1 Introduction

Throughout the asteroid belt, a class of objects that may be the remnants of exposed planetesimal cores have been observed. These M-type asteroids, of which 16 Psyche is the largest (Shepard et al., 2017), are generally characterized by their relatively featureless spectra in the visible and near-infrared (Bus & Binzel, 2002; DeMeo et al., 2009; Tholen, 1984) and high radar albedos (Shepard et al., 2008, 2010, 2015, 2017), which may indicate metal-rich surfaces. The masses and bulk densities of many of these bodies remain largely unconstrained, but a few have bulk densities determined to be much lower than that of iron, requiring either a high porosity or the inclusion of a substantial low-density component, like silicate rock. The bulk densities of M-type asteroids 16 Psyche, 21 Lutetia, 22 Kalliope, and 216 Kleopatra are , , , and kg/m³, respectively (Descamps et al., 2008; Elkins-Tanton et al., 2020; Marchis et al., 2021; Pätzold et al., 2011; Sierks et al., 2011). Notably, the estimate for the bulk density of Psyche is lower than the uncompressed densities of Mercury, Venus, and Earth and would require Psyche to be a rubble pile with a bulk porosity, defined as the average porosity of the entire body, of ∼52 vol% if purely composed of iron metal (Elkins-Tanton et al., 2020).

Although generally considered to be the parent bodies for iron meteorites (Cloutis et al., 1990; Magri et al., 1999), this taxonomic group of M-type asteroids appears to have a variety of surface compositions based on their wide range of radar albedos (Shepard et al., 2008, 2010, 2015) and variable appearance and depth of 3 m hydration features (Rivkin et al., 1995, 2000; Takir et al., 2016). Many magmatic iron meteorites show evidence for fractional crystallization indicative of differentiated parent bodies (Scott & Wasson, 1975) and their cooling rates imply a lack of an overlying insulating mantle (Yang et al., 2007, 2008, 2010). Therefore, a commonly invoked formation hypothesis for Psyche and other M-type asteroids is that these bodies are remanent stripped cores of differentiated bodies (Yang et al., 2007). For a Psyche-sized body, stripped cores could be produced by one or more hit-and-run collisions within the ∼1.5 Myr years of the solar system after which the newly exposed cores would continue to cool and could be further fractured by later impacts (Asphaug & Reufer, 2014; Asphaug et al., 2006; Yang et al., 2007). Several Psyche formation mechanisms have largely been motivated by this hypothesis and the unique possibility of an extant exposed planetesimal core in the solar system would provide unparalleled insight into the formation and evolution of planetary cores (Elkins-Tanton et al., 2016, 2020).

Current mass estimates of a handful of M-type asteroids imply low bulk densities in contrast to their high radar albedos (Shepard et al., 2008, 2010, 2015, 2017). These densities are sufficiently low to require bulk porosities in excess of vol% for an iron body. Porosities this large are generally observed exclusively in rocky, rubble pile asteroids, typically between 200 m and 10 km in diameter (Walsh, 2018) while Psyche has a volume equivalent diameter of  km (Shepard et al., 2021). Here, we identify the temperature limits for maintaining highly porous (>40 vol%) pure iron bodies with masses between and  kg by considering the effects of self-gravity and viscous closure of pore space. We show that high porosities can only be supported in an iron body that has previously cooled for a period of at least ∼10 Myr before disruption and reaccumulation as a rubble pile. In particular, a pure iron Psyche must have cooled below 800 K to retain a sufficiently high porosity to match its current bulk density.

2 Thermal Evolution Model

To determine the temperatures at which high porosities could be maintained in an iron body, we use a one-dimensional forward time, central space finite difference model of thermal conduction coupled with porosity evolution for a spherical geometry. The initial temperatures of these models represent the conditions at the time of the event which adds porosity to the iron body. We assume an isothermal initial condition which is a reasonable approximation for the thermal state of a body that was fully disrupted and reaccumulated thereafter. Each model has a constant surface temperature of 137 K, the average surface temperature of Psyche (Sori et al., 2017), and a zero flux central boundary condition. All models use the density of kamacite (7,780 kg/m³; Smyth & Mccormick, 1995) for the metallic iron component following Elkins-Tanton et al. (2020) and begin with a uniform initial porosity

Although porosity, in general, can be removed via plastic failure and/or viscous pore space closure, for a pure iron Psyche, the central pressure will be below 70 MPa which is not high enough to remove any substantial amount of porosity via plastic failure given the yield stress of iron is ∼175 MPa (G. R. Johnson & Cook, 1983). Any less massive iron bodies would have even lower central pressures, further reducing the possibility of pore closure by plastic failure. Therefore, in the absence of plastic failure, the final porosity structure will depend primarily on the thermal evolution of the body. The change in porosity is included as a function of pressure () and viscosity () following Fowler (1985).

(1)

The viscosity of iron is assumed to be Newtonian and vary via

(2)

where the activation energy  J/mol (Frost & Ashby, 1982) and the reference viscosity  Pa s at a reference temperature  K following Abrahams and Nimmo (2019). This reference viscosity is obtained by Frost and Ashby (1982) assuming diffusion creep and a grain size of 0.1 mm. As the true grain size is unknown, models with reference viscosities of and  Pa s are also explored (see Supporting Information S1). For a sufficiently high range of stresses within a Psyche-mass body ( MPa), the deformation of iron may lie in the power-law creep regime (Frost & Ashby, 1982). Hence, with the assumption of Newtonian rheology, we may potentially underestimate the strain-rate at higher pressures where power-law creep is appropriate, which would lead to the removal of less porosity than a more realistic scenario. This underestimation of the compaction rate and noting that the compaction viscosity is not identical to the shear viscosity (Cooper, 1990) suggest our results are upper limits for the temperatures at which porosities can be retained. This conservative estimate for maximum temperatures adds to the robustness of our results.

As the bodies cool, their viscosity increases exponentially following Equation 2. The rate of change of porosity is inversely proportional to the viscosity (Equation 1) and temperatures in our simulations monotonically decrease with time, so porosity evolution effectively ceases when a sufficiently low temperatures are reached, rather than simply becoming very slow. We therefore end models when the rate of change in porosity is less than 0.1% across a one-million-year period. We quantify further loss of porosity to be on the order of vol% and consequently the additional loss of porosity does not affect our results. Cratering events could add or remove porosity as well, however, these effects would likely be localized near craters and hence unlikely to produce the high bulk porosities of vol% that have been inferred from observations (Elkins-Tanton et al., 2020). Accordingly, we do not consider the effects of cratering events on porosity in this model.

Following Bierson et al. (2018), the thermal conductivity varies with porosity and is calculated at each grid point and timestep according to

(3)

which uses a parallel combination of the two conductivities assuming the heat transfer through the pore space is very inefficient and a constant thermal conductivity of iron  W/m K (Carslaw & Jaeger, 1959). This expression for thermal conductivity represents an upper limit of effective conductivities as the true value depends on pore geometry and can decrease more rapidly with porosity than indicated here (Henke et al., 2012). Overestimating the thermal conductivity results in more rapid cooling and enhanced preservation of pore space. Thus, our limits on the initial temperature required for the retention of porosity represent upper limits. Mass is conserved in our model as described in detail in Bierson et al. (2018) such that each grid point responds instantaneously to the contraction of any underlying grid point.

The numerical stability of the model requires that the length of each timestep, , must be less than the Courant number, (Courant et al., 1967). Using Fourier stability analysis, this stability limit for our model is determined to be where is the grid size and is the thermal diffusivity. To remain below this limit, is set as one tenth of the minimum Courant number and recalculated after each step, such that

(4)

3 Results

Using the mass estimate of Psyche from Elkins-Tanton et al. (2020), we model a pure iron Psyche with uniform initial temperatures of 600, 650, 700, 750, 800, 850, and 900 and initial porosities of 50, 60, 70, and 80 vol%. The initial temperatures and porosities are chosen to produce models with final porosities ranging from a few vol%, where most of the initial porosity is lost, up to within 1 vol% of the initial porosity. Initial porosities greater than ∼50 vol% are higher than typically expected, but their use here is meant to demonstrate that cool temperatures are required to produce a final bulk porosity of 52 vol% even when the initial porosities are unrealistically high. Generally, we find that for temperatures greater than 800 K, a pure iron Psyche cannot maintain the high bulk porosities required to match Psyche mass and density estimates. At temperatures greater than 800 K, porosity is annealed on the timescale of millions of years. As an example, the temporal evolution of bulk porosity for a pure iron Psyche with initial temperatures ranging between 600 and 850 K and an initial porosity of 70 vol% is shown in Figure 1. The models with initial temperatures of 600 and 700 K lose less than 2 vol% porosity, while models with the higher initial temperatures of 750, 800, and 850 K lose 10, 31, and 49 vol% porosity, respectively. Between 750 and 800 K, the ratio of internal pressures to the viscosity of iron becomes large enough to remove a substantial amount of the initial porosity.

Figure 2 shows the radial evolution of porosity for the pure iron Psyche models with initial porosities of 70 vol% and initial temperatures of 700, 750, 800, and 850 K. For initial temperatures below 700 K, the initial porosity structure remains largely unchanged (<2 vol%) due to negligible amounts of viscous pore space closure (see for example Figure 2a). For warmer temperatures (Figures 2b–2d), porosity is preferentially removed from the interior of the body where pressures are higher, and for temperatures of 800 and 850 K this results in a final structure where porosity is primarily limited to a porous surface layer of ∼12% and ∼6% of the radius, respectively (see Figures 2c and 2d).

The final bulk porosities for all of our pure iron Psyche models are shown in Figure 3 with a bilinear interpolation applied between model results. Initial bulk porosities are retained within 2 vol% for temperatures of 700 K and below, while warmer temperatures have final bulk porosities over 10–70 vol% below their initial porosities. We therefore find that for the estimated density of a pure iron Psyche to be explained by porosity alone, Psyche must have had a temperature below 800 K at the time porosity was added. The sharp corners in the contour line for 52 vol% porosity shown in Figure 3 are due to the rapid change in amount of porosity lost between models with initial temperatures of 700 and 800 K: as seen in Figure 1, minimal porosity is lost for models with initial temperatures of 700 K and cooler, while the models with an initial temperature of 800 K rapidly lose a large amount of porosity. In general, models which have higher initial porosities retain less of the initial bulk porosity at a given temperature and models with higher temperatures lose porosity on shorter timescales. Final bulk porosities of the models with reference viscosities of and Pa s can be seen in Figures S1 and S2 in Supporting Information S1. In these cases a pure iron Psyche must have cooled below 700 and 900 K, respectively, to retain 52 vol% bulk porosity.

For comparison, we ran similar models for Psyche-mass rocky bodies using the density and thermal conductivity parameters for silicates from Bierson and Nimmo (2019) and viscosity parameters determined for diffusion creep in dry olivine from Karato and Wu (1993). From these models we found that rocky bodies are able to retain high porosity at much greater temperatures, losing less than 1 vol% for temperatures up to the onset of partial melt at ∼1300 K for the range of initial porosities considered. Thus, layers composed primarily of rock should be able to retain substantial porosity at Psyche.

Iron bodies less massive than Psyche will have lower internal pressures and therefore would experience the removal of porosity to a lesser extent than a pure iron Psyche at similar temperatures. Using masses of , , , and  kg, we model smaller iron bodies with uniform initial temperatures of 750, 800, 850, 900, and 950 K and initial porosities of 70 vol% to compare with our pure iron Psyche models. Assuming these less massive bodies have the same bulk density as Psyche results in radii of 18.5, 31.6, 39.8, and 68.1 km, respectively. These masses were chosen to cover the effective diameter range of known M-type asteroids, which range from  km for Asteroid 413 Edburga (Shepard et al., 2015) up to km for Psyche (Shepard et al., 2021), when a high bulk porosity is assumed. There exists a bias in known masses toward larger bodies since their gravitational effects on passing asteroids are easier to observe, but for comparison, the estimated masses of 16 Psyche, 21 Lutetia, 22 Kalliope, and 216 Kleopatra are (Baer & Chesley, 2017; Elkins-Tanton et al., 2020), (Pätzold et al., 2011), (Descamps et al., 2008), and  kg (Descamps et al., 2011), respectively. The final bulk porosities of these models are shown in Figure 4 along with the pure iron Psyche models with the same initial porosity and temperatures and a bilinear interpolation between modeled results. Although high porosities (>40 vol%) can be maintained for temperatures up to 925 K for the least massive case considered, all of the modeled iron bodies must have had temperatures below 925 K before introduction of pore space to retain high amounts of porosity.

4 Implications for M-Type Asteroid Formation

Our models provide insights into formation of M-type asteroids composed of pure iron with a high bulk porosity. First, any surface remanent magnetization produced by an internal core dynamo on these bodies must predate their porosity structure. This relationship occurs because our models show an iron body must cool to at least 925 K to retain high porosities and the Curie temperature, the critical point below which a magnetic field can be recorded, for iron is 1,043 K (Garrick-Bethell & Weiss, 2010). The amount of cooling that iron bodies undergo to reach temperatures below 925 K serves as a lower limit since some disrupting impacts, potential sources of porosity for these bodies, could add heat in addition to adding porosity.

Additionally, the location of the magnetic pole produced by an internal dynamo may not be well preserved for a highly porous iron body since the event that added porosity would have had to occur after the body cooled through its Curie temperature. If a disrupting impact was the source of porosity, the impact may have caused reorientation of magnetized materials and may obscure locations of paleopoles. Although in some cases a disruptive impact could heat materials above the Curie temperature, minimal heat is lost during the process of disruption and reaccretion (Ren et al., 2021) and therefore many of these cases would be too warm to retain any of the gained porosity. However, if disruption was incomplete, the body would likely have higher temperatures at the center of the body accompanying a lower initial porosity near the center of the body than assumed here. In such a scenario, our isothermal models serve as upper limits for porosity retention in these bodies. For iron bodies that have been able to retain high porosity, impact heating would need to be highly localized with global temperatures remaining less than 800–925 K. In these cases remanent magnetization could remain largely intact, but magnetization within the region of highly localized heating could be lost.

Finally, as our models require disruption to occur after the body has cooled substantially, a highly porous iron body is expected to have fewer contractional tectonic features than an undisrupted body. Our models show <2 km of radial contraction in the models with temperatures below 700 K in comparison to 10–50 km of radial contraction in equivalent warmer models (most of this contraction is from removal of pore space instead of thermal contraction alone). The observed magnitude of radial contraction on a porous iron body could provide insights into both the temperature at the time porosity was added and the amount of porosity initially added to the body. More careful consideration of this relationship is left for future work.

5 Implications for the Formation of Psyche

For an intact differentiated planetesimal with a core of the same mass as Psyche, the timescale to cool below 800 K would be on the order of 100s of Myrs (Bryson et al., 2015; Tarduno et al., 2012). If the core was stripped of its overlying mantle, the timescale of cooling to below 800 K may be as short as ∼10 Myr (Scheinberg et al., 2016). The hit-and-run collisions described by Asphaug et al. (2006) occur during the accretion phase of planetary formation, which lasts until approximately 1.5 Myr post-CAI formation (Yang et al., 2007). Alternatively, the catastrophic impacts that disrupted parent bodies of ordinary chondrites occur at or near-peak temperatures for the bodies (Lucas et al., 2020), which are expected to be reached prior to 10 Myr post-CAI formation (Bouvier et al., 2007). As the cooling timescale required by a pure iron Psyche to retain a bulk porosity of ∼52 vol% extends beyond the hit-and-run collision and catastrophic impact epochs of the early solar system, a collision scenario for the formation of a highly porous, pure iron Psyche is not likely. Hence, a pure iron Psyche is not readily explained by a porous metal composition even if it had been previously stripped of its mantle. However, M-type asteroids smaller than Psyche would conceivably cool on timescales much shorter than 100s of Myrs, potentially permitting these small asteroids to be more readily explained as a porous metal bodies. Furthermore, the existence of small M-type asteroids composed of porous metal bodies would be consistent with the observation that these small M-type asteroids are less likely to have the 3 m spectral features indicative of hydrated minerals (Rivkin et al., 2000), which are difficult to reconcile with that of a pure metal body. Smaller M-type asteroids such as 216 Kleopatra ( kg; Descamps et al., 2011) could be remnants of stripped cores while larger bodies like 16 Psyche are more likely to be mixtures of silicates and metal.

If Psyche is a mixture of metal and silicates, it could resemble 21 Lutetia which was observed by the Rosetta mission in 2010 (Coradini et al., 2011; Pätzold et al., 2011; Sierks et al., 2011). Lutetia is an M-type asteroid like Psyche, but in some IR observations a weak 1 μm feature is observed that could be consistent with a chondritic surface composition (Birlan et al., 2006). It is notable, however, that this feature was absent from the Visible and Infrared Thermal Imaging Spectrometer spectra from Rosetta (Coradini et al., 2011). The bulk composition of Lutetia is still generally thought to be metal-rich with CB, CH, CR, and enstatite chondrites as potential meteorite analogs of the surface (Coradini et al., 2011). Although Psyche is also thought have a metal-rich composition, the thermal inertia and radar albedo of Psyche (de Kleer et al., 2021; Shepard et al., 2021) are greater than those of Lutetia (Coradini et al., 2011; Shepard et al., 2010), which could indicate a higher surface metal content for Psyche. If Psyche and Lutetia are observed to have similar structures and compositions, Psyche may therefore have a greater fraction of metal in its metal-silicate mixture than Lutetia.

Overall, our models show that a pure iron body must maintain a temperature of 925 K or below to retain the substantial porosity until present day. Small M-type asteroids could have cooled quickly enough to retain high porosity (>40 vol%). For such high porosities to be possible for a pure iron Psyche, the event which adds porosity must occur at least 10 Myr and 100s of Myrs after CAI formation for a stripped core and an intact planetesimal, respectively, which is difficult to reconcile with our current understanding of solar system evolution. If Psyche is observed to be pure iron, our models can provide predictions for the relative timing of observed near-surface magnetization. In the limiting case of an event adding porosity of 80 vol%, remanent magnetization by an internally generated core dynamo within the top ∼20 km of Psyche would have likely been emplaced prior to the porosity-adding event, even if Psyche was not fully disrupted. The depth to which remanent magnetization likely predates the porosity adding event increases when initial porosities less than 80% are considered.

The combination of metal and silicates could allow Psyche to have a relatively high metal content (∼40 vol%) and a lower bulk density (see Figure 3 in Elkins-Tanton et al., 2020) since rock can support high porosities for warmer temperatures and is an additional low density component. If Psyche is not observed to be purely iron, the existence of some metallic material at the surface in addition to the required lower density component may be the most direct explanation for the high radar reflectivity observed on regions of Psyche's surface (Shepard et al., 2017). Two structures that have been proposed to explain this are that Psyche is a mesosiderite-like body with both silicates and metal exposed on the surface (Viikinkoski et al., 2018) or that Psyche is a differentiated body with a thin rocky mantle that has experienced ferrovolcanic surface eruptions of iron (B. C. Johnson et al., 2020).

Conflict of Interest

The authors declare no conflicts of interest relevant to this study.