Human migrations and volcanic activity: Archaeomagnetic evidence of the probable abandonment of the Tingambato archaeological site due to the eruption of El Metate volcano (Mexico)

The archaeological site of Tingambato is one of the few evidences of the populations that settled in western Mexico during the Classic and Postclassic periods, before the development of the great Tarascan empire. Therefore, its study is fundamental to know both the characteristics of the culture that preceded the empire, as well as the phenomena that led to its formation. During the last decade, efforts have increased to reveal the history of this site. Thanks to the recent excavations, different archaeological materials belonging to the three construction phases of the place are available, which have served to define their main characteristics. In the present investigation, the magnetic characterization and dating of different archaeological materials belonging to the last occupational stage of Tingambato were carried out using archaeomagnetic methods. Some of these materials show evidence of exposure to fire after their elaboration. According to archaeological investigations, the ancient city of Tingambato was burned before being abandoned, so the ages obtained for four of the analyzed potsherds represent the first available dating for the abandonment of the site. Finally, these ages allow us to propose interpretations about the probable causes that led to its abandonment.


Introduction
Tingambato is one of the five open-to-public archaeological sites in the Michoacán state, Mexico.The human settlements that occupied the place date from the Classic and Epiclassic periods , before that the Tarascan empire, which is recognized as the most important culture of western Mexico (Carot 2005) and that dominated the region.Although local reports (see Punzo 2022 and references therein) suggest that archaeological vestiges in the site were well-known since the nienteenth century, archaeological research began in 1978 (Piña-Chan and Ohi 1982) and continues to the present.Since the pioneer studies, evidence of fire-such as remains of burnt floors, walls, and carbon layers-was found in the layer associated with the last occupation phase; after that affair, no more human occupation records were retrieved.Therefore, it is suspected that such fire (which could be provoked or accidental) was the cause of the total abandonment of Tingambato (Piña-Chan and Ohi 1982;Ohi 2005;Cruz and Landa 2013;Punzo 2016).This phenomenon of abandonment of a place after a fire has been observed in different pre-Hispanic cultures of Mesoamerica (e.g., Kelley 1990;Castañeda-López and Quiroz-Rosales 2004;Valencia-Cruz 2015;Torreblanca-Padilla 2015).
Within the research carried out in Tingambato by the "Archaeology and Landscape of the South-Central Area of Michoacán Project" (PAPACSUM) since 2013 (Punzo et al. 2016;Punzo et al. 2017), six units have been excavated in the archaeological site (Fig. 1a).One of the main purposes of the excavations has been to know the different constructive and occupational stages of the site, their characteristic material, and their corresponding ages.Inside excavation unit 5 (U5 in Fig. 1a), located in the north-eastern area of the archaeological site, additional evidence of the fire above referred was identified; more specifically in the limit between the bottom of layer B and the top of layer C (Fig. 1b), considering that layer A corresponds to the soil on the surface of the archeological site and B, C, D... etc., are increasingly deeper layers.
To associate an age to the fire event through the archaeomagnetic methods, this investigation presents the results obtained through rock-magnetism and archaeointensity experiments realized in distinct archaeological materials of Tingambato, which presents evidence of different degrees of exposure to fire.This is possible thanks to the foundations of archeomagnetism.According to them, the magnetic minerals within the archaeological ceramics are aligned with the geomagnetic field when they are exposed to high temperatures.For example, during the firing of an ancient ceramic, it records the characteristics of the geomagnetic field of that time.The Earth's magnetic field is variable over time and presents different patterns of behavior depending on the different latitudes around the planet; this phenomenon is called secular variation.Therefore, it is possible to compare the characteristics recorded in the ceramics with the properties of the current geomagnetic field and notice differences.Furthermore, if regional secular variation curves are available, it is possible to associate an age to the ceramic (for a detailed description of archaeomagnetism, see Carrancho et al. 2015).
The Earth's magnetic field is analyzed by its three components: declination, inclination, and intensity; each one presenting different patterns of secular variation for a specific temporality.For the ceramics, only one component of the geomagnetic field can be analyzed: the intensity, since the potsherds do not maintain the direction in which they were fired and the recovered inclination and declination of their magnetic particles are not representative of the direction of the ancient geomagnetic field.

Study area
The archaeological site of Tingambato (19°29′ 39.40′′ N, 101°51′ 30.98′′W), also known as Tinganio by the locals, is located in the municipality of the same name, in the state of Michoacán, west-central Mexico.The site reached its largest population size during the Epiclassic period (AD ~ 600-900) and was an important regional center in the Michoacán area before the development of the great towns founded by the Tarascan empire (since AD 1250), which was the last pre-Hispanic culture expanded in the area ahead of the Spaniard conquest.The archaeological site is made up of different structures such as pyramidal bases, a ball court, squares, courtyards, a civic area with multiple rooms, and two tombs (see Fig. 1a), all corresponding to a part of the religious and residential area of the ancient population.
Thanks to the archaeological research carried out at the site during the last decade (Punzo 2016;Punzo et al. 2016;Punzo et al. 2017;Punzo et al. 2019;Rangel-Campos 2018;Pérez-Rodríguez et al. 2021), different materials have been dated using radiocarbon, collagen, and archaeomagnetic methods.The dates, together with the stratigraphic description and the recognized differences in the ceramic and architectural styles, have been useful in defining three occupational stages: Tingambato I (AD 0-300), Tingambato II (AD 300-600), and Tingambato III (AD 600-900).Each phase is differentiated from the others on the site by the presence of fill layers between the floors of occupancy.The presence of charred materials among the stratigraphic layers associated with the Tingambato III phase is abundant.This characteristic has been associated with the burning of the city, after which no further evidence of human occupations has been found during the pre-Hispanic period; therefore, it is considered that the fire marks the end of the Tingambato settlement.
During the archaeological works realized by the PAPACSUM project in 2015, 2016, and 2017, six units were excavated in Tingambato (see distribution in Fig. 1a).The materials analyzed in the present research were recovered from excavation unit 5 (U5), situated in the northeast sector of the archaeological site.For the analysis of the materials obtained, unit U5 was divided into five segments: N050 E007, N048 E007, N046 E007, N044 E007, and N042 E007 (see Fig. 2).The segments are 2 × 2 m, except N042 E007, which has a width of 1 m.According to archaeological investigation, the stratigraphy of unit U5 is described with layers A, B, C (a tamped ground), and D, all of them associated with the Tingambato III stage.The main characteristics of the layers are described below.• Layer A is the most superficial stratum.It has a maximum depth of 38.5 cm and is composed of dark brown silty sediment (7.5 YR 2.5/3 very dark brown) of medium compaction, crumbly consistency, and massive structure.At the end of the layer, orange soil is observed with fragments of burnt soil and some charcoal traces.Ceramic and lithic materials were found in this layer.• The next stratum is layer B. It has a maximum thickness of 27.9 cm and is composed of a matrix of brown soil (7.5 YR 3/3 dark brown) mixed with reddish sediments (2.5 YR 5/8 red) and clods of burned mud.The layer has low compaction, a loose consistency, and a blocky structure.In a general way, the layer was defined as a conglomerate of the burned roof and wall fragments, product of the fire in the city.In this layer ceramic fragments, polished and carved lithic, and traces of ceilings imprints were recovered.Most of the ceramic material is burned.At the end of the layer, a tamped ground is observed, it was also registered in U2 and U4.• Layer C corresponds to a dark-colored tamped ground (Gley 1 5/10Y greenish gray) with a maximum thickness of 84.5 cm.The upper part of the stratum is covered by a coal layer with a maximum thickness of 0.5 cm.The concentration of carbon on the floor is an indicator that it was burned due to the fire that caused the collapse of a roof and walls found on layer B. The tamped sediment is highly compacted, sticky in consistency, and massive in structure.In this layer ceramic, carved lithic material, and imprints of ceilings and/or walls were recovered.• The last stratum described in U5 was layer D. It has a maximum thickness of 2 m, brown coloration (7.5YR ¾ dark brown), silty texture, medium compaction, a brittle consistency, and massive structure.In this layer, only a ceramic fragment and a wooden architectural element were recovered.

Materials selected and sample preparation
For the magnetic characterization, a total of nine potsherds and a roof fragment were selected: seven ceramic fragments and the roof portion came from layer B of U5, while the other two pottery fragments came from layer C. The labels used to identify the analyzed materials are composed as follows: for ceramics, the first letter indicates whether the material shows evidence of the fire, which was associated with the charred coloration of the materials; we used "B" for burned ceramics and "U" for unburned ones.The first letter used for the roof is "R."The second letter indicates the layer from which the materials were extracted, and the number at the end of the label is a sequential number assigned to the materials coming from the same layer.The UB1 potsherd was identified as a Baño Rojo Pulido ceramic style (Fig. 3a), and the UB2 potsherd as a Baño Rojo Escobillado ceramic style (Fig. 3b).The other five potsherds collected in layer B show different degrees of exposure to the fire (Fig. 3c-g), and for the most burned ceramics, the surface characteristics are unrecognizable.Only for the ceramic BB2, the Naranja Bruñido ceramic style could be identified (Fig. 3d).RB1 (Fig. 3h) contains both: remains of carbonized organic matter and organic matter well preserved.The remaining ceramic fragments correspond to potsherds collected in layer C (Fig. 3 i and j).The ceramic style Café claro was associated with the potsherd BC2 (Fig. 3j).
Table 1 synthetizes the characteristics of the ten materials analyzed and their precedence.
All the materials were cut into at least 8 fragments with surfaces of ~ 1 cm 2 , which were compressed into salt pellets with a non-magnetic hydraulic press for later be treated in archaeointensity and alternating field experiments.Before cutting the fragments employed in the intensity experiments, they were marked with a random reference system signalized with a set of parallel arrows on one of the flattening planes of the materials.The fragments were embedded in the salt pellets at different directions (±X, ±Y, ±Z) and the top of the pellets was marked with an arrow in the +X direction as a reference to maintain the alignment of the different fragments.

Magnetic characteristics of the analyzed materials
Small portions of the materials selected were pulverized in an agate mortar.Three hundred milligrams per sample were treated in a variable field translation balance (VFTB) for characterizing their magnetic mineralogy through the isothermal remanence magnetization (IRM), hysteresis, backfield, and magnetization vs temperature (M s -T) experiments.The stability of the magnetic signal, the type and size of the magnetic minerals, and the suitability of the samples for archaeointensity experiments were established.Data were analyzed with the RockMagAnalyzer 1.0 software (Leonhardt 2006) and the MATLAB tool HystLab (Paterson et al. 2018).
In addition, the magnetic components recorded in the samples were determined in one or two pilot specimens of each potsherd with an AGICO LDA 5 demagnetizer using the stepwise alternating field demagnetization (AFD) technique.Maximum fields between 160 and 200 mT were applied to achieve a demagnetization ≥ 90%, except for RB1 specimens, which were demagnetized by applying a maximum field of 40 mT.The analysis of the magnetic components was realized with the Remasoft software by Chadima and Hrouda (2006).

Isothermal remanence magnetization acquisition curves
The IRM acquisition curves were recorded in a magnetic field between 0 and 750 mT.The saturation (> 95%) of the IRM (SIRM) for the potsherds BB4 and BC1 is observed at ~ 250 mT (Fig. 4a), indicating the dominance of low coercivity magnetic minerals of titanomagnetite of single domain (SD) and/or pseudo-single domain (PSD).On the other hand, the main carriers of remanence for the potsherds  Finally, the potsherds BB5, BC2, and RB1 reach saturation at high fields (500-750 mT, Fig. 4c), pointing to high coercivity mineral phases as the main carriers of the remanence.
The medium and high coercivity minerals could signify the presence of minerals of magnetite of SD or PSD grain size with different degrees of replacement with Al, Mg, or Ti.

Hysteresis loops
The hysteresis loops were performed at a maximum field of (±) 800 mT (Fig. 5).The associated magnetic parameters-M s (saturation magnetization), M rs (saturation remanence), H c (coercivity), H cr (coercivity of remanence), and σ (shape; Fabian 2003)-were determined after the corresponding slope and drift corrections.The hysteresis parameters ratios M rs /M s and H cr /H c are displayed in the Day plot (Fig. 5k; Day et al. 1977).Most of the samples are in the PSD region or, according to Dunlop (2002), in the SD-MD (multidomain) (BB4, BB3, RB1, BC1, UB2, and UB1) and superparamagnetic-single domain (SP-SD) (BB1, BB5, BC2) mixture regions.Additionally, it is worth noting that potsherd BB2 has a high M rs /M s value (0.62) with a greater tendency to the SD region.However, one must consider the limitations of using the Day plot for the domain state magnetic mineralogy recognition (Roberts et al. 2018) and compare such information with the M s -T and IRM curves data for an integrated interpretation of the magnetic grain sizes.
The assemblage of the different coercivities of the magnetic mineral fractions was measured using the hysteresis loop shape parameter (σ) of Fabian (2003), for which the socalled wasp-waisted loops (Tauxe et al. 1996) have σ > 0 and the pot-bellied loops σ < 0. The shape parameter is reported in the lower right corner of each hysteresis loop (Fig. 5).Only the potsherds BB2 and BC1 (Fig. 5b and f), which additionally correspond to the potsherds with the higher and the lowest values for the ratio M rs /M s (Fig. 5k), respectively, have a wasp-waisted shape with values σ > 0.9.This reflects a great contrast between the low and high coercivity minerals than in the most homogeneous-coercivity distribution observed in the pot-bellied potsherds with values σ < 0.7.The wasp-waisted shape is commonly associated with the mixture of SD and large SP particles, while the pot-bellied shape is related to the mix of SD and small SP particles.

Magnetization vs temperature curves
Thermomagnetic curves were executed from room temperature until 600 °C in the air.Curie temperatures (T C ) were determined using the Moskowitz method (Moskowitz 1981).The samples were divided into five groups according to the magnetic phases observed in the heating and cooling steps: • Group A: For the specimens BB1, BB3, and BB4 (Fig. 6a-c) are observed three magnetic phases during the heating, while on the cooling stage, only two phases are present.The first heating phase is observed between 200 and 300 °C (high Ti-Titanomagnetite), the second at ~ 400 °C (associated with titanomaghemite due to its thermal instability), and the last is superior to 500 °C (corresponding likely to the Ti-low titanomagnetite mineral phase).The highest and medium mineral phases are present in the cooling process, but the mineral phase with the lower temperature is not reproduced.This set of samples does not show good reversibility; the differences between the heating and the cooling magnetizations for sample BB3 are ~ 60% and 100% for samples BB1 and BB4.
• Group B: Specimens BB2, BB5, and UB2 (Fig. 6d-f) have the same two mineral phases both in the heating and the cooling treatments.A high T C is observed between 540 and 560 °C, corresponding with magnetite or Ti-low titanomagnetite mineral phases.The samples additionally exhibit a medium T C between 300 and 440 °C, which could be related to a titanomaghemite phase in samples BB2 and BB5, possibly produced by the oxidation of the titanomagnetite, and with titanomagnetite in sample UB2.The potsherd UB2 is entirely reversible, while potsherds BB2 and BB5 show a difference of ~ 40% and ~ 20%, respectively, between the magnetizations at the beginning and the end of the experiment.
• Group C: The following set of samples, potsherds BC1 and BC2 (Fig. 6g and h), discloses two magnetic phases during the heating procedure; one at a low temperature (200-270 °C, high Ti-Titanomagnetite), and the other at a high T C (> 540 °C).However, during the cooling, only the high-temperature phase is observed.Potsherd BC1 exhibits a difference of 40% between heating and cooling magnetizations, and the difference is 80% for the potsherd BC2.Both curves are considered irreversible.• Group D: The data obtained for the roof fragment (RB1) shows only one T C during the heating process at 550 °C (Ti-low titanomagnetite mineral phase).Besides, during the cooling process appears a second T C (430 °C), related with an oxided titanomagnetite.The reversibility of the curve is low, with a difference between both magnetizations of ~ 40% (Fig. 6i).
• Group E: Finally, potsherd UB1 (Fig. 6j) shows a high T C magnetic phase at 560 °C (Ti-low titanomagnetite) and a low T C magnetic phase at 140 °C for the heating curve (probably related to the presence of goethite).In the cooling process, the high T C is still observed; nevertheless, the low T C disappears, and a new phase at 400 °C is formed (oxidized titanomagnetite).As in most of the sets described, the reversibility in sample UB1 is poor (the differences in heating and cooling magnetizations are ~ 40%).

Alternating field demagnetization
According to the magnetic component analysis realized with the method proposed by Kirschvink (1980), coercivity spectra of U5 materials is characterized by a main archaeomagnetic component accompanied by a secondary component of probable viscous origin (VRM).This component was erased at maximum fields of 20 mT and represents less than 20% of the total magnetization of the samples (Fig. 7), except for the specimens of the roof fragment (RB1, Fig. 7h), in which almost 80% of the total magnetization was erased at 10 mT.
The total magnetization of RB1 specimens was removed at maximum fields of 40 mT, evidencing that the main magnetization carriers are soft magnetic minerals such as Ti-rich titanomagnetites.
On the other hand, a particular behavior is observed in UB1 pilot specimens (Fig. 7i), where, besides the VRM component, a remagnetization process is appreciated between 10 and 40 mT.Both secondary components reach nearly 50% of the total magnetization, while the other 50% corresponds to the primary component of the sample, acquired during the elaboration of the sample.Additionally, in the pilot samples of samples BB4 and BC1, a high coercivity magnetization component is observed between 120 and 160 mT; this component covers 10% of the total magnetization of the sample.Excluding specimens coming from RB1, the characteristic remanent magnetization (ChRM) of the samples analyzed is well defined, characterized by a primary archaeomagnetic component towards the origin, and carried out by magnetic minerals with coercivities between 30 and 200 mT.
Analysis of the Tingambato potsherds' magnetic properties suggests, for all the samples, a coercivity range that covers the titanomagnetite minerals with different amounts of Ti (SIRM between 250 and 750 mT), which are considered as stable magnetization carriers.In the same sense, the hysteresis results plotted in the Day diagram suggest the predominance of SD-MD and SP-SD mixings grain sizes in the samples.The near-to-zero values of the shape parameter indicate an almost homogeneous coercivity distribution of the grain size in most of the samples.Therefore, a small contribution of the superparamagnetic minerals to the total magnetization is expected.However, although all the samples have a high T C corresponding with magnetite or Ti-low titanomagnetite phases, only the UB2 potsherd show good reversibility at 600 °C.

Archaeointensity results
For the archaeointensity determinations, the IZZI protocol (Tauxe and Staudigel 2004;Yu et al. 2004) was applied.This protocol is a variant of the Thellier methods (Thellier and Thellier 1959), in which the Aitken (Aitken et al. 1988) and Coe (Coe 1967) protocols are combined, i.e., an in-field, zero-field double heating step is followed by a zero-field in-field double heating step.Additionally, the IZZI protocol includes pTRM and pTRM tail check steps (Riisager et al. 2000;Riisager and Riisager 2001) to guarantee the quality of the archaeointensity determinations.
To reach the Curie temperatures of more than 90% of the magnetic minerals present in the samples and gradually replace the corresponding NRM with partial thermoremanences (pTRMs), thirteen temperature steps from room temperature until 560 °C were applied.Archaeointensity determinations were performed in an ASC Scientific TD48-SC furnace (reproducibility of ~ 2 °C between heating steps), and the laboratory field strength was fixed at 40.0 (± 0.5) μT along the salt pallets' cylindrical axis.The pTRM checks were carried out at 100, 250, 340, 400, 460, and 510 °C, while the pTRM tail checks (Riisager and Riisager 2001) determinations were applied at 250, 340, 400, 460, 510, and 560 °C.Once concluded the experimental procedure, archaeointensity determinations were realized with the Thellier-Tool 4.0 software (Leonhardt et al. 2004).
In the paleo/archaeomagnetic studies, it is well recognized that one of the main causes of curve tendencies in the Arai diagrams is the predominance of multidomain grains as the principal magnetic recorders in the samples (Levi 1977;Perrin 1998), which precludes a successful determination of the ancient geomagnetic field.To estimate the curvature (k') of the Arai diagrams, the method proposed by Paterson (Paterson 2011;Paterson et al. 2015) was used with the help of the MATLAB CircleFitByPratt (XY) function (Chernov 2020) based on Pratt's method (Pratt 1987).
Additionally, within archaeomagnetic studies, it is important to consider two additional facts for an adequate intensity estimation: the differences in the magnetization capacity depending on the cooling rate (CR) and the anisotropy effects.The first one was boarded applying three additional heating steps at the end of the IZZI protocol; all of them were carried out at 560 °C, but with different cooling times (Chauvin et al. 2000;Morales et al. 2006).The first and the third cooling took ~ 45 min (like the cooling realized during the steps of the IZZI experiment), while the second cooling was carried out for a longer time (~ 6 h).The magnetization in the specimens after each heating-cooling step was measured with a JR6 spinner magnetometer.The corresponding values of the first and second steps were compared to estimate the cooling rate correction factor (f CRC ).The comparison of the magnetizations acquired during the first and third steps was useful to ensure the thermal stability of the magnetic mineralogy.The cooling rate correction was applied only when the corresponding change in TRM acquisition capacity was below 15% and f CRC > 0 (Morales et al. 2009).
The second fact was addressed using an alternative method to the conventional one.Instead of determining a TRM anisotropy tensor (McCabe and Jackson 1985), which requires six additional heating steps to those already realized with the IZZI protocol and the cooling rate correctionwhich, in turn, could promote higher alteration of the magnetic mineralogy-the approach developed by Morales et al. (2009) was preferred.This technique only requires six specimens embedded in salt pellets in six different directions (see the section "Materials selected and sample preparation") and applying the laboratory field along the pellet's Z-axis during the archaeointensity experiment.Due to the orientation of the specimens in distinct positions, the possible bias generated by the anisotropy remanence is diminished.
Figure 8 presents representative Arai plots of the ten materials studied, along with their respective Zijderveld diagrams (the Arai plots of all analyzed specimens can be consulted in the Supplementary Material).As can be seen, the behavior of the BB4 specimens is characterized by a curved trend (Fig. 8d), and the tendency in its Zijderveld diagram does not go to the origin, therefore unsuccessful archaeointensity results were obtained.In the same sense, the specimens of samples RB1 (Fig. 8h) and UB1 (Fig. 8i) exhibit erratic behavior, probably caused by the low thermal stability of the samples.No archaeointensity results could be determined in these potsherds.The rest of the samples presented a favorable behavior and their archaeointensity determinations were realized.In the BC2 specimens (Fig. 8g), two magnetization components were determined during thermal demagnetization; the primary component associated with the time of the potsherd elaboration, and a secondary component that could be associated with the reheating of the sample during the fire in the city of Tingambato.For each magnetic component, an archaeointensity value was determined with the double-component method of Yu and Dunlop (2002).
To assess the quality of the archaeointensity data, the Thellier-Tool criteria (Leonhardt et al. 2004) modified by Paterson et al. (2014) was employed.The criteria establish two parameters set of different quality: class A (for the best quality data), where the number of Arai plot points used for archaeointensity calculous (N) is ≥ 5, the fraction of the NRM employed for the determination (f) is ≥ 0.35, the ratio of the standard error of the slope and the slope of the NRM-TRM plot (β) is ≤ 0.10, the quality (q) is ≥ 5, maximum angular deviation of NRM directions at each step acquired during paleointensity experiment (MAD) is ≤ 6, α parameter is ≤ 15, and the parameters to quantify the pTRM and pTRM tail checks are δCK ≤ 7, δpal ≤ 10, δTR ≤ 10, and δt* ≤ 9.For class B data, the threshold of the parameter values is more flexible: β ≤ 0.15, q ≥ 0, MAD ≤ 15, δCK ≤ 9, δpal ≤ 18, δTR ≤ 20, and δt* ≤ 99; the rest of the parameters have the same threshold values as in class A. All data with parameter values outside of classes A and B were classified as class C and their values were not used for the estimation of the mean intensity value per sample.Intensity values were determined for specimens of seven of the ten analyzed potsherds: 20 specimens are in class A; 16 in class B, and 12 in class C. Results can be consulted in Table 2.It is worth noting that Table 2 does not report intensity estimates for specimens of samples BB4, UB1, and RB1 due to the erratic behaviors observed in their Arai plots (see Supplementary material).
As mentioned above, the anisotropy mitigation approach proposed by Morales et al. (2009) was used in this investigation.To fulfill the premise of this approach, so that an archaeointensity can be considered reliable, the corresponding value must be obtained considering the six specimensembedded in salt pellets in six different directions-so that the possible bias generated by the anisotropy remanence is canceled out.Accordingly, successful archaeointensity determinations were obtained for specimens BB1, BB3, and UB2, and the primary component in specimens of BC2, these samples were used to archaeomagnetic dating.On the contrary, no archaeointensity results were obtained for specimens BB2, BB5, and the secondary component of BC2.
After evaluating the archaeointensity results with the selected criteria, similar intensity values were obtained for samples BB1 (43.34 ± 3.27 µT) and BB3 (44.36 ± 3.37

Archaeomagnetic dating
Archaeomagnetic dating requires the availability of a high concentration of information on the variations of the geomagnetic field.The data must be concentrated on the temporality of interest, according to the estimated age of the material to be dated, and for a specific area.This information is used to build a reference paleosecular variation curve that represents the behavior of the magnetic field, and the experimental data obtained for the studied materials are compared with the curve to obtain an age.
Ceramics with successful archaeointensity determinations for 6 of 6 specimens (BB1, BB3, BC2, and UB2) were dated with the regional intensity paleosecular variation curve published by García et al. (2021) using the archaeo_dating MatLab tool by Pavón-Carrasco et al. (2011).The use of a regional curve was preferred instead of a global model because global models tend to overestimate intensity values in Mexico, while regional curves present a better fit with intensity values recorded in the archaeological materials from the area (Pérez-Rodríguez et al. 2021).
BB1 and BB3 potsherds (with similar intensity values) give two different time intervals (Fig. 9a and b); the first from AD ~ 0 to 400, and the second from AD ~ 900 to 1200.The oldest temporal range can be discarded as a possible age since the stratigraphic position from which the material was obtained is associated with the last occupational stage of Tingambato (between AD 600 and 900).Therefore, the most probable age for BB1 is AD 936-1194, and for BB3, it is AD 958-1194.For the BC2 primary component, associated with the time of the potsherd elaboration, three temporal ranges were obtained (Fig. 9c).Again, the oldest temporal ranges can be discarded due to the provenance of the material, so the primary component has a most probable age of AD 769-1194.Finally, the most restricted temporal range was obtained for the UB2 potsherd (Fig. 9d).Considering the youngest age of the material as the most probable age, as in the previous materials, UB2 has an age of AD 935-1031.

Discussion and concluding remarks
Even now, after more than half a decade of investigations under the PAPACSUM project, it is hard to assess whether the occurrence of fire was the real cause of the abandonment of Tingambato.As it is well known, and as mentioned previously, the abandonment of a place after a fire has been Fig. 7 Representative vectorial (Zijderveld 1967) plots of unit U5 Tingambato materials ◂ Fig. 8 Representative Arai plots (Nagata et al. 1963) of the Tingambato unit U5 materials analyzed.In g, blue segment represents the adjustment line of the secondary magnetic component, while the red segment is the fit line of the primary magnetic component.The boxes in the Arai plots are their corresponding Zijderveld diagrams.The let-ters in the black circles are the classes used to categorize the quality of the paleointensity results (see details in the main text and in Table 2).For g, two circles were used, one for each magnetic component identified in the sample Table 2 IZZI archaeointensity results for the Tingambato potsherds.T MIN/MAX : minimum/maximum temperature; N: number of NRM-pTRM points employed for paleointensity determination; β: ratio of the standard error of the slope and the slope of the NRM-TRM plot; f: the fraction of NRM utilized; q: quality factor defined by Coe et al. (1978); MAD: maximum angular deviation of NRM end-point directions at each step acquired during paleointensity experiments; α: the angle between the vector average of the data selected for paleointensity calculation and the principal component of the data (Kissel and Laj 2004); δ(CK): difference between the pTRM check and original TRM value at a specified temperature normalized to the TRM; δ(pal): cumulative check error (Leonhardt et al. 2004); δ(t*): normalized pTRM tail (Leonhardt et al. 2004); δ(TR): relative intensity difference in pTRM-tail check; k': curvature parameter (Paterson 2011;Paterson et al. 2015); class: A, B, and C, see explanation in the main text; H RAW : raw paleointensity values; SE: standard error at specimen level; f CRC : cooling rate correction factor (f CRC values < 1 are discarded); H CORR : corrected paleointensity values by cooling rate factor; std: standard deviation at site level.Specimens' names and parameters with values out of the limits settled in the B parameters set are italicized, and their corresponding paleointensity values were not used for mean intensity calculation.Mean intensity values and standard deviations with (*) were calculated with less than 6 specimens; therefore, these potsherds were not dated Name observed in different Mesoamerican pre-Hispanic cultures (e.g., Kelley 1990;Castañeda-López and Quiroz-Rosales 2004;Valencia-Cruz 2015;Torreblanca-Padilla 2015).However, the idea of an incidental fire as a cause of the abandonment of a site seems hard to believe in.Setting fire to a site to be abandoned, whatever the reason for it (e.g., social, or environmental stress), together with the destruction of ornamental and utilitarian artifacts seems to be a more realistic hypothesis.Therefore, instead of trying to find the causes of the fire at Tingambato, estimating the time of its occurrence seems to be more feasible and useful.As mentioned in the section "Materials selected and sample preparation," the materials analyzed show evidence of different degrees of fire exposure.Of the four materials for which archaeointensity values could be successfully determined, fragments BB1 and BB3 present similar degrees of exposure to fire; they have evidence of having been reheated due to the smoky appearance of their surfaces, unlike the BC2 ceramic which seems to have been exposed to a higher temperature due to the blackish coloration that it presents on its surface.On the other hand, the UB2 fragment does not present any evidence of having been exposed to any source of heat after its elaboration.
The reheating suffered by the BC2 ceramic is evidenced in the Zijderveld diagrams elaborated for the analysis of the thermal demagnetization of the samples, where two magnetization components can be observed, and for which their respective intensity values were determined.Unfortunately, for the component associated with the reheating, no reliable intensity determinations could be obtained for the 6 treated specimens, so the average value obtained does not fulfill the acceptance criteria established, as the rest of the specimens, and was not dated.However, the intensity values are very similar in both components, 39.14 ± 2.89 µT for the primary  2), which may be indicative of a short time lapse between the registration of both components.Additionally, thanks to the inflection point associated with the presence of the two magnetization components, it can be inferred that the temperature reached by the sample during reheating was approximately 340 ºC.Among the different potsherds and the roof fragment analyzed in this investigation, those belonging to Layer C would represent a terminus post quem for the events registered above it-particularly for the occurrence of the fire.Similarly, those potsherds coming from layer B would represent a terminus ante quem for the events registered below it-again, for the occurrence of the fire.
Accordingly, the overlapping intervals AD 936-1194, AD 958-1194, and AD 935-1031 (associated with fragments BB1, BB3, and UB2, respectively) set an upper age for the occurrence of the fire, while the interval AD 769-1194 (associated with fragment BC2) sets a lower age for the occurrence of the fire.The stratigraphic position from which the potsherds were recovered, their corresponding intensity values, and their most probable age obtained with the archaeomagnetic method are presented in Fig. 10.As can be observed, archaeointensity values for the three potsherds from layer B are in good agreement within their uncertainty ranges.To set a mean terminus ante quem age for the fire, a mean archaeointensity value of 42.79 ± 1.91µT was obtained for BB1, BB3, and UB2.This value was used to date layer B in the same way as the individual potsherds (Fig. 11a and b).Three temporal ranges were obtained for potsherds of layer B: AD 2-54, AD 86-428, and AD 962-1194, the latter being the most probable (Fig. 11b).
Similarly, a comparison of the ages obtained for layers B and C may be useful to constrain the terminus post quem event, being the youngest age obtained for Layer B the upper limit for the age of the potsherd unearthed from Layer C, i.e., according to the ages obtained in the present study, the terminus post quem fire stage corresponds with the period between AD 769 and 962 (Fig. 11c).In one of the previous works carried out in the archeological site (Punzo 2016), two radiocarbon ages obtained for the temped ground of unit U4 associated with the fire yielded the temporal ranges AD 600-660 and AD 620-670.The organic matter for 14 C dating was obtained from a room beam and a carbon lenticule.Although both age intervals are older than the one obtained with the archaeomagnetic method, the results obtained with both methods are in good agreement considering the upper limit of the radiocarbon age and the lower limit of the archaeomagnetic date.
As mentioned in Pollard (2001Pollard ( , 2005)), evidence of fires was also found in the towns of Urichu and Erongarícuaro, both located ~ 17 km to the northeast of Tingambato, in a period corresponding to the abandonment of Tingambato (AD ~ 900).Additionally, this is an epoch of important social changes in the region, like the increase in the construction of buildings and people concentration in places like the Zacapu swamp and badlands (Migeon and Darras 1998;Migeon 2016), located 35 km north of Tingambato.All these facts seem to be indicative of external factors that forced changes in the population's dynamic.
On the other hand, it is important to consider the geographical context in which the study site is located.Tingambato is situated within the Michoacán-Guanajuato volcanic field, an area characterized by its constant volcanic activity since ~ 5 Ma (see Guilbaud et al. 2012 and references therein), so the landscape is dominated by a great variety of landforms of volcanic origin.To the west of the archaeological site (~ 15 km), the Metate volcano is located (19° 31′ 57.40″ N, 101° 59′ 23.96″ W).This is a shield volcano that, according to the results presented by Chevrel et al. (2016) began its activity in AD 1250 ± 30.Alternatively, Pérez-Rodríguez et al. (2020) proposed the time interval AD 990-1130 for the onset of its eruptive activity-contemporary interval with the abandonment of Tingambato.Field evidence suggests that the activity of this volcano was purely effusive (Chevrel et al. 2016), that is, characterized by very fluid lavas and low gas content.Therefore, it could hardly have had a direct impact on the population of the ancient city of Tingambato, like the total or partial abandonment of the site due to the fall of pyroclastic materials (i.e., bombs or ashes).However, the probable displacement of the population due to the growth of the volcano, the forest fires that it must have generated, and the imminent crisis that this phenomenon brought to society, could have been one of the main triggers for its abandonment.
Worth noting is that the archaeomagnetic dating carried out in this study was realized using only the intensity component of the magnetic field recorded in the analyzed potsherds.The ages obtained present uncertainties greater than 100 years (BB1: AD 936 ± 129; BB3: AD 1076 ± 118), except for the potsherd UB2 (AD 983 ± 48), and the age-restricted in the potsherd BC2 according to its stratigraphic position (AD 865 ± 96).Although the uncertainties are comparable to those obtained with other dating methods, the resolution in the ages obtained could be improved by using the Earth's magnetic field full vector, which, with proper treatment, could be recovered from the burned floor of layer C.
Although the success rate of the archaeointensity study presented here is around 50%, like that of most ancient Earth magnetic field intensity studies (see Calvo-Rathert et al. 2019 and references therein), the amount of data presented covers all the parameters that are currently used to provide greater certainty to the results obtained and to the ages derived from them.This favors the construction of interpretations integrating different phenomena and events recorded during the past with a higher temporal resolution, with which a better knowledge of the history of our ancestors can be achieved.

Fig. 1
Fig. 1 Location of the study area.a Distribution of the main structures of the Tingambato archaeological site.U1-U6 excavation units (modified from Punzo 2016).b Burned soil in excavation unit 5. c Example of the potsherds collected for the present study

Fig. 2
Fig. 2 Excavation unit 5 (U5).a Aerial view of the northeast section of the Tingambato archaeological site.Unit U5 can be seen in the right corner.b Division segments of unit U5.Modified from Punzo et al.(2017)

Fig. 3
Fig. 3 Potsherds collected in unit U5 and selected for magnetic characterization

Fig. 4
Fig. 4 Representative isothermal remanent magnetization (IRM) acquisition curves for the studied materials.a Samples dominated by low coercivity minerals.b Samples dominated by medium coercivity minerals.c Samples dominated by high-coercivity minerals

Fig. 5
Fig. 5 a-j Representative hysteresis loops of the ten materials analyzed.k Day plot with theoretical curves ofDunlop (2002)

Fig. 6
Fig. 6 Representative Ms-T curves of the Tingambato materials studied.Group descriptions are available in the main text

Fig. 10
Fig. 10 Schematic representation of the archaeointensity values of the potsherds with successful determinations and their uncertainties.The provenance layer of the material is labeled on the "y" axis.Below the intensity error bars are the most probable ages obtained by the archaeomagnetic method.The depth at which the potsherds are depicted, and the thickness of the layers are not indicative of their real positions and dimensions; they are presented only for illustrative purposes

Fig. 11
Fig. 11 Combined archaeomagnetic dating for potsherds of layer B. a Intersection of the mean intensity value with the paleosecular variation curve.b Age estimation with its probability density function.c Comparison of the probability density functions for the dating of potsherds from layer B (red color) with the probability density function for the dating of potsherd BC2 of layer C (blue color).The hatched section corresponds to the most probable age for the post quem event

Table 1
Characteristics of the potsherds selected for magnetic characterization µT), which are the highest values reached.While the primary component of the BC2 sample (39.56 ± 2.96 µT) and sample UB2 (40.66 ± 0.75 µT) have intensity values close to 40 µT.

Table 2
component and 41.57 ± 5.33 µT for the secondary component (see Table