet is permanently exposed to the stellar flux, while the other one is “dark.” (The “0/1” designation reflects the fact that the planet’s synodic angular frequency of rotation in this state is zero while the frequency of stellar pulses can be normalized to unity). The tidal lock in the 0/1 photospin resonance would be ideal for the BCR to work out, as the chain reaction on the illuminated surface is not broken by the planet’s rotation, and the BCR benefits from the total sequence of
$\sim \!{10^{10}}$
–
${10^{11}}$
stellar pulsations. The problem is, however, whether the tidal locking is achievable in available conditions and on available timescales (though large). This can be considered when data on properties of planets (now hypothetical) of RR Lyr stars are obtained in observations.
Concluding, among the considered possibilities, the tidal lock, implying the 0/1 photospin resonance, is most promising, as it may provide the unbroken BCR largest duration, potentially up to
$\sim \!{10^{10}}$
–
${10^{11}}$
stellar pulses.
Zones of BCR operability
It is instructive to define a “fertility zone” (FZ) of a given planetary system, as the zone in which the BCR is theoretically able to operate on planetary surfaces. The FZ represents the intersection (overlap) of the habitable zones at the star’s minimum and maximum luminosity. For a given fixed host star luminosity, the FZ is necessarily much thinner in radius than the HZ, to allow for a large (
$\sim $
50 K) amplitude of temperature variation within the boundaries of the liquid water interval (100 K). Therefore, at any fixed time epoch, there is a much smaller chance for a planet to reside in the FZ than in the broader HZ.
However, one should account for that (1) the ability to drive BCRs is needed on a relatively short time interval, (2) the host star’s luminosity (of the RR Lyr class) is subject to a slow secular change (Neilson et al., Reference Neilson, Percy and Smith2016), and therefore, the FZ slowly shifts in radius, thus radially scanning the system, including its eventual planets and planetesimal belts, and (3) the radial sizes of planetary orbits and planetesimal belts themselves also slowly change (advancing outward), because the pulsating host star suffers a permanent mass loss.
While in the Schwarzschild gap, the star slowly loses about a half of its initial mass (Catelan and Smith, Reference Catelan and Smith2015). From basic formulas for outcomes of adiabatic orbital evolution (see, e.g., Minton and Malhotra, Reference Minton and Malhotra2007), it follows that such a mass loss would result in doubling the sizes of planetary orbits. On the other hand, the luminosity of an RR Lyr star, when it is within the instability strip, is
$\sim $
1000 times greater than that of the Sun (Catelan and Smith, Reference Catelan and Smith2015), and its HZ is, therefore, somewhat enlarged with respect to the Solar one; see Figure 3. These two factors (the enlarged luminosity and the slowly expanding planetary orbits) favor placing the system’s eventual giant planets into the HZ and FZ. Indeed, if an RR Lyr variable, before it enters the instability strip, has its outer planetary subsystem architecture similar to that of the Solar system (whereas the inner subsystem could be lost during the red-giant stage), then, while the star is evolving within the instability strip, the giant planets would automatically enter the HZ. Indeed, a planet with initial orbital semimajor axis
$a\sim \!5$
AU (“Jupiter”) would obtain
$a\sim \!10$
AU and may, therefore, enter the host star’s HZ and FZ; see Figure 3.
What is more, while the host star evolves within the instability strip, a vast radial space of its planetary system would be slowly “scanned” by the FZ (whose boundaries stay relatively constant while the system’s planets radially migrate due to the host star’s continuous mass loss). Since the planetary system enlarges by a factor of two, it is able to radially move across the whole FZ while the host star evolves within the instability strip (what may take
$\sim \!{10^7}$
–
${10^8}$
years). Since the complete radial outward shift of the FZ is similar (in order of magnitude) to its initial size (which is similar to that of HZ, also in order of magnitude, being only somewhat smaller), this time is in order of magnitude comparable to the total time of the star residence within the Schwarzschild gap.
Therefore, the expected typical residence time for a planet within the FZ is about
${10^7}$
–
${10^8}$
years; this corresponds to
$\sim \!{10^{10}}$
–
${10^{11}}$
stellar pulses. As discussed above, if the chain reaction reproduction factor
$\alpha $
is high enough (say, greater than
${10^{ - 6}}$
), then this time would be sufficient for the BCR to work out. More definite estimates can be obtained by a detailed modelling, when bounds on the reproduction factor become established.
In Figure 3, potential HZs are shown for the modern Sun (left panel) and for the prototype star RR Lyr at its minimum (middle panel) and its maximum (right panel) luminosity. The graphs have been constructed using the interactive package TWAM (http://astro.twam.info/hz), developed in Müller and Haghighipour (Reference Müller and Haghighipour2014). The package realizes a number of theoretical HZ models (Kasting et al., Reference Kasting, Whitmire and Reynolds1993; Selsis et al., Reference Selsis, Kasting, Levrard, Paillet, Ribas and Delfosse2007; Kopparapu et al., Reference Kopparapu, Ramirez, Kasting, Eymet, Robinson, Mahadevan, Terrien, Domagal-Goldman, Meadows and Deshpande2013a, Reference Kopparapu, Ramirez, Kasting, Eymet, Robinson, Mahadevan, Terrien, Domagal-Goldman, Meadows and Deshpande2013b, Reference Kopparapu, Ramirez, SchottelKotte, Kasting, Domagal-Goldman and Eymet2014). In the given case, they all provide similar results; we have chosen that of Kopparapu et al. (Reference Kopparapu, Ramirez, SchottelKotte, Kasting, Domagal-Goldman and Eymet2014). For the prototype star RR Lyr, the minimum and maximum luminosity values are set to 50 and 100 (in solar units); the effective temperature is set to 6100 K (for the Sun it is 5780 K). The union of the RR Lyr’s HZs at the star’s minimum (middle panel) and maximum (right panel) luminosity represents the broadened united HZ of the pulsating star (because the water may maintain its liquid phase across the total HZ, although periodically at its borders). The intersection of these two HZs represents the star’s FZ because BCRs can be maintained solely when the water is in liquid state continuously in time. We see that the FZ occupies a significant part of the total HZ; and the chance for a planet to occur inside the FZ is only somewhat (about two times) less than the chance to occur inside the HZ.
Concomitant factors
Concomitant advantages of the stellar pulsations for the prebiotic processes can be multiple. For instance, cycling of evaporation on planetary surfaces may provide chiral purity in solutions with a slight excess of one isomer (Breslow and Levine, Reference Breslow and Levine2006; Breslow, Reference Breslow2011). This process can be maintained naturally on surfaces subject to a periodically variable radiation flux. Several concomitant advantages, specific for RR Lyr variables, are as follows.
UV insolation
The UV irradiation is essential for a number of prebiotic chemistries; see a review in Ranjan et al. (Reference Ranjan, Kufner, Lozano, Todd, Haseki and Sasselov2022). Moreover, the resistance to UV radiation may constitute a selection factor favoring the A, G, T, and C nucleic bases (Mulkidjanian and Galperin, Reference Mulkidjanian and Galperin2007): without such a factor, genetic information can be stored using a much broader variety of nucleic bases.
The spectra of stars of early spectral classes, which the RR Lyr stars belong to, are UV-biased. Moreover, luminosity variations of RR Lyr stars are accompanied by changes in their color: they are bluer at brightness maxima, and redder at brightness minima (Catelan and Smith, Reference Catelan and Smith2015, Ch. 6). It means that, during the BCR denaturation stage, the reaction product would be subject to an increased selective influence of UV radiation, and, during the BCR annealing/elongation stage, to a decreased one. Therefore, this specific spectral variability would tend to amplify the pre-Darwinian long-term selection toward a more stable product.
During the pulsation cycle, the effective temperature of the prototype RR Lyr star varies between
$ \approx $
6000 K and
$ \approx $
7000 K, at the star’s luminosity minimum and maximum, respectively, see table 1 in Fossati et al. (Reference Fossati, Kolenberg, Shulyak, Elmasli, Tsymbal, Barnes, Guggenberger and Kochukhov2014). It is easy to estimate (in the blackbody approximation for the star’s spectrum) that, due to this temperature variation, the star’s UV luminosity at the cycle maximum would additionally rise, though moderately: by
$ \approx $
3.1 times at 300 nm, and by
$ \approx $
5.5 times at 200 nm. However, the maximum temperatures (greater than 6500 K) are observed during a small part (about 20%) of the cycle; see the same table 1 in Fossati et al. (Reference Fossati, Kolenberg, Shulyak, Elmasli, Tsymbal, Barnes, Guggenberger and Kochukhov2014). Whether such UV modulations could be dangerous for the survival of biomolecules, requires a special study, taking into account the potential variety of planetary atmospheric properties.
Let us note that the periodic forcing can be particularly relevant at interfaces (mineral surfaces, water–air, water-ice), where material concentration and partial shielding may coincide with product selection. The periodic forcing, in principle, may either advance necessary reactions or, instead, enable degradation, for example, photocleavage. Of course, in any usable regime the cycle-averaged progress should exceed damage.
Synchronization in time and over areas
As already mentioned in the Introduction, it is important that the BCR processes are synchronized in time over vast planetary surface areas, thus precluding any escape of the reaction products. The periodic irradiation by the host star provides this kind of synchronization automatically.
Another favorable effect of a unique-source periodic driving may consist in the protocell cycle synchronization on irradiated surfaces. After the BCR works out, the variable radiation flux with suitable period and amplitude may drive primitive cell cycles. It is important that the protocell cycle would be synchronized in time over vast surface areas (i.e., not only locally), due to the uniformity of the radiation flux variations, and over large time intervals, while the host star evolves within the instability strip. The processes are thus synchronized in time both over vast areas and on large time intervals.
The cell cycle of bacteria is up to half a day (Crick and Orgel, Reference Crick and Orgel1973); the predicted minimal doubling time for cyanobacteria peaks at
$ \approx $
9–10 hours; see Figure 2C in Weissman et al. (Reference Weissman, Hou and Fuhrman2021). We see that, judging from the Bailey diagram in Figure 2, the primitive cyanobacteria may acquire their primordial (minimal) timescales of doubling in synchronism with stellar flux variations. In the globular cluster M15, the RRc stars have just appropriate periods of 9–10 hr, as illustrated graphically in Figure 2.
Metallicity
The RR Lyr variables are generally interpreted as metal-poor stars, belonging to the old Galactic Population II, with metallicities generally not exceeding
$ \approx - 1$
(Catelan and Smith, Reference Catelan and Smith2015; Smith, Reference Smith2004). On the other hand, metal-poor stars are generally not expected to possess planets, at least giant planets, see Mortier et al. (Reference Mortier, Santos, Sozzetti, Mayor, Latham, Bonfils and Udry2012); Boley et al. (Reference Boley, Wang, Zinn, Collins, Collins, Gan and Li2021) and references therein. In particular, [Fe/H]
$ \approx - 0.6$
is supposed to be the lower limit for planetesimal formation abundant enough to produce a few Earth masses of planet-forming material (Andama et al., Reference Andama, Mah and Bitsch2024). The theme on formation of second-generation planets, especially around a star that is currently relatively metal-poor, is a topic of ongoing research.
However, planets whose host stars are extremely metal-poor, with metallicities as low as from −0.4 to −0.6, are being discovered and observed; see Hellier et al. (Reference Hellier, Anderson, Cameron, Delrez, Gillon, Jehin, Lendl, Maxted, Pepe, Pollacco, Queloz, Ségransan, Smalley, Smith, Southworth, Triaud, Udry and West2014); Polanski et al. (Reference Polanski, Crossfield, Burt, Nowak, López-Morales, Mortier, Poretti, Behmard, Benneke, Blunt, Bonomo, Butler, Chontos, Cosentino, Crane, Dumusque, Fulton, Ghedina, Gorjian, Grunblatt, Harutyunyan, Howard, Isaacson, Kosiarek, Latham, Luque, Fiorenzano, Mayor, Mills, Molinari, Nagel, Pallé, Petigura, Shectman, Sozzetti, Teske, Wang and Weiss2021); Brinkman et al. (Reference Brinkman, Weiss, Dai, Huber, Kite, Valencia, Bean, Beard, Behmard, Blunt, Brady, Fulton, Giacalone, Howard, Isaacson, Kasper, Lubin, MacDougall, Murphy, Plotnykov, Polanski, Rice, Seifahrt, Stefánsson and Stürmer2023); Dai et al. (Reference Dai, Schlaufman, Reggiani, Bouma, Howard, Chontos, Pidhorodetska, Zandt, Murphy, Rubenzahl, Polanski, Lubin, Beard, Giacalone, Holcomb, Batalha, Crossfield, Dressing, Fulton, Huber, Isaacson, Kane, Petigura, Robertson, Weiss, Belinski, Boyle, Burke, Castro-Gonzalez, Ciardi, Daylan, Fukui, Gill, Guerrero, Hellier, Howell, Lillo-Box, Murgas, Narita, Pallé, Rodriguez, Savel, Shporer, Stassun, Striegel, Caldwell, Jenkins, Ricker, Seager, Vanderspek and Winn2023). On the other hand, observational evidence grows that a significant fraction of RR Lyr stars may be in fact metal-rich, with metallicities as high as [Fe/H]
$ \approx 0.2$
; see Iorio and Belokurov (Reference Iorio and Belokurov2021); Li et al. (Reference Li, Huang, Liu, Beers and Zhang2023); Bobrick et al. (Reference Bobrick, Iorio, Belokurov, Vos, Vuckovic and Giacobbo2024) and references therein. This fraction may be as high as
$10{\rm{\% }}$
of all the Galactic RR Lyr stars (Bobrick et al., Reference Bobrick, Iorio, Belokurov, Vos, Vuckovic and Giacobbo2024).
Therefore, many RR Lyr stars may have metallicities that allow the possession of planets. Besides, one should outline that the problem of relationship between the stellar metallicity and the planet presence is far from being resolved up to now.
Globular clusters
Many RR Lyr stars reside in globular clusters. On historical reasons, they are even sometimes called “cluster variables” (Catelan and Smith, Reference Catelan and Smith2015); nevertheless there is also a lot of them in the Galactic field. As we have argued above, RR Lyr stars may drive prominent biochemical processes on their hypothetical planets. On the other hand, a hypothesis was recently proposed and elaborated that globular clusters may “provide ideal environments for advanced civilizations that can survive over long times” (Di Stefano and Ray, Reference Di Stefano and Ray2016). There are several lines of evidence supporting this viewpoint; in particular, short interstellar distances favor the expansion and long-term survivability of globular-cluster civilizations; for details and discussion see Di Stefano and Ray (Reference Di Stefano and Ray2016); Kane and Deveny (Reference Kane and Deveny2018); Smith (Reference Smith2021); Grimaldi et al. (Reference Grimaldi, Lingam and Balbi2021). The study by Di Stefano and Ray (Reference Di Stefano and Ray2016) shows that locations inside globular clusters, where the long-term stability and habitability of planetary systems can be provided, do exist.
A pulsating Sun
Not only evolved stars, such as RR Lyr variables, are observed to pulsate with amplitudes and periods suitable for BCR. Appropriate conditions may provide some classes of young stars as well. A Gamma Doradus variable HR 8799 with its planetary system may provide a possible paradigm for the latter ones. According to Faramaz et al. (Reference Faramaz, Marino, Booth, Matrà, Mamajek, Bryden, Stapelfeldt, Casassus, Cuadra, Hales and Zurlo2021), the architecture of the planetary system of this young star is similar to that of the Solar system: the orbits of its four giant planets surround a warm dust belt similar to the asteroid belt in our system, and, from outside, they are surrounded by a cold belt similar to the Kuiper belt. The inner part of the system (inside the “asteroid belt”) contains a zone of potential habitability. The “asteroid belt” is radially limited by mean-motion resonances 4/1 (inner border) and 2/1 (outer border) with a giant planet; similar to the Solar system architecture. As stated in Faramaz et al. (Reference Faramaz, Marino, Booth, Matrà, Mamajek, Bryden, Stapelfeldt, Casassus, Cuadra, Hales and Zurlo2021), “simply put, the system of HR 8799 is a younger, broader, and more massive version of the solar system.” Although the amplitude of pulsations of HR 8799 is rather moderate as observed at present, it can vary as the star evolves.
A hypothesis was advanced in Willson et al. (Reference Willson, Bowen and Struck-Marcell1987) that young stars in the instability strip may evolve down the main sequence, loosing mass at a rate greater than
${10^{ - 9}}$
solar units per year, and the early Sun could evolve in such a manner. The mass loss occurs at the intersection zone of the “Cepheid instability strip” with the main sequence, at spectral types from early A to mid F; here, among others, δ Scuti pulsating stars reside. The mass loss is driven by pulsation and rapid rotation. According to Willson et al. (Reference Willson, Bowen and Struck-Marcell1987), the early Sun could have been such a pulsating star, which had arrived on the main sequence with mass up to twice its present value and lost its half over time about a gigayear. In particular, this scenario is capable to resolve the notorious “faint young Sun paradox,” as well as several less outstanding astrophysical problems, but it is not yet fully supported by relevant observational results, see a review and details in Minton and Malhotra (Reference Minton and Malhotra2007).
Conclusions
In this article, we have explored how large-amplitude variability of pulsating stars may affect the prebiotic evolution on their eventual planets. As we have seen, the RR Lyr type variables are particularly relevant, because their lightcurves possess amplitudes, periods, and time profiles that are potentially most suitable to drive prebiotic reactions. Indeed, the lightcurves of the RR Lyr variables possess suitable (for maintaining the BCR) saw-like time profiles (Figure 1) of the cycle, with a sharp initial jump followed by a slow decay.
In this context, we have introduced the notion of photospin resonance; the 0/1 photospin resonance (corresponding to the planetary tidal lock) turns out to be the most promising one within the discussed BCR scenario, as it may provide the unbroken BCR of largest duration (in reaction cycles). In addition to the well-known notion of the habitability zone (HZ), we have introduced the “fertility zone” (FZ) notion, to define the region (in a planetary system of a variable star) where BCRs are theoretically able to operate on planetary surfaces.
Concomitant factors of the BCR scenario can be multiple. Such factors include the role of UV insolation, synchronization of BCRs in time and over areas, metallicities of host stars, and their residence in Globular clusters.
How the proposed BCR scenario can be verified? This can be done directly via observations of RR Lyr stars on the subject of discovering planets in their HZs and FZs, and by seeking for biosignatures of such planets, if the latter are discovered.
Finally, it should be noted that, apart from RR Lyr variables, some other classes of pulsating stars (SX Phoenicis, δ Scuti, γ Doradus variables) may also provide advantages for BCRs. The BCR scheme, driven by stellar pulsations, may as well operate in planetary systems of young stars, such as HR 8799 and, speculatively, young Sun.
Acknowledgments
The author is grateful to the referees for useful remarks and comments.
Funding statement
This work was supported by ongoing institutional funding. No additional grants to carry out or direct this particular research were obtained.
Competing interests
No competing financial interests exist.
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