Implications for Chondrule Formation Regions and Solar Nebula Magnetism from Statistical Reanalysis of Chondrule Paleomagnetism

Recent advances in multiple scientific disciplines have demonstrated that protoplanetary disks typically display far more local scale structure than described in early analytical models (e.g., Hayashi 1981). Observations from the Atacama Large Millimeter/submillimeter Array have discovered dust concentrations, cavities, and gaps, known as substructures, in nearly every resolvable disk (Andrews 2020). The proposed formation mechanisms for these features, which may arise from multiple processes, include magnetohydrodynamical (MHD) instabilities and MHD-driven winds (Suriano et al. 2018; Takahashi & Muto 2018), sculpting by a planetary companion (Zhu et al. 2013, 2014), and solid aggregate accretion at condensation fronts (Ros et al. 2019).

In parallel, numerical MHD models of nebular gas dynamics indicate that protoplanetary disks may be partitioned into regions with distinct angular momentum transport and turbulence properties based on factors such as local ionization and density (Armitage 2019; Lesur 2021a). Further, although past studies have argued for quiescent, laminar gas dynamics in a "dead zone" near the midplane of inner disk (Gammie 1996), more recent simulations have identified processes such as the Hall shear instability (HSI), which can generate significant local magnetic field amplification even in weakly ionized disk regions (Kunz 2008; Bai 2017; Béthune et al. 2017). Specifically, studies of the Hall effect have indicated that alignment or anti-alignment between a net vertical disk field and the disk rotation vector can, respectively, produce steady or episodically, locally enhanced magnetic fields (Lesur et al. 2014; Bai 2015; Simon et al. 2015).

Finally, experimental measurements of ancient magnetic fields and their variability in the solar system accretion disk have become available through the paleomagnetism of chondritic meteorites. Although these challenging experiments have been attempted for several decades (Butler 1972), recent studies have benefited from newly developed instrumentation and more expansive knowledge regarding the ferromagnetic mineralogy of chondrite components (see the review in Weiss et al. 2021). One series of paleomagnetic studies has targeted individual chondrules. A study of the well-preserved Semarkona LL chondrite recovered disk magnetic field intensity estimates from eight dusty olivine-bearing chondrules, which are expected to carry high-fidelity magnetizations due to their high concentrations of submicrometer Fe grains (Figure 1; Lappe et al. 2013; Fu et al. 2014b). Subsequently, two studies of chondrules from CR and CO chondrites found evidence for weak (<8 μT) and substantial (∼100 μT) magnetic fields in their chondrule formation regions, respectively (Fu et al. 2020; Borlina et al. 2021). At the same time, recent paleomagnetic measurements of the magnetization carried by fine-grained chondrite matrices have produced evidence for magnetic fields between <0.15 μT and >100 μT recorded by CM chondrites, the CV chondrite Allende, and the ungrouped chondrites WIS 91600 and Tagish Lake (Cournede et al. 2015; Bryson et al. 2020a, 2020b; Fu et al. 2021).

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Taken together, these experimental results suggest large spatial and/or temporal variations in disk magnetic field strength. Specifically, the contrasts between the high paleointensities recorded by CO chondrules and CV chondrites, the moderate paleointensities from LL chondrules, and the weak fields from other carbonaceous chondrites cannot be easily explained by measurement uncertainties or differences in the timescale of magnetic field acquisition, which determines whether the total or net vertical magnetic field in the disk is recorded (see Section 5). Instead, they are consistent with magnetic field self-organization effects or variations in local inward gas accretion rates arising from differences in disk density (Riols & Lesur 2018; Suriano et al. 2018). As a result, these paleomagnetic observations have been cited as support for the existence of substructures in the solar nebula and generally corroborate the view of protoplanetary disks as complex environments with highly variable local conditions (Borlina et al. 2021; Fu et al. 2021).

However, interpretations of chondrite-derived paleointensities are based on paleomagnetic data analysis tools adapted to terrestrial rocks that formed in a stable geomagnetic field and are rarely limited by sample availability. In the case of single-chondrule paleointensity studies, both Fu et al. (2014b) and Borlina et al. (2021) first computed a single intensity for the nebular magnetic field using a simple mean of individual chondrule measurements. Next, both studies noted that each chondrule likely underwent many rotations around a spin axis during remanence acquisition, implying that the recorded magnetic field is attenuated by a factor cos θ, where θ is the angle between the spin axis and the average background magnetic field. Integration across all values of θ shows that, on average, rotation results in a 50% decrease in the recorded paleointensity relative to the true ambient field (Fu et al. 2014b). Therefore, these authors, along with those of other chondrite paleomagnetism studies (Bryson et al. 2020b), have reported the paleointensity as two times the raw mean.

Considering the uncertainties in the chondrule formation process, the typically small number of individual chondrule samples, and the potentially time-variable solar nebula magnetic field, this simple procedure to obtain a paleofield estimate may not be appropriate. First, chondrules may not have been spinning. Although previous studies have suggested that chondrules cooling over a timescale of hours are unlikely to despin due to gas drag at radii ≥2 au (Fu & Weiss 2012; Fu et al. 2014b), these calculations assumed that the gas density during magnetization acquisition was close to minimum mass solar nebula (MMSN) values. In fact, peak gas densities may have been enhanced during chondrule formation events to ≥10 Pa (Alexander 2004), which is at least two orders of magnitude higher than MMSN gas densities in chondrite formation regions. Meanwhile, the dust density during chondrule formation may have been enhanced by much large factors of ≥10⁶ (Alexander et al. 2008), potentially adding significantly to the total vapor density at high temperatures. The degree of gas density enhancement during the tail end of chondrule cooling when magnetization is acquired is expected to be lower than these peak values; however, the precise density is uncertain, due to the unknown mechanism of chondrule formation, which affects the rate of gas decompression (Fu et al. 2018). If chondrules did not spin during magnetization acquisition, the two times correction applied to chondrule paleointensities would result in overestimation of the true nebular field strengths.

Second, as discussed above, the magnetic field strength in the solar nebula may have been variable at timescales longer than that of chondrule formation but shorter than that of secular disk evolution. Therefore, even considering the modified probability density function of recorded field values due to chondrule spin (Figure 2), individual chondrule paleointensities may not represent repeated sampling of the same underlying magnetic field value plus uncertainty. In this case, the simple mean approach used in previous paleomagnetic studies can still be used to estimate the time-averaged value. However, a variable ambient field would inflate the observed variance of chondrule paleointensities. Evaluating the resulting data in a constant ambient field framework would lead to misattribution of this variance to other sources, such as analytical uncertainty. Further, analyzing paleointensities using the correct assumptions about the ambient field behavior can result in more physically meaningful parameters, such as the peak-to-peak amplitude of variation. Finally, as we will derive below, the distribution of chondrite-recorded paleointensities is not symmetrical around the mean for a number of disk scenarios, leading to asymmetric, non-Gaussian confidence intervals that are not captured via traditional paleomagnetic statistical analysis.

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Third, the necessarily small number of measured chondrules and the significant uncertainty associated with each derived paleointensity complicate the required statistical analysis. At the most basic level, the analytical uncertainty of each chondrule, which typically stems primarily from noise in the demagnetization sequences fitted to derive the paleointensity, should be propagated into the overall mean paleointensities. Moreover, computing the uncertainty on the mean paleointensity using a simple standard error may be inappropriate, due to the combination of the small number of samples and the highly non-Gaussian underlying distributions due to chondrule rotation (Figure 2). Small sets of chondrule paleointensities therefore do not produce normally distributed average values according to the central limit theorem. Consequently, a simple comparison based on the mean and standard error is not rigorously suitable for determining whether chondrules from different meteorites formed in distinct magnetic field environments (Borlina et al. 2021; Fu et al. 2021).

In this work, we introduce a Bayesian method for estimating the intensity and time variability of nebular magnetic fields and their uncertainties, based on meteorite data. This approach is valid for any scenario where an expected probability density function for a chondrule-recorded magnetic field intensity can be derived. We apply the method in the case of both a constant nebular field strength and sinusoidally varying magnetic fields, noting that scenarios with fields typically of constant intensity punctuated by short transitions between reversal states are mathematically also covered by the constant nebular field case. Using this framework, we obtain new estimates of nebula field strengths and uncertainties and discuss the implications of these paleointensities for the occurrence of nonideal MHD phenomena in the solar nebula.

Further, we apply the same statistical framework to determine the relative likelihood of different hypothesized conditions during the formation of LL and CO chondrules. Specifically, we find that the existing paleointensity data provide positive to strong support for chondrule rotation during cooling. Finally, we discuss additional considerations for comparing the paleomagnetic records from distinct meteorite groups, including the effects of the magnetization acquisition mechanism and time.

Several recent studies have introduced Bayesian frameworks to analyze common data types found in paleomagnetism (Heslop & Roberts 2018a, 2018b, 2019). These techniques, although more complex to implement than traditional frequentist approaches, have several advantages, including a balanced treatment of competing hypotheses and a lack of implicit assumptions about sample sizes. We implement a similar framework for the analysis of meteorite-derived paleointensities using the two related concepts of Bayesian hypothesis testing and Bayesian parameter estimation. An implementation of these calculations in Python is provided on the Harvard Dataverse (Fu & Steele 2023).

Following the derivation and notation found in pp. 126–130 of Raftery (1995), Bayes' theorem states that the likelihood that model M₁ is correct given a set of observations D, also known as the posterior likelihood of M₁, is given by:

$$\begin{eqnarray}&&p\left({M}_{1}| D\right)=\tfrac{p\left(D| {M}_{1}\right)}{p\left(D\right)}p({M}_{1}).\end{eqnarray} \tag{ 1 }$$

The relative likelihoods of the two possible models M₁ and M₂ are then given by:

$$\begin{eqnarray}&&\tfrac{p\left({M}_{1}| D\right)}{p\left({M}_{2}| D\right)}=\tfrac{p\left(D| {M}_{1}\right)}{p\left(D| {M}_{2}\right)}\tfrac{p({M}_{1})}{p({M}_{2})}.\end{eqnarray} \tag{ 2 }$$

A value for this ratio much larger or smaller than 1 would imply that the data strongly favor M₁ or M₂, respectively. If no prior information is available to favor either model, which we adopt for all analyses below, then p(M₁)/p(M₂) = 1 and the relative likelihood of the two models simply becomes the ratio of the likelihoods $p\left(D| {M}_{1}\right)/p\left(D| {M}_{2}\right)$ (e.g., Heslop & Roberts 2018a). This ratio, also known as the Bayes factor, can be directly evaluated if the likelihood of observing the data D within each model framework M_x can be computed. If a model contains a single unknown parameter, such as the instantaneous ambient magnetic field magnitude during chondrule cooling (B_amb), the total likelihood for observing the given data, also known as the marginal likelihood, is given by integrating the likelihood across all possible values of B_amb:

$$\begin{eqnarray}&&p\left(D| {M}_{1}\right)={\int }_{0}^{\infty }p\left(D| {M}_{1},{B}_{\mathrm{amb}}\right)p\left({B}_{\mathrm{amb}}\right){{dB}}_{\mathrm{amb}},\end{eqnarray} \tag{ 3 }$$

where the integral ranges from 0 to ∞ because only the absolute value of the nebular field can be inferred from meteorite observations. Given the lack of robust, predictive models for nebular magnetic fields, we assume an uninformative prior for $p\left({B}_{\mathrm{amb}}\right)$, treating it as a constant that does not affect the ratios of model likelihoods. The quantity $p\left(D| {M}_{1},{B}_{\mathrm{amb}}\right)$ denotes the likelihood of observing the experimental data set assuming a model (for example, a scenario with a constant background field and rotating chondrules) and a given value for B_amb. To relate B_amb to a single observed paleointensity D_i , we must first define the expected magnetic field recorded in the chondrule, hereafter B_rec, which may differ from B_amb due to chondrule rotation (Figure 2). As long as a probability density function for B_rec can be written down for a given value of B_amb, the quantity $p\left({D}_{i}| {M}_{1},{B}_{\mathrm{amb}}\right)$ can be calculated as the integral of the likelihood of observing the ith paleointensity D_i across all possible values of B_rec:

$$\begin{eqnarray}&&p\left({D}_{i}| {M}_{1},{B}_{\mathrm{amb}}\right)=\tfrac{1}{{\sigma }_{i}\sqrt{2\pi }}{\int }_{0}^{\infty }\ p\left({B}_{\mathrm{rec}}| {B}_{\mathrm{amb}}\right){e}^{-{\left({D}_{i}-{B}_{\mathrm{rec}}\right)}^{2}/{\sigma }_{i}^{2}}{{dB}}_{\mathrm{rec}},\end{eqnarray} \tag{ 4 }$$

where the difference between the observed D_i and the expected value B_rec is assumed to follow a normal distribution, with standard deviation equal to the analytical uncertainty of the observation σ_i (Figure 2(D)). Previous investigations of paleointensities recorded in modern lavas have revealed Gaussian and nearly Gaussian error distributions (Muxworthy et al. 2011). The likelihood of observing all data points for a given B_amb is then:

$$\begin{eqnarray}&&p\left(D| {M}_{1},{B}_{\mathrm{amb}}\right)=\displaystyle \prod _{i=1}^{N}p\left({D}_{i}| {M}_{1},{B}_{\mathrm{amb}}\right),\end{eqnarray} \tag{ 5 }$$

where N is the total number of observations. Integration of Equation (5) over all possible values of B_amb would then provide the total likelihood for a given model M₁, and performing the same calculation for an alternative model M₂ would allow us to compute the Bayes factor (Equation (2)). Bayes factors greater than 3 and 20 correspond to 75% and 95% probability in favor of M₁ and are generally termed "positive" and "strong" evidence in its favor, respectively (Raftery 1995).

Further, the expression $p\left(D| {M}_{1},{B}_{\mathrm{amb}}\right)$, which is proportional to the posterior distribution of B_amb given the uninformative priors used, can be used in the framework of maximum likelihood estimation to identify the modal value of B_amb and its confidence interval (Glickman & Dyk 2007). Evaluating this distribution for two different data sets, for example from two different chondrite groups, can determine whether one population is likely to have originated in a stronger or weaker magnetic field. Analogous to other applications of Bayesian model selection in paleomagnetism (e.g., Heslop & Roberts 2018a), and unlike previous statistical treatments of chondrule paleointensities, the methods described here avoid asymmetry between a null and alternative hypotheses, are applicable to small sample sizes with no assumptions related to the central limit theorem, and explicitly treat highly non-Gaussian probability distributions.

2.1. Mapping the Ambient Field (B_amb) to the Recorded Paleointensity (B_rec)

The evaluation of Equation (4) requires an expression for $p\left({B}_{\mathrm{rec}}| {B}_{\mathrm{amb}}\right)$, which is the probability of recording a certain paleointensity in a given ambient field, appropriate to the assumptions about ambient field variability and the rotation state of chondrules. As discussed in Section 1, the HSI in combination with aligned net vertical magnetic fields can result in temporally near-constant magnetic fields throughout the planet-forming region (Lesur et al. 2014). Further, a group of chondrules that formed in a single event, such as a planetesimal collision, and subsequently accreted onto the same parent body would sample the same ambient magnetic field, even if the disk field fluctuates over longer timescales (Metzler 2012). For nonrotating parent bodies or chondrules in such a constant ambient magnetic field, B_rec is simply equivalent to B_amb:

$$\begin{eqnarray}&&p\left({B}_{\mathrm{rec}}| \tfrac{{{dB}}_{\mathrm{amb}}}{{dt}}=0,\omega =0\right)={B}_{\mathrm{amb}},\end{eqnarray} \tag{ 6 }$$

where the ω = 0 condition specifies nonspinning bodies. We note that because chondrules only record the field magnitude, episodically reversing magnetic fields are also well described by Equation (6) if, analogous to the Earth's geodynamo, the duration of the polarity transitions is short compared to the time spent in a steady configuration in each polarity. MHD simulations show that this is the case for the Hall-effect-dominated regime (Bai 2015); therefore, our treatment of the "constant" magnetic field regime includes scenarios involving episodic reversals. Body rotation during magnetization acquisition implies that B_rec is the projection of B_amb on the spin axis. Assuming an isotropic distribution of spin-axis orientations results in a constant distribution of B_rec (Fu et al. 2014b, Equation (7)):

$$\begin{eqnarray}p\left({B}_{\mathrm{rec}}| \tfrac{{{dB}}_{\mathrm{amb}}}{{dt}}=0,\omega \ne 0\right)=\left\{\begin{array}{cc}\tfrac{1}{{B}_{\mathrm{amb}}} & 0\leqslant {B}_{\mathrm{rec}}\leqslant {B}_{\mathrm{amb}}\\ 0 & {B}_{\mathrm{rec}}\gt {B}_{\mathrm{amb}}\end{array}\right..\end{eqnarray} \tag{ 7 }$$

Time-varying magnetic fields can result in a variety of distributions for the instantaneous magnetic field strength. As a first step, we adopt a sinusoidal magnetic field pattern (${B}_{\mathrm{amb}}={B}_{\mathrm{peak}}\sin t$) to approximate this situation, where B_amb and B_peak refer to the instantaneous and peak field values, respectively (Figure 2). Future numerical MHD studies of the inner disk may provide more specific models of the time-varying magnetic field.

To translate B_amb to the observable paleointensity B_rec, we assume that each chondrule formed at a distinct, randomly chosen time, which is consistent with the observation of mutually resolved chondrule ages in LL and CO chondrules (Kita & Ushikubo 2012; Fukuda et al. 2022). We further assume, based on MHD simulations, that the timescale of the B_amb variability, which is on the multi-orbital timescale, is much longer than the hours-to-days window of chondrule formation (Hewins et al. 2005; Bai 2015). The probability density function of B_rec is then higher where the value of $\sin t\$ remains relatively constant for a range of t. This flatness or horizontalness of the ${B}_{\mathrm{amb}}\left(t\right)$ function can be quantified as the derivative of the inverse function ${\sin }^{-1}{B}_{\mathrm{amb}}/{B}_{\mathrm{peak}}$, which yields after normalization:

$$\begin{eqnarray}&&p\left({B}_{\mathrm{rec}}| {B}_{\mathrm{amb}}={B}_{\mathrm{peak}}\sin t,\omega =0\right)=\tfrac{2}{\pi {B}_{\mathrm{peak}}\sqrt{1-{B}_{\mathrm{rec}}^{2}/{B}_{\mathrm{peak}}^{2}}}.\end{eqnarray} \tag{ 8 }$$

This probability density function, which is only defined for positive B_rec less than B_peak, because only the field magnitude is recorded, is applicable to nonspinning bodies cooling rapidly in a sinusoidally varying field.

For a parent body or chondrule rotating rapidly in such a variable field, the recorded magnetic field is equally distributed between 0 and the instantaneous absolute value of the ambient field (B_amb; Equation (7)), which itself is distributed according to Equation (8). The probability density function of B_rec is a combination of these two independent random processes. To achieve a given final value of B_rec, the instantaneous magnetic field B_amb as sampled from Equation (8) must be greater than or equal to B_rec. When this condition is met, B_rec occurs with a probability given by the product of Equations (7) and (8):

$$\begin{eqnarray}&&\begin{array}{l}p\left({B}_{\mathrm{rec}}| {B}_{\mathrm{amb}}={B}_{\mathrm{peak}}\sin t,\omega \ne 0\right)\\ \ =\,{\displaystyle \int }_{{B}_{\mathrm{rec}}}^{{B}_{\mathrm{peak}}}\tfrac{2}{\pi {B}_{\mathrm{peak}}\sqrt{1-{B}_{\mathrm{amb}}^{2}/{B}_{\mathrm{peak}}^{2}}}{{dB}}_{\mathrm{amb}}.\end{array}\end{eqnarray} \tag{ 9 }$$

This integration yields the result for a spinning body in a sinusoidally varying field:

$$\begin{eqnarray}&&\begin{array}{l}p\left({B}_{\mathrm{rec}}| {B}_{\mathrm{amb}}={B}_{\mathrm{peak}}\sin t,\omega \ne 0\right)\\ \ =\,\tfrac{2}{\pi {B}_{\mathrm{peak}}}{\tanh }^{-1}\sqrt{1-\tfrac{{B}_{\mathrm{rec}}^{2}}{{B}_{\mathrm{peak}}^{2}}}.\end{array}\end{eqnarray} \tag{ 10 }$$

After establishing the probability density functions appropriate for the different magnetic field and body rotation scenarios, the remaining portion of Equation (4) describes the observed paleointensity values D_i and their uncertainties σ_i . For most chondrule paleointensities, we simply use the published values provided in Fu et al. (2014b) and Borlina et al. (2021; Figure 3). One complication is the finding that two dusty olivine-bearing chondrules in Semarkona do not carry a resolvable high-temperature component of magnetization (Fu et al. 2014b). The strong initial magnetizations (>3 × 10⁻¹² A m²) of all studied chondrules are well above the detection threshold of the SQUID microscope and the statistical limit for reliable paleointensities (Berndt et al. 2016). We therefore interpret these chondrules as having cooled in a weak solar nebula magnetic field and adopt a paleointensity of 0 for these two chondrules. The authors of the original study did not perform a recording limit experiment using laboratory-imparted magnetizations on these chondrules, which, if done, would have provided a confidence interval. We therefore average the uncertainties on the remaining five chondrules with recovered paleointensities to establish the typical uncertainty and use this value (8.7 μT) for the standard deviation of the two chondrules lacking high-temperature magnetizations (Equation (5)). For chondrules where the paleointensities and uncertainties of separate subvolumes are reported individually, we take the mean of the two paleointensities and the quadratic mean of their uncertainties, as is appropriate for observations with independent errors. Finally, all dusty olivine-derived paleointensities published to date are based on room temperature normalization methods, specifically anhysteretic remanent magnetization normalization, which can lead to bias compared to the true recorded magnetic field intensity (Figure 2(A)). Fortunately, dusty olivine-dominated magnetic mineralogies appear to require a relatively small and consistent correction (Lappe et al. 2013), which is already incorporated into the original studies on LL and CO chondrules (Fu et al. 2014b; Borlina et al. 2021). We therefore do not further incorporate uncertainties due to paleointensity calibration error in our results, but note that the paleointensity calibration uncertainty would be potentially significant for comparing dust olivine results to those derived from other lithologies.

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3.1. Testing for Chrondrule Rotation

We first apply the Bayesian model selection framework to examine the relative support for rotating and nonrotating chondrule scenarios. For a constant ambient magnetic field, the candidate probability density functions describing B_rec in nonspinning and spinning scenarios are, respectively, a Dirac delta function centered on B_amb and Equation (7). We find that the separate ensembles of LL and CO chondrules show positive to strong preferences for the spinning model with Bayes factors of 10.7 and ∼10⁵⁵, respectively, corresponding to probabilities of p = 0.91 and ≫0.99 for the spinning model where $p={BF}/\left({BF}+1\right)$, given a Bayes factor BF. Using a less tightly constrained Gaussian function for the nonspinning case, with a standard deviation equal to the reported standard error of the mean, results in Bayes factors of 3.7 and 719.8, corresponding to p = 0.79 and 0.998 (Figures 3 and 4). In both cases, the very large Bayes factors for the CO chondrules stem from the fact that the mean field value lies far outside the 2σ range of most individual chondrule results, thereby providing evidence against repeated sampling of a single underlying value.

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For the sinusoidally varying B_amb scenario, the greater similarity between the spinning and nonspinning probability density functions of B_rec results in a less decisive preference between the two hypotheses (Figure 3). Assuming that both sets of chondrules were either spinning (Equation (7)) or nonspinning (Equation (8)), the relative probability of spinning to nonspinning is the product of the two Bayes factors calculated individually for each set, which are equal to 1.7 and 2.2 for LL and CO chondrules, respectively. Their product results in a combined Bayes factor of 3.81, which implies a probability of 0.79 for the spinning case and is usually termed "positive support" (Raftery 1995).

In summary, the existing chondrule paleointensity data sets provide positive to strong support (p = 0.79 to ≫0.999) for chondrule rotation during magnetization acquisition, depending on the assumed degree of background field variability. We therefore find that the factor of 2 used in previous studies to account for rotation was very likely correctly applied and assume chondrule rotation for subsequent analysis in this work.

3.2. Testing for Magnetic Field Variability

3.3. Magnetic Field Strengths in Chondrule-forming Regions

When analyzed using our Bayesian statistical framework, single-chondrule paleointensities from LL and CO chondrites provide positive to strong (p = 0.79 to >0.99) evidence that the chondrules spun during cooling, validating previously published ambient disk magnetic field strength estimates that adopted this assumption.

Further, the persistence of chondrule rotation during the entire cooling interval, despite gas drag, implies a relatively low-density formation environment. The e-folding timescale of spin down (t_drag) in the relevant free molecular regime is (Fu & Weiss 2012):

$$\begin{eqnarray}&&{t}_{\mathrm{drag}}=\tfrac{4{\rho }_{c}}{5f{\rho }_{g}\overline{c}}{r}_{c},\end{eqnarray} \tag{ 11 }$$

where ρ_c is the chondrule density, assumed to be 3200 kg m⁻³ (Friedrich et al. 2015), ρ_g is the gas density, f is a momentum transfer factor of order 1, $\overline{c}$ is the mean gas particle velocity, and r_c is the chondrule radius. This despinning timescale can be compared to the cooling timescale to constrain the ambient gas conditions that permit continual spinning during magnetization acquisition. Assuming that LL and CO dusty olivine-bearing chondrule magnetizations are blocked between approximately 600°C and 400°C (Fu et al. 2014b), and that the cooling rates in this time interval ranged between 1°C and 10°C per hour (Schrader et al. 2018a), we adopt 20–200 hr for the cooling timescale.

Given these cooling rates and initial chondrule spin velocities of ∼100 revolutions per second (Tsuchiyama et al. 2003), approximately 15 e-foldings of despinning are necessary to slow rotation sufficiently such that the chondrule does not complete several revolutions during the magnetization acquisition interval. Setting the total despinning time (≈15t_drag) equal to the cooling timescale and adopting MMSN parameters (Hayashi 1981), we find the maximum permissible gas density for the smaller and therefore faster despinning CO chondrules is 4 × 10⁻⁸ kg m⁻³ and 4 × 10⁻⁹ kg m⁻³, assuming the shortest and longest cooling times, respectively. These values correspond to minimum formation radii of 4 and 8 au, which are likely to be outside the H₂O snow line in the solar nebula (Min et al. 2011; Mulders et al. 2015). This formation location is also beyond the orbit of early Jupiter forming near the snow line, according to most models (e.g., Walsh et al. 2011), although the radius of Jupiter formation remains uncertain (Öberg & Wordsworth 2019). The same analysis for the larger LL chondrules yields minimum radii of 2–5 au. These large inferred radii contradict scenarios where chondrules formed in the inner solar system and were subsequently transported outward (e.g., Shu et al. 1997; van Kooten et al. 2021).

If the mechanism of chondrule formation resulted in the compression of ambient gas even during late-stage cooling, these constraints for a maximum nebular gas density would require an even lower initial gas density and correspond to an origin farther in the outer solar system. A denser disk than the MMSN, which would be required if substantial disk mass was lost via winds (Lesur 2021b), would likewise increase these minimum formation radii. Given these considerations, the minimum formation radii given above should be regarded as generous lower bounds, with chondrule formation likely taking place at greater radii. Although our inferred formation locations of >2 and >4 au for LL and CO chondrules are still plausible, given the current understanding of chondrite reservoir origins (e.g., Kleine et al. 2020), they also motivate consideration that chondrules formed in anomalously low-density regions, potentially related to magnetic field instabilities (see Section 1; Riols & Lesur 2018; Suriano et al. 2018).

After establishing that chondrules spun during cooling, the corrected probabilities for the ambient nebular field strength derived from the Bayesian framework support two possible, self-consistent scenarios. First, the chondrule paleointensities are compatible with temporally constant magnetic fields in the solar nebula with well-defined, three times higher strengths in the CO formation region. Alternatively, chondrule paleointensities are also consistent with time-variable magnetic fields with, given the uncertainties, marginally higher peak-to-peak amplitudes in the CO formation region. We assess the likelihood of each scenario and discuss the implications for MHD dynamics in the planet-forming regions.

Beginning with the case of temporally stable magnetic fields, such a regime may exist near the midplane of the inner disk, either due to low ionization suppression of all MHD instabilities or strong HSI in the case of an aligned net vertical magnetic field and the nebular rotation vector (see Section 1; Turner & Sano 2008; Lesur et al. 2014). Further, these stable fields, if they are responsible for mass and angular momentum transport, are expected to be stronger at smaller orbital radii for a given rate of inward disk accretion (Equation (19) in Wardle 2007; Equation (14) in Bai 2015).

The greater degree of CO chondrite hydration and affinities to the CC isotopic reservoir imply that they originated at larger orbital radii (Schrader & Davidson 2017). The stronger ambient magnetic field recorded by CO chondrites therefore appears to contradict the expectation of weaker magnetic fields in the outer solar system.

As one potential explanation, highly efficient disk winds leading to the rapid loss of disk gas may imply much higher mass accretion rates in the outer disk, necessitating stronger magnetic fields at larger radii (Lesur 2021b). Assuming highly approximate formation radii of 2 and 5 au for the LL and CO chondrite groups in accordance with the chondrule spinning analysis above and cosmochemical constraints (Sutton et al. 2017), the three times difference in their nominal constant magnetic field strengths corresponds to a factor of 40–100 times greater mass accretion rate at the larger radius, depending on the wind-driven accretion-rate scaling law adopted (Wardle 2007, Equation (19); Lesur 2021b, Equation (18)). Assuming an upper bound for the LL chondrule paleointensity yields a minimum difference of 12–23 times faster inward disk accretion in the CO formation region. The conservation of mass would require the extremely efficient wind-driven loss of disk material between 2 and 5 au. For comparison, models of disk winds predict that the disk-averaged ratio of wind-ejected to accreted mass is of order unity, which corresponds to only a two times difference in accretion rates between 2 and 5 au (Lesur 2021b). We therefore find that winds in an unperturbed disk cannot explain the much stronger magnetic fields and therefore higher implied accretion rates in the CO chondrule formation region.

As a more likely scenario under the temporally constant magnetic fields assumption, recent nonideal MHD simulations suggest that protoplanetary disk magnetic fields may undergo flux concentration, forming au-scale regions of enhanced magnetic fields, rapid wind-driven mass loss, and low surface density (Béthune et al. 2017; Riols & Lesur 2018; Suriano et al. 2018; Cui & Bai 2021; Aoyama & Bai 2023). These regions may form under a range of nonideal regimes with or without the presence of giant planets. Given that any chondrules formed within a gas planet's Hill radius are unlikely to escape to the eventual asteroid belt, CO chondrules are more likely associated with planet-free magnetic flux concentrations or regions of enhanced magnetic fields immediately outside planet-carved gaps. The locations of planet-free enhanced regions are not reliably predictable (Aoyama & Bai 2023); therefore, the formation of CO chondrules in such an environment does not yield strong constraints on their formation radius.

Both planet-bearing and planet-free flux concentrations can host magnetic fields stronger by a factor of several compared to background values, which is consistent with the factor of ∼3 stronger paleointensities observed in CO chondrules compared to LL samples. No other studied process to date is known to sustain such a strong local enhancement of the magnetic field. Therefore, if LL and CO chondrules record temporally constant magnetic fields, the observed paleointensities support the existence of such nonideal MHD-induced magnetic flux concentrations in the solar nebula. On the other hand, these regions may also host lower dust densities (e.g., Suriano et al. 2018), potentially implying a smaller supply of chondrule precursors.

In the case of time-variable magnetic fields, only a limited number of phenomena are known to produce magnetic field fluctuations in the likely Hall-effect-dominated, inner (≤10–30 au) disk midplane regions relevant to chondrule formation (Armitage 2019). According to recent models of Hall-effect-driven instabilities, time-variable, or "bursty," magnetic fields in the inner solar system can occur in the case of anti-aligned net vertical fields and rotation vector (Simon et al. 2015), although a higher but still permissible level of ambipolar diffusion can potentially suppress this "bursty" behavior. In a fiducial "bursty" case, with a midplane plasma beta parameter of $\beta \propto {\left|B\right|}^{-2}={10}^{5}$, the predicted peak-to-peak amplitude of the azimuthal–radial magnetic field geometric mean ($\sqrt{{B}_{\phi }{B}_{r}}$) is 5–19 μT at 5–2 au (Simon et al. 2015). Assuming an approximate B_ϕ /B_r ratio of ∼10 (Bai 2017), this range implies a total magnetic field of ∼15–60 μT. Although this field is somewhat weaker than the 50–130 μT peak field estimated from LL and CO chondrule paleointensities, the modeled field strengths are expected to vary based on model assumptions, such as the strength of the net vertical magnetic field. The inferred amplitude of the ambient field variability is therefore compatible with current predictions from Hall-effect-regime MHD models. The anti-alignment of the net vertical field in the early solar system and the resulting "bursty" behavior is also consistent with the lack of planets with orbital radii ≤0.3 au due to the low solid particle-to-gas ratio predicted for this scenario (Simon 2016).

We have focused the above discussion of time variability on the Hall-effect-induced, "bursty" magnetic field phenomenon, due to the availability of quantitative simulations and predictions for fluctuation timescale and amplitude. However, we note that most existing simulations are run for only 10²–10³ orbits. This implies that other phenomena that produce field variability at timescales much longer than that of chondrule formation, but much shorter than that of secular disk evolution, remain understudied and may also result in similar chondrule paleointensity variations.

Comparisons to paleomagnetic data from other chondrite groups may provide additional insights into the variability of solar nebula magnetic fields. A study of CR chondrites, which may have originated from large orbital radii, revealed that the analyzed chondrules formed in a magnetic field of less than ∼8 μT. However, the younger age of these chondrules, approximately 3.7 Myr after calcium-aluminum-rich inclusion (CAI) formation, complicate direct comparisons to LL and CO data (Fu et al. 2020). As discussed above, the disk accretion rate and magnetic field strength likely remained relatively constant during much of the disk lifetime, although with potential short-term fluctuations inferred from stellar accretion variability (Fischer et al. 2022), and decayed precipitously over several 0.1 Myr during gas dispersal (Simon & Prato 1995; Wolk & Walter 1996). The presence of submicrometer dust in CR chondrites, which are rapidly ejected from a gas-free solar system due to radiation pressure, demonstrates that substantial nebular gas remained until the time of their formation. However, the average CR chondrule crystallization age of ${3.7}_{-0.2}^{+0.3}$ Myr after CAIs overlaps with estimated ages for the depletion of solar nebula gas in at least the inner solar system between 3.8 and >4.0 Myr after CAIs (Schrader et al. 2017, 2018b; Wang et al. 2017). The weaker magnetic fields recorded by CR chondrules compared to LL and CO samples can therefore be attributed to either the secular evolution of the disk or formation in a distinct, likely more distant disk region.

Besides CR chondrules, nebular field paleointensities have been obtained from bulk samples, which are a mixture of fine-grain matrix and chondrule material, of CM and CV chondrites and the ungrouped carbonaceous chondrites WIS 91600 and Tagish Lake (Cournede et al. 2015; Bryson et al. 2020a, 2020b; Fu et al. 2021). These records are likely chemical remanent magnetizations, which necessitate additional considerations before they can be compared to LL and CO chondrule results. Chondrules acquired magnetization as they cooled over hours to days, while aqueous alteration that causes chemical remanence can occur over 1–10⁶ yr timescales, which is typically longer than the orbital period (Hewins et al. 2005; Dyl et al. 2012; Schrader et al. 2018a; Ganino & Libourel 2020). Because the parent body effectively rotates around the normal of its orbital plane during each revolution around the Sun, only the component of the magnetic field normal to the disk plane, referred to as the net vertical magnetic field, is likely to be recorded in bulk samples. MHD models consistently show that the azimuthal component of the disk magnetic field is typically stronger than the net vertical field by a factor of 10 within several scale heights of the midplane (Bai 2017; Béthune et al. 2017; Lesur 2021b), although, as discussed above, the local magnetic flux concentration due to instabilities can lead to au-scale regions of very strong vertical fields (Riols & Lesur 2018). Due to this amplification of the total field compared to the net vertical field, the average chondrule paleointensities are expected to be at least several times stronger than those of the bulk-sample-derived values in the same disk region.

In addition to this complication, the magnetization ages of WIS 91600 and Tagish Lake are not precisely known, as the magnetization-carrying minerals have not been directly dated or firmly associated in an alteration sequence with dated minerals. As such, we cannot reject the possibility that their recorded magnetizations date from the final several 10⁵ yr of the solar nebula, during which ambient magnetic fields have decayed significantly (Wang et al. 2017). Future studies that further constrain the ages of chemical remanent magnetization acquisition in these chondrite groups—using either additional radiometric ages or detailed mineralogical studies that establish the timing of magnetic mineral formation relative to dated phases—would permit more detailed comparisons of these data with other nebular paleointensities.

The nebular magnetization of the CV chondrite Allende is carried by the iron sulfide mineral pyrrhotite, which formed from the sulfidation of troilite and Fe–Ni metal on the parent body (Krot et al. 1998). Fe sulfides in Allende are commonly associated with magnetite, which have been directly dated using the ¹²⁹I–¹²⁹Xe system to 3.0 ± 0.5 Myr after CAIs (Pravdivtseva et al. 2013). Similarly, ages from fine-grained matrix and dark inclusions that host the strongest magnetizations range between 3.0 and 4.1 Myr after CAIs (Swindle et al. 1983; Pravdivtseva et al. 2003; Fu et al. 2014a). Combining these ages, Allende material likely acquired magnetization before the final several 10⁵ yr of the solar nebula.

Allende's strong paleointensity of ≥40 μT can therefore be compared directly with LL and CO results (Fu et al. 2021). Because the chemically acquired magnetization in Allende likely sampled only the net vertical field (see above), these values correspond to total field intensities equal to or potentially greater than the ${106}_{-18}^{+88}$ μT constant field or ${128}_{-11}^{+307}$ μT peak time-variable field associated with CO chondrules. If, as argued above, CO chondrules formed in a zone of local magnetic flux concentration, CV chondrules likely formed in the same or similar zones of enhanced fields. Meanwhile, weakly magnetized meteorites, like CM chondrites, if their magnetization was acquired prior to the end of the solar nebula, may represent formation in a distinct subregion of the outer solar system not subject to local flux concentration. Although the existence of such au flux perturbations implies that ambient field intensity cannot be mapped simply to formation radius, paleointensities recovered from many chondrule groups may be used to establish affinities and provide evidence for formation in a shared disk subregion.

We use a Bayesian model selection and parameter estimation framework to interpret single-chondrule paleointensities from LL and CO chondrites. This analysis is more appropriate for sparse, non-Gaussian data and permits quantitative evaluation of assumptions such as chondrule rotation and ambient field variability. We find positive to strong support for chondrule rotation during cooling, validating previous paleomagnetic studies and suggesting that LL and CO chondrules formed in low-gas-density environments outside of 2 and 4 au, respectively, in an MMSN scenario.

We show that two magnetic field environments are compatible with the chondrule paleointensities: (1) temporally constant magnetic fields, potentially with occasional, rapid reversals, of ${34}_{-14}^{+36}$ μT and ${106}_{-18}^{+88}$ μT in the LL and CO chondrule formation regions, respectively; or (2) time-variable magnetic fields, with peak amplitudes between ${49}_{-21}^{+97}$ μT and ${128}_{-11}^{+307}$ μT in the two chondrule formation regions. In the time-variable magnetic field scenario, the two chondrule groups may have also formed in a single magnetic field environment with peak fields of ${112}_{-20}^{+81}$ μT. The temporally constant field scenario suggests the formation of CO chondrules in regions of locally concentrated magnetic field flux, as modeled by recent MHD simulations. Adopting the time-variable magnetic field scenario, the amplitudes of field fluctuations are consistent with "bursty" magnetic fields driven by the HSI in a disk where the net vertical magnetic field was anti-aligned with the disk rotation vector.

The currently available data are insufficient to distinguish between these constant field and variable field scenarios. However, using the Bayesian statistical framework described here, additional paleointensities from individual chondrules or bulk paleointensities of different meteorites from the same disk region can be combined with existing data sets to explicitly test the degree of time variability in the nebular field.

We thank J. Bloxham and A. Brenner for discussions. We also thank two anonymous referees for comments that improved the manuscript. Support for RRF was provided by the Alfred P. Sloan Foundation. The Python code in a Jupyter notebook for performing the Bayesian likelihood analysis described here is available as a supplementary file. The user can input chondrule paleointensities and any relationship between the recorded paleointensity and ambient field during chondrule formation $[{B}_{\mathrm{rec}}\left({B}_{\mathrm{amb}}\right);$ see the text].