Magnetic Field Signatures of Intermediate-sized Impact Craters on Mars

Magnetic field signatures over impact craters provide constraints for the history of the Martian dynamo. Due to limitations of the spatial resolution of magnetic field models, previous studies primarily focused on large impact craters (mostly ≥ 500 km in diameter). To fill the impact crater age gaps of previous studies, we investigate the magnetic field signature of 23 intermediate-sized craters (150–500 km in diameter) on Mars using both MAVEN data and a magnetic field model. Ten impact craters located in the South Province, the unmagnetized primordial crust, exhibit no or weak magnetic field signatures. The other 13 impact craters produce stronger magnetic anomalies, with the ratio of the averaged magnetic field inside and outside the craters (B in/B out) ranging from 0.4 to 1.2. The B in/B out values exhibit correlation coefficients of −0.54, −0.57, and −0.69 with the diameters of craters, calculated from the MAVEN data, the crustal field model at the surface, and 150 km altitude, respectively. A B in/B out larger than 1.0 usually appears in craters with smaller diameters, which is also demonstrated by the forward modeling in this study. Furthermore, the results of the forward modeling indicate that the craters of stronger magnetizations show a larger B in/B out. According to this, the Martian dynamo can be associated with the magnetization of craters of different ages, and the characteristic time of the dynamo can be limited. Our study supports the hypothesis that the Martian dynamo weakened or ceased at ∼4.0 Ga and a late dynamo was perhaps active at ∼3.7 Ga.


Introduction
Mars is found to have no global magnetic field today, but it does have locally strong crustal magnetization, which indicates the existence of a dynamo in its early history (Acuna et al. 1999(Acuna et al. , 2001)).The Martian dynamo is closely related to the interior compositional and thermal evolution of Mars, of which the duration could provide valuable insight into the mechanism underlying the Martian dynamo (Schubert et al. 2000).Meteorite impact craters and volcanoes are used to place constraints on the timing of the dynamo.Evidence from the ALH84001 meteorite suggests the acquisition of magnetizations in an Earth-like paleofield prior to 4.1 Ga, or possibly 3.9 Ga (Weiss et al. 2002;Steele et al. 2022).The absence of a crustal magnetic field above giant impact basins, such as Hellas, Arygre, Utopia, and Isidis, is interpreted as the cessation of the Martian dynamo when these structures formed (Arkani-Hamed 2004;Mohit & Arkani-Hamed 2004;Lillis et al. 2008;Roberts et al. 2009).Recently, the Zhurong rover took magnetic field measurements on the surface of the Utopia basin and recorded extremely weak magnetic fields on the meter-to-kilometer scale (Du et al. 2020(Du et al. , 2023)).The magnetic field signatures over volcanoes and lava flows indicate the possible existence of the Martian dynamo in the late Noachian or early Hesperian period (Lillis et al. 2006;Langlais & Purucker 2007;Hood et al. 2010;Milbury et al. 2012).However, surface ages of the volcanic region cannot accurately constrain their magnetization ages due to deep magma intrusions (Lillis et al. 2013a).
Impacts reset the magnetization of the crust, making craters ideal magnetic markers for determining the history of the core dynamo (Lillis et al. 2013a).Geological processes associated with impact craters are essential to understand the evolution of the Martian magnetic field.These processes include excavation, shock, and heating, which play significant roles in altering the magnetization of the crust (Melosh 1989;Lillis et al. 2008Lillis et al. , 2013a;;Collins et al. 2012).First, impact excavates a transient cavity and spreads a considerable amount of crust material across the surrounding area.Second, heating can cause thermal demagnetization of magnetic minerals when they are exposed to temperatures exceeding the Curie point.Crustal minerals may acquire thermoremanent magnetization (TRM) if they cooled below the Curie temperature in the presence of an ambient field.Third, shock waves produce high pressure, which can demagnetize the crust or create shock remanent magnetization (SRM) in the absence or presence of an ambient field (Cisowski & Fuller 1978;Gattacceca et al. 2007).
Furthermore, the inversion of the magnetic field measured in orbit to determine the three-dimensional distribution of crustal magnetization is challenged by inherent nonuniqueness (Blakely 1996;Lillis et al. 2010Lillis et al. , 2013a)).Previous studies have highlighted this issue and proposed alternative methods for calculating the magnetization probability distribution, such as the Monte Carlo Fourier domain magnetic model utilized by Lillis et al. (2013b), and the direct derivation of the visible Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.magnetization model by Vervelidou et al. (2017).These analyses of magnetic fields and magnetizations of impact craters with diameters larger than 300 km have inferred that the Martian dynamo ceased between 4.1 and 4.0 Ga.Mittelholz et al. (2020) suggested that the absence of magnetic fields above major large craters could be due to the excavation of a significant fraction of the magnetizable crust material or episodes of weaker dynamo activity.The authors identified magnetic field decreases from low-altitude MAVEN data over a ∼35 km fresh crater emplaced on the magnetized pyroclastic flow of Lucas Planum, indicating an active dynamo at ∼3.7 Ga.
Smaller impact craters located in young or old terrains can provide additional information for the magnetization history of Mars.Previous studies focused on large craters due to limitations in observation altitude and magnetic field model resolution.The MAVEN spacecraft has conducted magnetic field measurements at altitudes as low as 150 km (Jakosky et al. 2015).This makes it possible that magnetic field signatures above craters with the same or a larger scale than 150 km can be identified (Blakely 1996).Furthermore, Langlais et al. (2019) combined Mars Global Surveyor (MGS) magnetometer, MGS electron reflectometer, and MAVEN magnetometer data to build a new model of the Martian internal magnetic field with an improved spatial resolution (the L19 model).With the availability of low-altitude magnetic field data and highresolution magnetic field models, it is now possible to statistically investigate the magnetic field signature of intermediate-sized impact craters (150-500 km in diameter).
In this study, we focus on intermediate-sized impact craters that are unlikely to have excavated the entire magnetization layer.We calculate the ratio of the circumferential averaged magnetic field inside and outside the impact craters (referred to as B in /B out ) and analyze the relationship between the crater diameters and the B in /B out .Furthermore, we use the statistical approach of Lillis et al. (2010) to perform forward modeling of the magnetic field over impact craters emplaced on the random magnetized crust.By combining the isochron model ages (Robbins et al. 2013) with the B in /B out of the intermediatesized impact craters, we aim to provide more details for the possible history of the Martian dynamo.

Crater List, Dating, and Crustal Magnetic Field
Based on the spatial resolution of the L19 model (∼160 km) and the perigee of MAVEN (∼150 km), 50 impact craters are chosen with diameters ranging from 150 to 500 km from the worldwide named craters (Robbins et al. 2013).Their locations are shown in Figure 1.Robbins et al. (2013) mapped the rims of impact craters with diameters 150 km from a global database of Mars impact craters (Robbins & Hynek 2012).The crater size-frequency distributions were estimated for each impact crater by which cumulative crater densities, N (D), were extracted.The model ages of these impact craters were then given by isochron fitting using both the systems of Neukum et al. (2001) andHartmann (2005).We primarily use the Hartmann isochron age for the 50 impact craters in this work.
The magnetic field data in this study are obtained from MAVEN and the L19 model to investigate the magnetic field signatures of impact craters.We use the 1s-resolution MAVEN magnetometer data with altitudes between 140 and 180 km at nighttime.The L19 model provides a spatial resolution of ∼160 km at the surface with order 134.

Magnetic Field Signature above Impact Craters
We classify the impact craters into three categories according to the magnetic signature from satellite data and the L19 model.The first type (type I, black circles in Figures 1(a) and (b)) includes 10 craters, with magnetic field magnitudes 10 nT at 150 km altitude.The type I impact craters are located south of 30°S with a longitude range from west of Hellas basin to Argyre basin, known as the South Province (Arkani-Hamed & Boutin 2012a, 2012b).Table 1 lists these impact craters, together with their center positions (east longitude, latitude), topographic rim diameters, absolute model age, and mean magnetic field magnitude at 150 km altitude within rims |B| mean .The topography map, magnetic field map of MAVEN observations, and the L19 model, as well as the corresponding circumferentially averaged magnetic field profiles of the impact crater Galle, are illustrated in Figure 2. The impact crater Galle, located on the east of the Argyre basin (Figure 2(a)), exhibits weak magnetic fields inside/outside the impact crater at the surface and 150 km altitude (Figures 2(b) and (c)).When MAVEN crossed over Galle, no remarkable change in magnetic signals is captured (Figures 2(d) and (e)).The other nine craters exhibit similar signatures (see Appendix A).
There are 40 impact craters whose magnetic field intensities are much stronger than those of the 10 type I craters.Thirteen impact craters (type II) have stronger magnetic anomalies, and variations of field strength B t and sign of radial component B r at low-altitude MAVEN tracks (Mittelholz et al. 2020) are coherent with the topographies of the impact craters.For the remaining 27 of the 40 impact craters (type III), some have no topography-related orbital magnetic field signatures, possibly due to the influence of other surrounding large-scale magnetic anomaly bodies (e.g., magma intrusions).Furthermore, there are few or no MAVEN nighttime tracks close to some smaller craters, making it difficult to obtain their magnetic field signatures.Lack of magnetic field measurements at lower orbits and lack of knowledge about the craters' subsurface structures mean it is unsuitable to simply use the B in /B out upon the type III craters to represent their impact-related magnetization features.Therefore, we do not consider analyzing the magnetic field signatures of type III craters and using them to constrain the time of the core dynamo in this study.Table 2 lists the properties of the 13 impact craters: center position (east longitude, latitude), topographic rim diameters, absolute model age, and B in /B out , where B in /B out is the ratio of the averaged magnetic field within 0.5 radii distance (B in ) to that between 1.5 and 2 radii distance (B out ).The B in /B out is a proxy of magnetization variation caused by impact (Lillis et al. 2013a).The B in is the averaged magnetic field caused by post-impact magnetization.The B out represents the magnetic field of pre-impact magnetization, which avoids the effect of unrelated demagnetized regions.The magnetic fields of the Antoniadi impact crater are relatively lower than its surroundings (Figures 3(d) and (f)), with B in /B out values of 0.62 (L19 model at the surface altitude), 0.41 (L19 model at 150 km altitude), and 0.47 (MAVEN data).The B in /B out values of Antoniadi indicate that the possible demagnetization is related to the impact process.However, the magnetic field signature of the Becquerel impact crater is different from the Antoniadi crater.The magnetic field intensity close to its center is stronger than that close to its rim from MAVEN data (Figure 4 Although the B in /B out of a crater can reflect the demagnetization associated with impact, it also cannot exclude the influence of crater size.

Relationship between Magnetic Anomaly and Crater Size
Combined with the B in /B out of the 13 impact craters (see Table 2), we find that larger B in /B out values mostly occur in small impact craters, probably because the short wavelengths attenuate rapidly with altitude and small amplitudes close to their center may be not visible from orbit, or small fractions of the magnetized crust beneath the small impact craters are excavated.The ratio B in /B out of the 13 type II impact craters depends on their magnetizations and sizes.Figure 5 plots the B in /B out calculated from MAVEN data (red circles), the L19 model at 150 km (green rectangles), and at the surface altitude (blue diamonds), as well as the linear fitting (lines).There is a statistical decline in the B in /B out as a function of the crater diameters.The correlation coefficients between B in /B out and the diameters of impact craters are calculated to be −0.54,−0.57, and −0.69 based on the magnetic field model at the surface and 150 km altitude as well as the MAVEN data, respectively.This is consistent with previous studies, which concluded that the magnetic fields are weaker over large impact basins than small ones (Acuna et al. 1999;Nimmo & Gilmore 2001).Langlais & Thebault (2011) found that impact craters with diameters >200 km display a slope of −0.23 between magnetic field intensity and impact crater diameters.In this study, we use the B in /B out instead of the magnetic field intensity because the pre-impact magnetizations of impact craters are different and a small impact cannot alter the entire magnetization layer.
We note that the B in /B out values are relatively dispersed for impact craters with diameters less than 250 km, ranging from 0.4 to 1.2.This suggests smaller craters could occur in old (young) areas with the Martian dynamo active (ceased).The errors of the L19 model or MAVEN observations for these small impact craters cannot simply be excluded.Therefore, we perform a forward calculation of the relationship between the interior remanent magnetization (IRM) and B in /B out of impact craters in the next section.

Magnetization Layer Setting beneath the Impact Crater
Forward modeling of the magnetic field of an impact crater (Figure 6) is carried out based on the Fourier domain magnetization model of Lillis et al. (2010).An initial three-dimensional simulation domain and magnetization vector assigned to each pixel with a magnitude randomly assigned from a Gaussian distribution are defined.Negative magnetization indicates antiparallel alignment with the assigned magnetic axis direction.Then the magnetization is Fourier transformed to spatial frequency space and a Gaussian filter is applied to   For our model, the region of impact demagnetization consists of a complete demagnetization region (caused by thermal demagnetization and complete shock demagnetization)  and a partial demagnetization region (caused by partial shock demagnetization).Thermal (de)magnetization is restricted to basin interiors and mostly affects the crust closer to a crater's center (Louzada et al. 2011;Vervelidou et al. 2017).According to the spatial distribution of the shock pressures and temperatures around impact craters (Hood et al. 2003;Mohit & Arkani-Hamed 2004;Arkani-Hamed 2005;Lillis et al. 2013b), we set thermal (de)magnetization and complete shock (de)magnetization below the excavation depth, covering the area within 0.5 basin radii (red area in Figure 6(b)).The excavation depth-to-transition diameter ratio is 0.1 (Melosh 1989;Nimmo & Gilmore 2001;Langlais & Thebault 2011).The partial (de)magnetization region extends vertically to the bottom of the magnetization layer within 1.5 basin radii in the horizontal direction (yellow area in Figure 6(b)).When the dynamo is inactive, the magnetization of the completely demagnetized zone is set to zero, while that of the partially demagnetized region decays with the increasing distance from the impact crater center.When the dynamo is active, the magnetization in the completely demagnetized zone is dominated by IRM (TRM and SRM), depending on the dynamo field intensity.In this case, IRM is set to be 5-40 A m −1 .Meanwhile, the partially demagnetized zone consists of two parts.One is the original magnetization, which increases with the distance from the impact crater center.The other is IRM, which decreases with the distance from the impact crater center.

Magnetic Field Signature at Orbital Altitude
The resulting magnetic field is calculated by summing the vector field of each horizontal layer using the method of Blakely (1996).Then, a 20 × 20 (100 × 100 km) smoothing window is used to match the resolution of the L19 model.Figure 6(a) illustrates the forward result of a completely demagnetized impact crater with a diameter of 500 km.With an increasing altitude, the magnetic field signature associated with the impact crater becomes weak and not visible at 400 km altitude (Figure 6(c)).For smaller impact craters (150-250 km), their larger B in /B out values indicate that their demagnetization signatures are less pronounced than those of larger craters at the same altitude.To study the upper limit of altitude for identifiable magnetic field signatures of smaller impact craters, we calculate the B in /B out of smaller impact craters (150, 200, and 250 km in diameter) with different observation altitudes.Figure 29 in Appendix B shows a rise in the B in /B out as the observation altitude increases.The B in /B out above the smaller craters is larger.For the 250 km diameter craters, the B in /B out of ∼0.5 at 200 km altitude indicates that demagnetization signatures can be observed at this altitude.The 200 km diameter craters have a similar B in /B out at 100 km altitude.The B in /B out of the 150 km diameter craters is ∼0.6 at 50 km altitude and ∼0.7 at 150 km altitude.For impact craters smaller than 200 km, the B in /B out is relatively large but still less than 1.0 at 150 km altitude, meaning that the demagnetization signatures above these craters may be visible at this altitude.
We calculate the B in /B out for impact craters with different diameters and (de)magnetizations.Figure 7 shows the dependence of B in /B out on the crater diameters under different IRMs, which are set at 0-40 A m −1 with an increment of 10 A m −1 .The curves are the mean values of 100 simulations with different crater diameters, and the shaded areas denote the standard deviations.Figure 7 shows a decline in the B in /B out as the crater diameter increases.The different colors in Figure 7 show the effect of IRM, with larger B in /B out as the IRM increases.For impact craters with diameters less than 250 km, the averaged B in /B out exceeds 1 when the IRM is set at 30 A m −1 , while the averaged B in /B out of the impact craters with diameters larger than 400 km is always less than 1, even though the IRM is set at 40 A m −1 .So we cannot simply say that the Antoniadi crater is demagnetized with smaller B in /B out values but the Becquerel crater is magnetized with larger B in /B out values in Section 2.2 because of different diameters.Instead, the B in /B out of these 13 type II craters are supposed to be classified to eliminate the effect of their diameter and compared with the result of forward modeling.

Possible Martian Dynamo Timeline
Magnetic field signatures over impact craters are valuable records of dynamo activity at the time they formed.We now utilize the B in /B out and absolute model age of intermediatesized craters to extend knowledge of the Martian magnetic field history and to establish a relatively clear dynamo timeline (Figure 8).
We simply classify the 13 type II impact craters (Antoniadi, Dollfus, Tikhonravov, Newton, Koval'sky, Kepler, Mutch, Flammarion, Henry, Becquerel, Denning, Savich, and Schoner) into three categories (red, green, and blue in Figure 8), which exclude the effects of diameters on the B in /B out .We calculate the averaged B in /B out values from 100 individual simulations of the forward modeling.The light blue, green, and red shaded areas represent the averaged B in /B out of the forward modeling of impact craters 350, 300, and 150 km in diameter.The upper and lower edges of these areas are calculated when IRM = 10 and 0 A m −1 .Considering the backfilled magnetized ejecta and secondary magnetization, we think that the IRM of 10 A m −1 indicates a weak Martian dynamo during the formation of the impact crater.The areas serve as proxies to determine whether the Martian dynamo becomes weak or ceases during the impact crater formation.
Figure 5.The B in /B out over 13 type II impact craters whose magnetic field signatures are associated with their topography, from different data: modeled B at 150 km altitude (blue diamonds), modeled B at the surface altitude (green rectangles), and MAVEN magnetic field from 140 to 180 km (red circles).The linear regression lines are shown in corresponding colors together with the 95% confidence interval (shaded area).
For the three impact craters in blue, Tikhonravov (model age 4.07-4,19 Ga) and Dollfus (model age 4.01-4.12Ga) have a larger B in /B out than the theoretical calculation over a 400 km impact crater, whereas Antoniadi (model age 3.75-3.88Ga) has a smaller B in /B out .Similarly, focusing on the craters depicted by red, the B in /B out over Kepler (model age 4.00-4.12Ga), Denning (model age 4.08-4.22Ga), and Schoner (model age 4.05-4.18Ga) are larger than the theoretical calculation over a 200 km impact crater.However, the B in /B out of Mutch (model age 3.80-3.92Ga), Savich (model age 3.85-3.97Ga), Henry (model age 3.84-3.99Ga), and Flammarion (model age 3.84-3.96Ga) are smaller (red in Figure 8).The B in /B out value of impact craters formed 4.0 Ga earlier is significantly distinct from that formed 4.0 Ga later.The clear dividing line at ∼4.0 Ga implies that the Martian dynamo may decline or even shut off at this time.This result is consistent with the conclusion of the dynamo cessation by 4.1-4.0Ga (Lillis et al. 2013a;Vervelidou et al. 2017).The B in /B out values of the five intermediate-sized impact craters (Antoniadi, Savich, Mutch, Henry, and Flammarion) are similar to those calculated from the forward modeling with IRM 0-10 A m −1 .Their ages range from 3.93 to 3.83 Ga; therefore, the core dynamo also remained inactive during the period.Besides, the absence of magnetic fields over large basins (Hellas, Arygre, and Isidis) from 3.91 to 3.99 Ga also confirms the inference.
The B in /B out of the Becquerel crater is larger than the B in /B out of the forward model with IRM 10 A m −1 .Its model age ranges from 3.56 to 3.69 Ga with a median of 3.64 Ga.The state of the core dynamo at this period is also unclear.Mittelholz et al. (2020) speculated that an active dynamo may present at 4.5 and 3.7 Ga, which could explain the rise of B in /B out over Becquerel.However, we must note its Neukum isochron age is 3.81-3.91Ga (Table 2).Considering the uncertainty of the crater age and that it is only one crater, whether the dynamo was active at ∼3.7 Ga needs more statistics of young impact craters and/or surface observations in the future (Du et al. 2023).
The B in /B out of Newton (model age 3.89-3.99)is larger and likely magnetized, which is not consistent with the above dynamo timeline.Lillis et al. (2013a) thought that the "soft" rim of Newton has very likely been reworked, so the isochron fitting age could be considered a last resurfacing age.Similarly, the rim of Koval'sky is blurred and heavily eroded, making the  isochron age younger.Further on, the magnetic field signature of Koval'sky from MAVEN shows a clearer decrease and the B in /B out has a small value of 0.19 (see Figure 12 in Appendix A and Table 2).Therefore, the actual B in /B out of Koval'sky is uncertain and possibly between 0.2 and 0.7.
Turning to the type I impact craters shown in Table 1, we observe that their diameters are small (between 150 and 250 km) and their magnetic field signatures are not associated with topographies.The weak magnetic fields may be related to the surrounding source bodies, which overlap parts of the craters but are not affected by the crater formation.The source bodies are likely created and magnetized after the formation of the craters (Arkani-Hamed & Boutin 2012a).The B in /B out values cannot accurately assess the magnetization variation of type I craters.Instead, all type I craters have weak averaged magnetic fields inside and outside less than 10 nT, indicating that the surrounding primordial crust may be unmagnetized (Schubert et al. 2000;Arkani-Hamed & Boutin 2012a, 2012b;Arkani-Hamed 2019).The South Province probably formed in the later stages of accretion of Mars, when the core dynamo was not active because of giant Borealis impact.It took ∼100 Myr to create a strong core dynamo that magnetized the Martian crust (Arkani-Hamed 2019).

Summary and Discussion
In this paper, we focus on the magnetic field signature of intermediate-sized craters, which excludes the possibility that large portions of magnetic minerals have been excavated, as a large crater does (Mittelholz & Johnson 2022).A total of 23 intermediate-sized craters with diameters between 150 and 500 km were chosen, of which 10 exhibited weak magnetic fields while the other 13 had topography-related orbital magnetic field signatures.We observed that the 10 type I craters with weak magnetic fields inside and outside were all located in the South Province, where the crust is ancient and likely unmagnetized at formation.Further on, we calculated the B in /B out of the other 13 impact craters using MAVEN data and the L19 model.We observed that the B in /B out ranged from 0.4 to 1.2, with a decline as the diameter increases.The wide range of B in /B out for the small impact craters (<250 km in diameter) suggests that these impact craters could occur in old areas with the Martian dynamo active or young areas with it ceased.Besides, the decline in the B in /B out as a function of the crater diameter indicates that the B in /B out of intermediate-sized impact craters is associated with the diameter.
We also used a simple forward modeling of the magnetic field (Lillis et al. 2013a) to calculate the magnetic field over impact craters in orbit.Based on the model, the B in /B out is affected by both IRM and impact crater diameter.For smaller impact craters, the B in /B out values are large, even more than 1.The B in /B out becomes large as the IRM increases.The strength of the IRM is mainly affected by the state of the Martian dynamo at the time of crater formation.
Combined with the isochron model ages of the impact craters, we constrained the possible Martian dynamo history: it could decrease or cease at an isochron age of ∼4.0 Ga, and likely restarts or becomes active again at ∼3.7 Ga.
Impact craters with diameters smaller than 150 km are not involved in this study because MAVEN did not observe topography-related orbital magnetic field signatures at that resolution.Furthermore, some impact craters with diameters 150-250 km are not considered in this study because the B in /B out values are around 1.0 at 150 km altitude.It is difficult to determine whether the B in /B out is due to the influence of the internal remanent magnetization or the surrounding large-scale magnetic anomalies.Consequently, the ages of the 13 identified type II impact craters do not cover the entire evolution of the Martian core dynamo.The limited number of available impact craters results in a low resolution when crater Figure 8. B in /B out of 13 type II impact craters in the study, calculated from the L19 model at 150 km altitude (circles) and MAVEN data (diamonds).The impact craters are divided into three categories according to their diameters: 150-200 km (red), 280-320 km (green), and 350-400 km (blue).The light red, green, and blue shaded areas represent the averaged B in /B out calculated from the forward modeling of impact craters 150, 300, and 350 km in diameter, respectively.The upper and lower edges of these areas are calculated from IRM = 10 A m −1 and 0 A m −1 .The horizontal position of these circles and error bars correspond to the best absolute model ages and uncertainties.The text symbols are contractions of the impact crater names listed in Table 2.The isochron age ranges of the Hellas (He), Isidis (Is), and Arygre (Ar) impact basins are marked by the orange bar.The aqua bar highlights the possible dynamo cessation inferred by 4.1-4.0Ga (Lillis et al. 2013;Vervelidou et al. 2017).The purple star suggests a likely active late dynamo at 3.7 Ga (Mittelholz et al. 2020).
ages are used to constrain the Martian dynamo.Due to the absence of pressure and temperature information associated with the impact events, the forward modeling we employed may not precisely reflect the real magnetization of impact craters.Besides, the reliability of B in /B out depends on the L19 model/MAVEN data on craters.The constraints on the history of the Martian dynamo are limited by the altitude of the satellite and the model age of the impact craters.In the future, measured magnetic field data at lower orbits, even on the surface, and more accurate crater magnetization models, as well as reliable model ages of smaller and younger impact craters, are required to consider the magnetic field signatures of more impact craters to constrain the evolution time of the Martian dynamo.
(d)), but opposite from the L19 model (Figure 4(f)).The other 11 craters are shown in Appendix A. The corresponding B in /B out values of Becquerel are 0.66 (L19 model at the surface altitude), 0.96 (L19 model at 150 km altitude), and 1.05 (MAVEN data).Note that the radius of the crater, 165.2 km, is at the limitation of the spatial resolution of the L19 model.The different ratios make it hard to determine the (de)magnetization of Becquerel.

Figure 1 .
Figure 1.(a) Global Mars Orbiter Laser Altimeter (MOLA) topography showing the named intermediate-sized impact craters (solid lines).Impact craters with weak magnetic fields are marked with black (type I).Impact craters with magnetic field signatures related to topographies are shown in red (type II).Other impact craters without consideration in the study are indicated by white.(b) Magnetic field map predicted at the surface altitude (Langlais et al. 2019).

Figure 2 .
Figure 2. MOLA topography of the Galle crater (D = 223.0km).The modeled B t at (b) the surface altitude and (c) 150 km altitude.(d) B t and (e) B r along MAVEN nighttime tracks at 140-180 km altitude close to the Galle crater.The rim of the Galle crater and the distance of two radii from its center are marked by solid and dashed lines, respectively.(f) The circumferentially averaged magnetic field over concentric circles of increasing radius from the center.The three colored curves represent three sources of data.
obtain the simulated crustal magnetization distributions.In our study, we set the random magnetization layer as 2000 × 2000 × 40 km with 400 × 400 × 40 grids.The magnetization polar angle is 45°, while the horizontal and vertical coherence wavelengths are 1000 and 20 km.The magnetization is limited to ±40 A m −1 , which is comparable to the estimates of Mars by Arkani-Hamed (2002) and Nimmo & Gilmore (2001).

Figure 3 .
Figure 3. MOLA topography of the Antoniadi crater (D = 400.8km).The modeled B t at (b) the surface altitude and (c) 150 km altitude.(d) B t and (e) B r along MAVEN nighttime tracks at 140-180 km altitude close to the Antoniadi crater.The B t and B r as functions of distance are indicated by the gray curves along the tracks.The rim of the Antoniadi crater and the distance of two radii from its center are marked by the solid and dashed lines, respectively.(f) The circumferentially averaged magnetic field over concentric circles of increasing radius from the center.The three colored curves represent three sources of data.

Figure 4 .
Figure 4.The same as Figure 3 but for the Becquerel crater.

Figure 6 .
Figure 6.Forward modeling of crustal fields of impact craters.(a) A forward result of a completely demagnetized impact crater 500 km in diameter.The random area is 2000 × 2000 × 40 km with 400 × 400 × 40 elements.The magnetization is between −40 and 40 A m −1 , of which the polar angle is 45°.The calculated magnetic fields at the surface altitude are superimposed on the magnetization distribution.The gray lines represent magnetic lines of force.(b) Schematic of the forward magnetization model profile.(c) The magnetic field profile over the impact crater in the plane y = 0 at three altitudes: surface (blue), 150 km (green), and 400 km (red).

Figure 7 .
Figure 7.The relationship between B in /B out and the diameter of impact craters based on the forward modeling of five interior remanent magnetizations: 0 A m −1 (orange), 10 A m −1 (navy blue), 20 A m −1 (yellow), 30 A m −1 (green), and 40 A m −1 (blue).The solid lines represent the averaged B in /B out of 100 individual calculations while the shaded areas indicate standard deviation regions.

Figure 10 .
Figure 10.The same as Figure 9 but for the Tikhonravov crater.

Figure 11 .
Figure 11.The same as Figure 9 but for the Newton crater.

Figure 13 .
Figure 13.The same as Figure 9 but for the Kepler crater.

Figure 12 .
Figure 12.The same as Figure 9 but for the Koval'sky crater.

Figure 15 .
Figure 15.The same as Figure 9 but for the Schoner crater.

Figure 14 .
Figure 14.The same as Figure 9 but for the Mutch crater.

Figure 16 .
Figure 16.The same as Figure 9 but for the Savich crater.

Figure 17 .
Figure 17.The same as Figure 9 but for the Flammarion crater.

Figure 18 .
Figure 18.The same as Figure 9 but for the Henry crater.

Figure 19 .
Figure 19.The same as Figure 9 but for the Denning crater.

Figure 20 .Figure 22 .
Figure 20.MOLA topography of the Secchi crater (D = 216.7 km).The modeled B t at (b) the surface altitude and (c) 150 km altitude.(d) B t and (e) B r along MAVEN nighttime tracks at 140-180 km altitude close to the Secchi crater.The rim of the Secchi crater and the distance of two radii from its center are marked by the solid and dashed lines, respectively.(f) The circumferentially averaged magnetic field over concentric circles of increasing radius from the center.three colored curves represent three sources of data.(The complete figure set (9 images) is available.)

Figure 21 .
Figure 21.The same as Figure 20 but for the Schmidt crater.

Figure 23 .
Figure 23.The same as Figure 20 but for the Lowell crater.

Figure 24 .
Figure 24.The same as Figure 20 but for the Phillips crater.

Figure 25 .
Figure 25.The same as Figure 20 but for the Green crater.18

Figure 26 .
Figure 26.The same as Figure 20 but for the Darwin crater.

Figure 27 .
Figure 27.The same as Figure 20 but for the Proctor crater.

Figure 28 .
Figure 28.The same as Figure 20 but for the Lohse crater.

Figure 29 .
Figure 29.The relationship between B in /B out in completely demagnetized impact craters and the observation altitude based on forward modeling with 150 km (orange), 200 km (blue), and 250 km (green) diameters.

Table 1
List of 10 Type I Craters with Weak Magnetic Fields, along with the Position of the Center (East Longitude, Latitude), Diameter, Absolute Model Age, and Mean Magnetic Field Magnitude within Their Rims