Upper Limits to Long-Term Variability of Solar-Type Stars from Observations of the Open Cluster M67

Abstract

Variations in the luminosity of the Sun on timescales of thousands to millions of years could potentially be responsible for terrestrial climate variations in the Phanerozooic geological period (last 540 million years). In this paper, I consider a method that utilizes observations of an open star cluster with approximately the age of the Sun, specifically M67, with data taken from Geller (2015). The idea is to measure the width of the main sequence in the region of solar-type stars, here generously defined to be about spectral class G0 - K1. This width gives an estimate of the dispersion in absolute magnitude of nominally solar-type stars. The sample used consists of 170 solar-type main sequence stars which are not known to be binaries. With this sample, I form an empirical measurement of the width of the main sequence, which is compared with a theoretical expression from Spangler (2025). The measured spread is consistent with a value of sigma, the normalized Gaussian variability of the primary and (if present) secondary star, in the range of 0.100 - 0.135. However, the expected value of sigma from purely photometric noise is estimated as 0.101. Generous upper limits to the intrinsic variability contribution to the inferred width are sigma in the range 0.058-0.089. These limits are not totally devoid of interest in a paleoclimatic context. However, major improvements in the technique are possible with the use of existing data sets from space astronomy missions such as Gaia and Kepler.

Figures (6)

• Figure 1: The Hertzsprung-Russell diagram for all "single" stars in M67 (NGC2682). The abscissa is $(B-V)$ and the ordinate is $m_V$. Blue data points represent data from Geller15 which are known to be cluster members, and not demonstrated to be binaries. The solid red curve is a theoretical isochrone for a cluster with the age and chemical composition of M67, and nominally corrected for distance, absorption, and reddening.
• Figure 2: The empirical probability density function $p_x(x)$ for the solar-type part of the main sequence for M67 (blue data points). The red curve with solid red guide points gives a fit of a theoretical probability density function $p_x(x)$ to the measured function for M67. The model incorporates a Gaussian width $\sigma=0.117$ and a binary fraction $A=0.353$ (defined in Section 4).
• Figure 3: Residual magnitudes $x=\Delta m$ values for stars with $|x_i| \leq 0.30$ in the sample, plotted versus color $(B-V)$. The purpose of this plot is to look for possible trends. The line is a linear regression to the sample, and possesses a slope of 0.376 magnitudes of $V$ per magnitude of $(B-V)$ .
• Figure 4: Slice through $\chi^2_{\nu}$ space for fit to empirical $p_x(x)$ function. This is a slice for fixed $\sigma=0.117$, as a function of the binary fraction $A$. The arbitrary offset parameter $x_{off}=-0.010$. The green, red-dashed, and black-dotted lines correspond to $\chi^2_{\nu} = 1.530, 1.700, 1.811$, respectively, which represent probabilities of 5%, 2%, and 1%.
• Figure 5: Slice through $\chi^2_{\nu}$ space for fit to empirical $p_x(x)$ function. This is a slice for fixed $A=0.30$, as a function of the Gaussian width $\sigma$. The arbitrary offset parameter $x_{off}=-0.010$. The green, red-dashed, and black-dotted lines have the same significance as in Figure 4.