Screening Earth Analog Exoplanets on the Basis of a Predicted Nitrogen over Phosphorus Ratio

Since the first discovery in 1995, data for over 5300 exoplanets have been documented in the NASA archive, revealing a vast diversity. Identifying life-enabling analogs of the Earth among this rapidly expanding catalog is of major interest. The stability of liquid water at the planetary surface defining the concept of the habitable zone (HZ) around the host star, may be necessary for the emergence of life as we know it but not sufficient. The practically constant atomic ratio nitrogen:phosphorous = 16:1 in oceanic surface layers of our planet Earth was discovered by Redfield in 1934. It corresponds to phytoplanktonic biomass in suspension and appears optimal to fertilize phytoplankton development and therefore the food pyramid of marine life. Loladze and Elser have shown that it corresponds to a homeostatic protein:RNA ratio and is therefore “rooted in the stoichiometry of the foundational structures of life.” I show that according to the recent theory of the chemical differentiation of planets, this optimal ratio is also an intrinsic chemical property of our planet Earth uniquely determined in the solar system by its average orbital radius. On that basis, I propose a criterion of fertility within the HZ of a stellar system, which when applied to screen the public database allows us to sort out an extended list of up to 74 Earth analogs. The latter and its future extensions could provide priority targets for focused detection techniques.


Introduction
The practically constant ratio of nitrates to phosphates in oceanic surface layers of our planet Earth was discovered by Redfield (1934Redfield ( , 1936Redfield ( , 1958)).It corresponds to phytoplanktonic biomass in suspension, which is also characterized by an average atomic ratio close to N:P = 16:1.This ratio appears optimal to fertilize phytoplankton development and therefore the food pyramid of marine life.Whether this ratio of nutrients was the cause or is a consequence, of the emergence of photosynthetic organisms has been debated (Falkowski & Davis 2004;Bertrand & Legendre 2021) and remains an open question.It has been shown that it corresponds to a homeostatic protein:RNA ratio and is therefore "rooted in the stoichiometry of the foundational structures of life" (Loladze & Elser 2011).I propose here that this optimal ratio is also an intrinsic chemical property of our planet Earth, and that a criterion based on a critical distance to the host star should allow for a refined screening of exoplanets databases for Earth analogs.

Theoretical Background
Recently we proposed a novel scenario for the early chemical differentiation of gaseous protoplanetary disks and therefore later condensed planets (Toulhoat & Zgonnik 2022): at a radial distance d from the energetic photons emitting protostar, ionized atoms are stabilized in orbit by Lorenz forces.The local mass fraction X(A, d) of ionized element A at distance d is shown to obey the following law:

PS
where X SS (A) is the initial mass fraction in the nebula fixed by nucleosynthesis, IP(A) is the first ionization potential of element A, k B is the Boltzmann constant, R PS is the protostar radius, and T G (d) is the local temperature of the partially ionized gas.Terms are referred to as "differentiation factors."In what follows, for convenience, we refer to this scenario as the "photophysical model." For our solar system, the average compositions of planetary surface rocks sampled thanks to past space missions to Mercury, Venus, the Moon, and Mars, compared to those of the Earth and meteorites show that: for d < 1 au: while for d 1 au (that is beyond the Earth): It follows that our home planet is in orbit at a very special position: a cusp point on the radial distribution profile of any element in the solar system.Considering further radial distribution profiles of atomic ratio ρ A/B of element A over B, one gets: for d < 1 au: and for d 1 au: and δ AB IP = IP(B) − IP(A), where M A is the atomic weight of element A. Terms X SS (A)/X SS (Si) are the well-known relative abundances in the solar photosphere referred to the element Silicium.The protosolar radius was inferred from the differentiation factors of noble gases on Earth as R PS ≅ 1.5R S , where R S is the present-time solar radius (Toulhoat & Zgonnik 2022).
As shown in the Appendix, Section A.1, the cusp point at d = 1 au is a minimum for δ AB IP < 0 and a maximum for δ AB IP > 0.

Radial Distribution Profiles of Atomic Ratios across the Solar System
Table 1 displays numerical values allowing us to figure out the predicted profiles for the main elements involved in biomasses, C, H, O, N, and P.
Table 2 displays ρ A/B (1) for selected combinations and the main features of the corresponding radial profiles.The N/P profile across the solar system is shown in Figure 1.
The most striking result is ρ N/P (1) = 16.01,identical within the error margin to the Redfield ratio: this means that according to our photophysical model, our planet Earth would have offered from the beginning precisely the ratio of essential nutrients necessary for the emergence and blooming of marine phytoplankton.This step in the evolution of life on Earth meant the onset of O 2 -producing photosynthesis and further a basis for the food pyramid of marine life up to the present time.
Moreover, as shown in Figure 1, ρ N/P (d) increases rapidly for orbits closer to the Sun, or farther than the Earth, so that for Mercury, Venus, and Mars for instance, the availability of Phosphorus decreases with respect to Nitrogen.Too hot or too cold surface temperatures on these planets, and the apparent absence of liquid water, may be sufficient to explain their sterility.However, the imbalanced ρ N/P (d) could be another crucial detrimental factor to life-enhanced habitability as opposed to what prevailed for the Earth (i.e., an appropriate greenhouse effect and a protecting ozone layer).
Notice that ρ A/B (d) predicted for the protoplanetary disk holds in first approximation as averages for the bulk planets in orbit at distance d.For the Earth, chemical differentiation should generally further occur between the surface and bulk of planets due to various matter transfer processes.As underlined in our previous report (Toulhoat & Zgonnik 2022), measurements of the average elementary composition of the planet Earth are available only for superficial samples.The photophysical model predicts bulk values, and their comparison with surface values yields elemental partition coefficients between bulk and surface.The latter appear correlated with thermochemical properties, indicative of a progressive radial chemical differentiation of the Earth in response to an outward reactive flow of hydrogen driving the most electronegative elements (O and the halogens) to end up at the surface as acidic water, further neutralized into the oceans.In turn, the gradient of oxygen chemical potential thus established tends to drive the outward radial migration of elements according to their affinity for oxygen.The average atomic ratio N:P on the surface is 0.04, quite far from the predicted bulk N:P of 16, which coincides with the Redfield ratio.However, the latter is actually the molar ratio nitrates:phosphates (NO 3 − :PO 4

3-
), the most oxidized species for nitrogen and phosphorus, and "hard anions" highly soluble in liquid water.Oceans are connected with the Earth's interior mostly through ridges at the junctions of oceanic floor tectonic plates, where deep vents deliver permanent hydrothermal fluids (HF).These fluids have been sampled and subjected to chemical analysis after numerous scientific expeditions in the past 40 years since the discovery of these deep-seafloor vents.The open-access MARHYS 2.0 database collects this information systematically for circa 3400 samples so far (Diehl & Bach 2020).Selecting HF samples, NO 3 − and PO 4 3-contents at the μmol kg −1 level yield an average N:P ratio of 16.6, in conformity with the Redfield ratio, which is otherwise very much documented in oceans seawater.CO 2 and N 2 contents, which at the mmol kg −1 level include the vast majority of C and N atoms in HF samples are available simultaneously for 113 samples.These data yield an average C: N ratio of 29.7, consistent with the model prediction of 30.1.Then, as the majority of O atoms the HF samples outside water itself are reported as conveyed by dissolved gases CO 2 , SO 2 , and the SO 4 2-anion, at on average 50.4,38.2, and 13.1 mmol kg −1 levels, it is easy to compute an average C:O molar ratio of 2.3 also very consistent with the model prediction of 2.3.For the contents in other elements or species available in the database, such agreements between predicted and experimental ratios are generally not obtained, except for combinations of lanthanides, well documented simultaneously for up to 260 samples: average Cerium:X ratios, where X is another lanthanide (at the pmol kg −1 levels) appear well correlated with model predictions (R 2 = 0.85).Although the precise speciation of these elements is not reported, it is expected that their oxidation number in hydrothermal fluids is +3, generally the most stable for rare earths.As for noble gases, Only He, Ar, and Ne contents are reported simultaneously for at most circa 60 samples.Average molar ratios of these gases dissolved in hydrothermal fluids do not match the * higher or lower than 16 would correspond, respectively, to P-limited or N-limited growth conditions (Loladze & Elser 2011).Besides, recent experimental measurements of variability in the cellular stoichiometry of diverse classes of marine eukaryotic phytoplankton under sufficient nutrient conditions reveal a distribution of N:P with an average N:P = 16.2 and a standard deviation of 5 in living cells (Garcia et al. 2018).Therefore, as a conservative ansatz, I propose that values of r ( ) d d

N P Star
* in the interval [11,22] can be expected favorable to the emergence on an exoplanet of life as we know it.
In this section, I provide first a criterion to determine d * for any star system, allowing us to screen confirmed exoplanets with respect to their distance to the cusp point for chemical differentiation.Next, for the subclass of stars liable to host fertile exoplanets, considering the reported distance of an exoplanet to its star d Star , the criterion r Î ( )  procedure provides a probability of occurrence of such Earth analogs and sorts out promising targets.
Generalizing the validity of Equation (2) to any star system, the critical distance d * is the solution of Equation (6): where R PS is the protostar radius, in the same unit of length as for d * , and ( ) T R G PS is the gas temperature at the protostar surface.Assuming by analogy with our solar system R PS ≅ 1.5R star , where R star is now the actual stellar radius, as documented in the exoplanet archive, we can estimate The database (NASA Exoplanet Archive 2023) contained 34111 rows at the downloading date, corresponding to 5322 distinct confirmed exoplanets.A subset of 2633 distinct exoplanets allowed to calculate d * in this way, including 1867 for which T Pl eq , the estimated equilibrium temperature of the exoplanet, was also reported.For calculating T Pl eq , it appears that research groups may use different Bond albedos A B .Adopting uniformly the classical formula: with A B = 0.3 and T star = T Stellar Surface , allows us to complete to 2633 the subset of available exoplanets equilibrium temperatures, while T Pl_rec eq remains linearly correlated to the reported T Pl eq (R 2 = 0.988).then, another linear correlation (R 2 = 0.947) appears in Figure 2 between ( ) d d Log 10 Star * and T Log 10 Pl_rec eq .For the solar system, the same plot shows a perfect regression line (R 2 = 1).Actually, this broad dependence is expected as the critical distance d * depends also on R star and T star according to Equations (6) and (7).As both d * and T Pl_rec eq are mathematical constructions on the basis of the same data set, the following exact expression holds: B , 9 3 .This is exactly verified, for exoplanets as well as planets of our solar system.Therefore, the spread as well as outliers in Figure 2  is compatible with the presence of water in the liquid state at the surface of the exoplanet, provided it is augmented by an atmosphere providing a suitable greenhouse effect.
With ò HZ a small real number, in Equation (11) provides thus a first screen of exoplanets within their host star HZ:  A1 and A2 in the Appendix list these 49 exoplanets proposed as Earth analogs.Interestingly, the average value of predicted N:P ratios is 17.2 and the rms deviation is 2.9, very close to the average Redfield ratio on Earth: this is obviously a consequence of the random character of the first selection * .Besides this sample, the proportion of fertile habitable exoplanets in the database according to these criteria is 2.4%, as obtained through the numerical approach detailed in the Appendix, in Section A.3.This fraction produces already a significant number if extrapolated to the estimate of stars number in the Milky Way only.Improved detection techniques might increase this fraction in the future.This estimate should be also compared to 7.6%, the fraction of confirmed exoplanets within the HZ with ò HZ = 0.2 obtained by integration of the fitted log-normal distribution represented in Figure A1: it may be concluded that on average merely around 30% of exoplanets within the habitable zone as defined here are potentially fertile.
In Figures A2, A3, and A4 of the Appendix are plotted, respectively, for the 49 exoplanets listed in Tables A1 and A2 A2 and A3 show that the most fertile habitable planets identified reside between 10 and 100 pc from the Earth, so within the Orion arm of our Galaxy.A3.The average uncertainties on the screening variables were almost unchanged following this extension of ranges.The numerical estimate of the proportion of fertile habitable exoplanets now increases to 4.5%.However, as this numerical estimation is based on the actual distributions of stellar host radii and surface temperatures reported in the exoplanets database, it should be updated in time, as more exoplanets will be confirmed and included in the database, and as detection techniques will improve.
As presented in Tables A4 and A5, most exoplanets found as potentially fertile within the HZ have best masses well above that of the Earth.Only 4 surface gravities over those of 49 exoplanets may be deduced when planet mass and radius are reported together, ranging from 0.41 to 417 Earth surface gravities (g).The corresponding densities range from 0.37 to 498 g.cm −3 (Earth average density 5.513 g.cm −3 ).It is clear that the knowledge of these properties would allow us to further refine the initial screen by ranges of equilibrium temperature followed by ranges of N:P ratio, in order to identify real analogs of our planet Earth.Unicellular life might in principle accommodate a very large range of surface gravity, but probably not evolve toward large multicellular organisms above some threshold.The existence of open gray circles, planets in the solar system; regression lines: gray broken line for exoplanets and gray line for the solar system (in inset: equations and squared coefficients of correlation, in gray and black, respectively).Crossing points of regression lines with horizontal axis are 217.3K for exoplanets and 264.0 K for the solar system (very close to Earth's equilibrium temperature according to Equation (8)).
liquid seawater should not be compatible with average planet densities below 1 g.cm −3 .In the subset of 4 habitable and potentially fertile exoplanets documented enough, only Kepler-1661 b (1.13 g and 1.62 g.cm −3 ) and Kepler-22 b (6.36 g and 14.72 g.cm −3 ) meet these complementary criteria.

Corrections for Host Star Metallicities
So far, I incorrectly assumed that protostellar nebulae have very similar chemical compositions although they are enriched in elements beyond H and He as the universe is aging.Stellar metallicity ratios [M/H], or mostly [Fe/H], are indeed reported in the NASA Exoplanets archive so that it is possible to examine the influence of corrections on the photophysical model predictions.The Hypatia catalog has compiled so far relative abundance data for nearly 10,000 stars within 1000 pc of the Sun, including a number of known exoplanet hosts (Hinkel et al. 2014).The elemental abundance data may be considered as usually expressed: where A X * is the relative abundance of element X in the star * system, characterized by a metallicity M, and A X SS is the relative abundance of reference in the solar system.The metallicity ratio is generally reported as: Fe H

*
Another usual alternative definition of the metallicity ratio is: . 17 * are more or less correlated for most stars and elements, with a 2D distribution clearly centered at the origin, the latter characterizing the solar system's analogs.
Let us now consider a set of host stars for which with M spanning some interval.Screening potentially fertile exoplanets contained in habitable zones of these host stars will involve Equations (A12) and (A13) with prefactors now changed from: in the cases when both abundances of N and P relative to H are available for the host star, or, when only ⎡ ⎣ ⎤ ⎦ Fe H * is available and * and ⎡ ⎣ ⎤ ⎦ P H * have to be guessed assuming a linear

10
, 20 where α is another empirical constant allowing consistency between databases: indeed α ≈ 1.06 is the average ratio between ⎡ ⎣ ⎤ ⎦ Fe H * reported in the Hypatia Catalog compared to those reported in the NASA archive for the same host stars.
Notice that the relative abundance of Si in the host star ⎡ ⎣ ⎤ ⎦ Si H * cancels in these expressions, so that this information is not needed for our purpose.Depending on the screening interval chosen (e.g., [11][12][13][14][15][16][17][18][19][20][21][22]), corrections for metallicities to predicted exoplanets N:P ratios may either include them in or exclude them from, the set of potentially fertile.However, error margins to these corrections will increase the uncertainties on the predictions and must be also evaluated.* , and 5 unreported).
The exact Equation ( 19) can be used for 3 cases only, HD 92788 b, HD 24040 c, and HD 82943 b, but with corrections to N:P of 1.0, −2.25, and 0.0 from 21.1, 17.4, and 16.2, respectively, these exoplanets remain in the range of potential fertility, as listed in Tables A1 and A2.
Tables A4 and A5 present also host star metallicities and corrections to N:P according to Equation (19) for the lists of potentially fertile and habitable exoplanets given in Tables A1  and A2.
In conclusion for this section, in view of the metallicity data available, corrections to the first screening assuming Fe H Fe H SS * for all host stars appear minor and do not change significantly the current list of potentially fertile habitable exoplanets.However, for future studies, it should be applied systematically, using at least Equation (20) and at best Equation (19).
Table 3 lists the subset of five sub-Jovian fertile exoplanets found in HZ with mass documented.T Pl_rec eq spans a 28 K interval around the average 249.8K, a value close to 264.8 K, the latter corresponding as mentioned above, to the Earth in the absence of the current greenhouse effect.

Conclusions
In this report, a theoretical basis is proposed for identifying potentially fertile exoplanets inside the habitable zone of their host star.It consists of a set of simple analytical equations calling astronomical parameters associated with confirmed exoplanets.An analysis of the NASA exoplanet archive allows us to conservatively estimate that at least 2.4% of the latter are at the moment suitable as Earth analogs.Corrections for host stars metallicities, when available, do not change significantly the list obtained assuming solar relative abundances of elements.However, such corrections should be applied as far as possible in order to confirm the screening.Moreover, reasonable ranges of exoplanet average densities and surface gravities should also restrain the screen.
In the future, improved space probing techniques might be focused as a priority in the directions of these potentially fertile exoplanets, in order to detect any signal compatible with life.
The criterion r * for this first screening might be somewhat extended to include forms of life adapted to more N-or P-depleted conditions.
Besides, future robotic and manned missions to Mars (predicted ρ N/P = 68.5)might allow us to verify experimentally the predictions that its soil in equilibrium with water and exposed to sunlight, either cannot support the development of phytoplankton species or will select species with N:P much higher than the Redfield ratio.
where T star , R star , and d Star are the effective temperature and radius of the host star and the distance from the exoplanet to the host star, respectively, and according to column notations in the database: Furthermore: When several independent characterizations of the same exoplanet are reported in distinct rows of the NASA exoplanet archive, it is legitimate to retain the one affected by the lower uncertainties, thus producing the lower relative errors from Equations (A4) to (A6).
Tables A1 and A2 present the results of these estimations of error margins for the potentially habitable and fertile exoplanets identified according to the procedure detailed in the main text.Averages of the latter were used to indicate error margins in Figures A2 to A4.

A.3. Numerical Evaluation of the Fraction of Fertile Exoplanets within the Habitable Zone
Let us express ρ − and ρ + according to Equations (4) and (5) of the main text, transposed from the solar system to any system by rescaling the unit of length: and: encompassing 4647 distinct exoplanets for which T Star has been reported.This distribution may be characterized as a skewed normal law spanning the interval [0.0037 au, 0.015 au] peaked at 0.007165 au, slightly above R Protosun (dashed vertical line).It shows that 92% of exoplanet-hosting stars so far detected are warmer than our Sun, and the current sampling of cooler host stars underestimates their occurrence.
Application of Equation (A12) or (A13) at = 1 predicts the atomic ratio N/P at the cusp point so that the lower limit of R d PST * corresponding to the upper limit of the screening interval for this ratio can be determined.For the interval [11,22] it is 0.0065446 au, and over 99% of host stars of exoplanets detected so far have R d PST * above this limit and therefore may host fertile exoplanets in their habitable zone.
In order to evaluate the probability of an exoplanet being potentially fertile within the habitable zone, let us consider two bidimensional grids with nodes spaced at regular intervals along variables > 0.0065446 at nodes of the first grid and ρ + at nodes of the second one from Equations (A12) and (A13), respectively.A logical test returning 1 if the value for an "active" node belongs to the chosen screening interval (e.g., [11,22]) and 0 if the value is outside, will allow us to count the active nodes in a grid.The ratio of active to total number of nodes is an estimate of the required probability for one grid, expected to converge rapidly as the number of nodes in the grids is increased.The average over both grids holds for the probability ( ) P 1 HZ  over the whole HZ.These grids cover practically all realizations of   A.5. Supplementary Figures    A1 and  A2 (filled circles).This graph shows that detected exoplanets predicted fertile within the Habitable Zone are mostly located inside the Orion arm of our Galaxy.Dotted line: average N/P atomic ratio.Dashed lines limit the range of selection [11][12][13][14][15][16][17][18][19][20][21][22].Error bars on N/P correspond to 24%, the calculated average relative uncertainty indicated in Table A2.

12
The Astrophysical Journal, 958:124 (15pp), 2023 December 1 Toulhoat  A1 and A2 (filled circles).Vertical error bars correspond to the average uncertainty of 0.07 This graph shows further that detected exoplanets predicted fertile within the Habitable Zone are mostly located inside the Orion arm of our Galaxy.A1 and A2 (filled circles).This graph shows that the predicted (N/P) appears randomly valued between 11 and 22, with an average of 17.2, within the Habitable Zone as defined by < computed thanks to the rescaled Equations (4) and (5), namely, Equation (A12) if d Star < d * or (A13) if d Star d * , will test its potential fertility.Screening the online NASA exoplanet archive (NASA Exoplanet Archive 2023) according to this

Figure 1 .
Figure1.Atomic ratio N/P in the solar system as a function of the distance to the Sun, in au, as predicted by the "photophysical model."On Earth, this ratio is predicted to coincide with the Redfield ratio (N/P = 16) characterizing oceanic phytoplankton.

Figure 2
Figure 2 confirms that our solar system is not singular in the known universe.It shows moreover that current exoplanet detection techniques are relatively blind to planets very distant from their stars and therefore cold, i.e., analogs of Jupiter and

Figure**
are rather well distributed within the HZ.As mentioned in TablesA1 and A2, average absolute and relative uncertainties are 0.07 for ( ) d d Log 10 Star * and 24% for r ( ) d d N P Star * .The corresponding error bars are indicated in Figures A2 to A4. FigureA4in particular reveals that some exoplanets listed in Tables A1 and A2 might actually orbit outside the range assumed habitable, and/or exhibit outside the interval of fertility[11, 22].Conversely, some exoplanets might have escaped this too stringent screen combining habitability and fertility: they might be identified however by extending the screening intervals to < ., augmenting the initial ranges slightly above the average uncertainties.This procedure resulted in a complementary list of 25 exoplanets reported in Table

Figure
Figure 2. Plots of ratios are basically functions of two random variables d d star * and T Star .Figure A5 demonstrates that these variables are uncorrelated (R 2 = 0.0263).Figure A6 presents the distribution of R d PST *

*
: predicted nitrogen over phosphorus atomic ratio; fertility criterion taken as r Î .: absolute and relative uncertainties, respectively, in %, affecting the previous column (see Section A.2 of this appendix for their estimations).

Figure A1.
Figure A1.Distribution of the random variable Figure A1.Distribution of the random variable

Figure A2 .
Figure A2.Predicted atomic ratio Nitrogen over Phosphorus as a function of the host star distance from Earth (in parsec) for the 49 exoplanets listed in TablesA1 and A2(filled circles).This graph shows that detected exoplanets predicted fertile within the Habitable Zone are mostly located inside the Orion arm of our Galaxy.Dotted line: average N/P atomic ratio.Dashed lines limit the range of selection[11][12][13][14][15][16][17][18][19][20][21][22].Error bars on N/P correspond to 24%, the calculated average relative uncertainty indicated in TableA2.

Figure A4 .
Figure A4.Predicted atomic ratio nitrogen over phosphorus as a function of Figure A5.Plot of random variable R d PST * against random variable

Figure A6 .
Figure A6.Distribution of the random variable R d PST * for the NASA exoplanet archive.The sample involves 4657 distinct exoplanets (some sharing the same host star).The filled circles and the continuous line trace the observed histogram.Dotted curve: best-fit normal law (μ = 0.00716467, σ = 0.00025).The dashed vertical line locates the Sun.

Table 3
Sub-Jovian Fertile Exoplanets Located in the Habitable Zone(ò HZ = 0.2 ) of their Host Stars Star (au): distance of the exoplanet from the host star; r ( ) Pl_rec eq : planet equilibrium temperature according to Equation (8); Planet mass: best plant mass in units of the Earth mass; density in g cm −3 ; gravity in g (Earth's * (au): critical distance from the host star; d * ; T

Table A1
List of the First 24 of 49 Fertile Exoplanets Found in the Habitable Zone (HZ) Star (au): distance of the exoplanet from the host star; ., R.U.: absolute and relative uncertainties, respectively, in %, affecting the previous column (see Section A.2 of this appendix for their estimations).List of the 25 Last of 49 Fertile Exoplanets Found in the Habitable Zone (HZ) * (au): critical distance from the host star; d * : predicted nitrogen over phosphorus atomic ratio; fertility criterion taken as r Î Star  * ; A. UNote.Rowid: row number in the NASA exoplanet archive; d * (au): critical distance from the host star; d Star (au): distance of the exoplanet from the host star ;