Survival-Weighted Buried-Volume Fraction

• The paper introduces the survival-weighted buried-volume fraction to measure how much biologically loaded ejecta survives joint entry heating and radiation exposure.
• It employs a spherical fragment model that removes an outer shell to determine the protected interior, linking required burial depth directly to flight time.
• Population averaging over a size-spectrum demonstrates that fragment size and transfer duration critically affect survival, highlighting Mars as a competitive donor channel.

The survival-weighted buried-volume fraction, denoted $F_{\mathrm{bur}}$ in prose and $F$ in several equations, is a population-level viability measure introduced in "Natural Panspermia to Earth: Quantitative Limits on Donor Classes, Transport, Survival, and Establishment" (Turyshev, 5 Apr 2026). It quantifies “what fraction of the biologically loaded low-shock ejecta volume remains viable after the joint entry-and-radiation requirement is imposed.” Within the paper’s Earth-directed transport-survival framework, it is not a single-rock survival probability, but a spectrum-averaged penalty applied to the low-shock spall population after accounting for entry heating and ablation, long-duration radiation, carrier size distribution, and flight time. Its principal function is to convert an existence argument—whether some fragment could survive—into a competitiveness measure for an entire ejecta population.

1. Conceptual role within the transport-survival problem

The paper formulates natural panspermia as a transport-and-establishment hypothesis rather than a theory of abiogenesis, and $F_{\mathrm{bur}}$ is introduced to sharpen that distinction (Turyshev, 5 Apr 2026). A stage-by-stage survivability analysis can show only that a sufficiently large carrier might remain viable. The buried-volume fraction instead asks how much of the biologically relevant material in the low-shock ejecta population survives once the joint burial-depth constraint from atmospheric entry and radiation exposure is imposed.

This shift in emphasis is central to the paper’s methodology. The relevant quantity is not the number of surviving rocks, but the surviving fraction of biologically loaded volume. Large carriers may exist yet contribute little to the total biological inventory if the size spectrum is steep; conversely, numerous small fragments may dominate counts while contributing negligibly to viable buried volume. $F_{\mathrm{bur}}$ addresses this by weighting the survival calculation by fragment volume.

A common misconception is to interpret $F_{\mathrm{bur}}$ as a generic survival probability for all ejecta. The paper explicitly restricts it to the biologically loaded low-shock population. Launch shock is treated separately through the low-shock spallation requirement, and $F_{\mathrm{bur}}$ is evaluated only after that launch filtering has already occurred.

2. Single-carrier burial geometry

The single-fragment model begins from a spherical carrier of radius $R$, for which only material buried deeper than a required depth $d_{\min}(t_{\mathrm{fl}})$ is protected (Turyshev, 5 Apr 2026). The protected buried-volume fraction for one carrier is

$$f_{\mathrm{bur}}(R,t_{\mathrm{fl}})= \Theta\!\big(R-d_{\min}(t_{\mathrm{fl}})\big) \left(1-\frac{d_{\min}(t_{\mathrm{fl}})}{R}\right)^3 .$$

The Heaviside factor $\Theta(R-d_{\min})$ imposes a hard cutoff. If the fragment is smaller than the required protection depth, it has no surviving protected interior and contributes zero. When $F$0, the cubic factor is the fraction of spherical volume remaining after removal of a sacrificial outer shell of thickness $F$1. Under the paper’s assumption that biological cargo is distributed through fragment volume, the surviving biological fraction is therefore the ratio of the inner protected sphere to the full sphere.

The paper describes this construction as conservative for survival. It states that “A surface-biased biosphere would fare worse than Eq.~(\ref{eq:fbur}); the present model is therefore conservative for survival.” This matters because any departure from volumetric loading toward near-surface loading increases vulnerability to both entry losses and radiation damage.

The geometric simplicity of $F$2 is deliberate. It isolates the core physical bottleneck: survival requires not merely a large object, but sufficient protected interior after cumulative filtering by depth-dependent hazards.

3. Population averaging and size-spectrum weighting

The formal definition of the survival-weighted buried-volume fraction is a size-spectrum average of $F$3 (Turyshev, 5 Apr 2026):

$F$4

Here $F$5 is the largest lightly shocked carrier radius available in the biologically viable ejecta tail, and the assumed carrier spectrum is

$F$6

The factor $F$7 follows directly from combining the power-law count spectrum with volumetric biological loading. The numerator is therefore proportional to

$F$8

which the paper interprets as number of fragments at radius $F$9, multiplied by biological cargo per fragment, multiplied by surviving fraction.

The appendix gives the corresponding analytic structure. Defining $F_{\mathrm{bur}}$0 and $F_{\mathrm{bur}}$1, one has $F_{\mathrm{bur}}$2 if $F_{\mathrm{bur}}$3. Otherwise,

$F_{\mathrm{bur}}$4

with

$F_{\mathrm{bur}}$5

This is simply the integral form obtained by expanding $F_{\mathrm{bur}}$6. The construction is phenomenological rather than a universal fragmentation law; the paper states that it is used to isolate the competition between the required protected depth and the maximum lightly shocked carrier size available.

4. Minimum protected-depth envelope

The central control variable for $F_{\mathrm{bur}}$7 is the minimum protected-depth envelope (Turyshev, 5 Apr 2026):

$F_{\mathrm{bur}}$8

This definition states that the required burial depth is whichever is larger: the entry survival floor or the radiation survival depth accumulated over flight time. The paper gives the entry floor as

$F_{\mathrm{bur}}$9

that is, approximately $F_{\mathrm{bur}}$0–$F_{\mathrm{bur}}$1 cm.

For radiation shielding, the paper provides a piecewise envelope:

$F_{\mathrm{bur}}$2

The implication is direct: at short transfer times, shielding requirements are mm–cm scale and the entry floor can dominate; at $F_{\mathrm{bur}}$3–$F_{\mathrm{bur}}$4 yr, the radiation requirement rises to $F_{\mathrm{bur}}$5–$F_{\mathrm{bur}}$6 m and becomes the controlling term; at $F_{\mathrm{bur}}$7 yr, the required shielding exceeds meter scale by a wide margin. The paper therefore uses $F_{\mathrm{bur}}$8 as the bridge from flight time to carrier architecture.

5. Dependence on flight time, carrier size, and spectrum slope

Because $F_{\mathrm{bur}}$9 grows with transfer duration, $F_{\mathrm{bur}}$0 decreases as $F_{\mathrm{bur}}$1 increases (Turyshev, 5 Apr 2026). The paper makes this dependence explicit in several threshold statements. For lightly shocked carriers with size $F_{\mathrm{bur}}$2 m, $F_{\mathrm{bur}}$3 is “effectively driven to zero” once transfer times enter the common martian-meteorite range of $F_{\mathrm{bur}}$4–$F_{\mathrm{bur}}$5 yr. For carriers around $F_{\mathrm{bur}}$6 m, survival in that regime is already $F_{\mathrm{bur}}$7. It also states that “for $F_{\mathrm{bur}}$8, $F_{\mathrm{bur}}$9 at $F_{\mathrm{bur}}$0 m is only of order $F_{\mathrm{bur}}$1, and it falls to zero as $F_{\mathrm{bur}}$2 approaches $F_{\mathrm{bur}}$3 m.” Only meter-to-multi-meter lightly shocked carriers retain appreciable $F_{\mathrm{bur}}$4 on Myr timescales.

The slope $F_{\mathrm{bur}}$5 controls how strongly the size spectrum concentrates biological inventory in small fragments. The paper states that steeper spectra make survival worse: larger $F_{\mathrm{bur}}$6 shifts more biological inventory into smaller fragments, smaller fragments are more strongly penalized by $F_{\mathrm{bur}}$7, and therefore $F_{\mathrm{bur}}$8 decreases as $F_{\mathrm{bur}}$9 increases. This is the population-level reason that a viability claim based on a rare large stone can overstate the competitiveness of an entire channel.

Launch conditions enter only indirectly into these trends because $R$0 is computed after the low-shock spallation filter. The paper places biologically viable launch below $R$1 GPa and treats the biologically relevant ejecta as a low-shock tail. Thus $R$2 does not summarize all launch physics; it summarizes the fate of the subset already admitted by launch-shock survivability.

6. Mars-specific averaging and donor-class significance

The paper gives Mars special treatment by separating a fast tail from the slow, common transfer regime (Turyshev, 5 Apr 2026). It defines

$R$3

with

$R$4

and

$R$5

The corresponding averaged buried-volume fractions are

$R$6

$R$7

These enter the effective martian transport factor as

$R$8

This decomposition is important because the paper concludes that “the biologically privileged fast-transfer tail avoids the severe $R$9 suppression imposed on the much more common Myr-transfer regime.” Mars is therefore treated not as a single channel but as two biologically distinct channels: a rare $d_{\min}(t_{\mathrm{fl}})$0–$d_{\min}(t_{\mathrm{fl}})$1 yr tail with modest shielding requirements, and a common $d_{\min}(t_{\mathrm{fl}})$2–$d_{\min}(t_{\mathrm{fl}})$3 yr regime in which $d_{\min}(t_{\mathrm{fl}})$4 collapses unless carriers are meter-class or larger.

Within the broader donor hierarchy, $d_{\min}(t_{\mathrm{fl}})$5 is one of the main reasons early Mars remains the only quantitatively serious external hard-panspermia alternative. The paper states that sibling birth-cluster donors have transport ceilings $d_{\min}(t_{\mathrm{fl}})$6 and require $d_{\min}(t_{\mathrm{fl}})$7; galactic-field donors satisfy $d_{\min}(t_{\mathrm{fl}})$8 and $d_{\min}(t_{\mathrm{fl}})$9; intergalactic donors are effectively excluded because capture and survival both fail. In these cases, $$f_{\mathrm{bur}}(R,t_{\mathrm{fl}})= \Theta\!\big(R-d_{\min}(t_{\mathrm{fl}})\big) \left(1-\frac{d_{\min}(t_{\mathrm{fl}})}{R}\right)^3 .$$0 is not the sole suppressor, but it contributes to the conclusion that beyond Mars the combined transport cost becomes overwhelming.

7. Relation to the transport kernel and interpretive significance

The paper’s central Earth-directed transport kernel is written as (Turyshev, 5 Apr 2026)

$$f_{\mathrm{bur}}(R,t_{\mathrm{fl}})= \Theta\!\big(R-d_{\min}(t_{\mathrm{fl}})\big) \left(1-\frac{d_{\min}(t_{\mathrm{fl}})}{R}\right)^3 .$$1

Within this formalism, $$f_{\mathrm{bur}}(R,t_{\mathrm{fl}})= \Theta\!\big(R-d_{\min}(t_{\mathrm{fl}})\big) \left(1-\frac{d_{\min}(t_{\mathrm{fl}})}{R}\right)^3 .$$2 is conceptually tied to the product of $$f_{\mathrm{bur}}(R,t_{\mathrm{fl}})= \Theta\!\big(R-d_{\min}(t_{\mathrm{fl}})\big) \left(1-\frac{d_{\min}(t_{\mathrm{fl}})}{R}\right)^3 .$$3 and $$f_{\mathrm{bur}}(R,t_{\mathrm{fl}})= \Theta\!\big(R-d_{\min}(t_{\mathrm{fl}})\big) \left(1-\frac{d_{\min}(t_{\mathrm{fl}})}{R}\right)^3 .$$4, because it encodes the fraction of low-shock biological volume surviving the joint burial-depth requirement imposed by those filters. The paper explicitly notes that $$f_{\mathrm{bur}}(R,t_{\mathrm{fl}})= \Theta\!\big(R-d_{\min}(t_{\mathrm{fl}})\big) \left(1-\frac{d_{\min}(t_{\mathrm{fl}})}{R}\right)^3 .$$5 is not literally a factor in $$f_{\mathrm{bur}}(R,t_{\mathrm{fl}})= \Theta\!\big(R-d_{\min}(t_{\mathrm{fl}})\big) \left(1-\frac{d_{\min}(t_{\mathrm{fl}})}{R}\right)^3 .$$6 as written. Rather, it functions as a population-level summary statistic for what the paper identifies as the dominant geometric survival bottleneck.

Its interpretive significance lies in distinguishing physical possibility from quantitative competitiveness. The paper argues that panspermia discussions often overfocus on whether life can survive in principle, while neglecting whether enough of the relevant ejecta population survives to make the channel competitive. $$f_{\mathrm{bur}}(R,t_{\mathrm{fl}})= \Theta\!\big(R-d_{\min}(t_{\mathrm{fl}})\big) \left(1-\frac{d_{\min}(t_{\mathrm{fl}})}{R}\right)^3 .$$7 addresses that gap by embedding burial geometry, flight-time-dependent shielding, and size-spectrum effects in a single volume-weighted measure. This suggests that the metric is less a generic survivability observable than a comparative discriminator among donor classes under Earth-specific constraints.

In the paper’s bottom-line synthesis, fast Mars transfers remain plausible because $$f_{\mathrm{bur}}(R,t_{\mathrm{fl}})= \Theta\!\big(R-d_{\min}(t_{\mathrm{fl}})\big) \left(1-\frac{d_{\min}(t_{\mathrm{fl}})}{R}\right)^3 .$$8 stays modest; common Mars meteorite timescales strongly suppress survival unless carriers are meter-scale or larger; and long interstellar or intergalactic transfer is crushed by the combined demands of shielding, capture, and time. On that basis, $$f_{\mathrm{bur}}(R,t_{\mathrm{fl}})= \Theta\!\big(R-d_{\min}(t_{\mathrm{fl}})\big) \left(1-\frac{d_{\min}(t_{\mathrm{fl}})}{R}\right)^3 .$$9 is one of the central technical devices supporting the conclusion that early Mars is the only external hard-panspermia source that remains quantitatively serious for Earth’s actual origin history (Turyshev, 5 Apr 2026).