The stability of the 430-day period of Betelgeuse

2024 December 9

Mark Kidger

There has been great interest in the claims of a 430-day period in the light curve of Betelgeuse, which is important to the analyses of the fundamental parameters and future evolution of the star. As such, it is critical to study and to validate its reality and long-term stability as a feature of the light curve. While this cyclic behaviour is evident in the published light curves at certain epochs, particularly in the years before its deep 2021 minimum, it is less evident at other epochs. Although there is general agreement that the 430-day period exists, different studies find different values of the exact period and identify other, shorter periods that are suggested to be overtones of the fundamental pulsation period of the star. However, a problem with archival analysis that has not been treated in studies is the extremely heavy weighting of recent data: 81 per cent of data cover just 41 per cent of the length of the archival light curve, while 31 per cent of the data cover just the last seven and a half years of an archive 130 years long. This leads to recent behaviour being erroneously interpreted as fully representative of historical behaviour. Examination of the historical light curve, based on the American Association of Variable Star Observers archive, which includes an important quantity of British Astronomical Association Variable Star Section data, allows both the length of light curve studied and the number of data used to be expanded greatly, with an attempt made to assess the impact of the heavy weighting of recent data. The study presented here shows that, although there is strong evidence of a dominant period of 430±60 days in the light curve over the last 46 years, the period has changed considerably, both in amplitude and in value, while at some epochs other periods have dominated the variations of Betelgeuse.

Introduction

The 2021 minimum of Betelgeuse has created great interest in this star and considerable speculation. Suggestions made at the time in the popular press which echoed the news of the fade (Overbye, 2020),¹ that the star is in an unstable pre-supernova stage, seem to have been unjustified, as the consensus to date has been that it is in core helium-burning and probably as many as 100,000 years from explosion (e.g., Joyce et al., 2020).² However, there is considerable doubt about fundamental aspects of the star and its basic parameters, including its mass, distance, diameter, and evolutionary stage.

At the lower limit of estimates of the mass, it would not be massive enough to initiate core-collapse and would only be able to generate a low-luminosity, electron collapse supernova (Woosley & Heger, 2015).³ At the upper limit, it would be close to being able to generate a black hole as remnant. One reason why modelling its past and future evolution is complicated is that there is some evidence, particularly in the unexpectedly fast rotation velocity of the star, that Betelgeuse may have merged with a binary companion of as much as four solar masses in the last few million years, thus greatly modifying its original state (Wheeler et al., 2017).⁴

The distance of Betelgeuse is quite uncertain, which considerably affects estimates of other fundamental parameters such as its luminosity and diameter. Betelgeuse is at the absolute limit of distance at which, despite modern advances in techniques, we can measure a parallax with ground-based telescopes. A value of 520 light-years was generally accepted for many years (e.g., Moore, 1978), although extremely uncertain.⁵ In the 1990s, ESA’s Hipparcos mission measured the parallax from space, giving a rather smaller distance of 430 light-years (Perryman et al., 1997).⁶ Although Hipparcos was a game changer in stellar dynamics, its data are and were not without issues, with rather large uncertainties; they have hence been largely superseded by Gaia. Unfortunately, Gaia data is severely limited for stars brighter than magnitude 6, which saturate its detectors, and stars such as Betelgeuse that are brighter than magnitude 3 cannot be measured. The 430-light-year distance measured by Hipparcos came with an error of around 20 per cent, meaning that there was a one-in-three chance that the true distance was not even in the range between 350 and 510 light-years. This, in turn, leads to an uncertainty of a factor of more than two in the calculated luminosity of the star, which causes uncertainty in its estimated mass, which causes uncertainty in its future evolution, etc. A later reworking of the Hipparcos data gave a best value of 520 light-years, with a likely range from 450–590 light-years (van Leeuwen, 2007).⁷

Other attempts have been made to determine the distance of Betelgeuse by measuring the parallax with big radio observatories. However, the fact that the apparent diameter is an order of magnitude greater than its parallax complicates this task greatly. To measure the parallax, one must measure a small displacement of the centre of brightness of the star’s disc, which may not correspond to the physical centre. One such attempt was made with the Very Large Array (VLA), giving a best distance estimate of 640 light-years and a likely range from 500–790 light-years (Harper et al., 2008).⁸ Probably, the best estimate to date of the distance is one of Harper et al. (2017),⁹ who combined the Hipparcos and radio estimates into a combined value of 724 light-years, with a one-sigma range from 613–880 light-years.

The light curve

Betelgeuse is a well-known semi-regular variable that is normally in the magnitude range of about 0.3–0.9. Although there are frequent suggestions of a period of ~6 years in the historical light curve, its reality has never been firmly established.

There are various issues with studying the light curve from archival data. First, the errors on the data are rather large (typical dispersion ±0.35 magnitudes), while the normal amplitude of variation is quite small. Second, the sampling of the light curve in the American Association of Variable Star Observers (AAVSO) database is extremely uneven, with the data heavily weighted to recent epochs. Thus, recent light-curve behaviour will have a disproportionate weight in any periodicity analysis. In particular, ~10 per cent of all data in the AAVSO database were obtained during and shortly after the Great Minimum of 2021. Hence, the time around the Great Minimum massively weights any analysis of recent behaviour, and so great care must be taken in any analysis that includes this interval.

A feature of the light curve in recent years is the presence of oscillations that, apparently, increased in amplitude in the years before the Great Minimum. These have been the subject of various studies, giving somewhat contradictory results, albeit in part because of using quite limited datasets, heavily weighted to the years immediately prior to the Great Minimum.

Guinan et al. (2019) find a 430±19-day period in the light curve over approximately the last 25 years.¹⁰ They also find periods of 242 days and 6.06 years in the light curve from 1998–2021. Other authors, though, have claimed that the 430-day period was only strongly present in data from 2016–2020 (Sigismondi, 2020),¹¹ While Jadlovský et al. (2023) find two characteristic periods of 2,190±270 (6.00 years) and 417±17 days, both similar to the main radial-velocity periods.¹²

In contrast, Saio et al. (2023) note that there are four approximate periods in the light curve, of 2,200, 420, 230, and 185 days.¹³ They identify the long period with the fundamental oscillation mode of the star and the three shorter periods as the first, second and third overtones of radial oscillation. They further suggest, from modelling, that the current mass of Betelgeuse is ~11 solar masses and that the original mass of the star was ~19 solar masses: this current mass is towards the lower end of the accepted range of values, but just above the limiting mass found by Woosley & Heger (2015) for a star to become a core-collapse supernova.³ What is particularly interesting in the analysis of Saio et al. (2023) is that they find that,¹³ contrary to the generally accepted view to date, Betelgeuse is in the final stages of carbon burning. Once this phase ends, Betelgeuse would be expected to become a Type IIp supernova within a few decades: in no case would the star be more than a few centuries from supernova if it is now burning carbon.

Here, we use a much longer light curve consisting of 46 years of V-band photometry from the AAVSO database to study the dominant periods in the light curve and their variation, and also attempt to address the impact of the weighting issue to assess to what extent recent data affect the overall results.

The AAVSO database

The database (Figure 1) consists of 46,099 observations made in V, CV and visual (vis.) bands, obtained since 1893 Dec 10 (a total of 40 upper limits and 326 outliers identified by AAVSO have been excluded from this total). An issue that we find immediately is that the visual estimates split into a bright group and a faint group, offset from each other by ~0.5 magnitudes. This is one of the complications in light-curve analysis of this star, although similar trends can be seen in both the bright and the faint data (note that it is the bright group that agrees closely with photoelectric and CCD photometry). Similarly, the dispersion in the data (±0.33 magnitudes) is such that the star’s rather low-amplitude variations are difficult to separate from the scatter.

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