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

This paper analyzes the luminosity dispersion of 170 solar-type stars in the open cluster M67 to establish generous upper limits on their long-term intrinsic variability, finding that while current results offer some paleoclimatic relevance, future studies using space-based data from missions like Gaia and Kepler could significantly improve these constraints.

Original authors: Steven R. Spangler

Published 2026-03-25

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CC BY 4.0

Original authors: Steven R. Spangler

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Question: Is the Sun a "Fickle Friend"?

Imagine the Sun as a giant, steady lightbulb in the sky. For a long time, scientists have wondered: Does this lightbulb ever flicker or dim significantly over thousands or millions of years?

Why does this matter? Because if the Sun's brightness changes by even a tiny amount (say, 1% or 2%), it could be the reason Earth's climate swings between ice ages and scorching heatwaves over geological history. We know the Sun has short-term cycles (like the 11-year sunspot cycle), but we don't know if it has "long-term mood swings" that last for centuries or millennia.

The problem is that we can't wait 1,000 years to watch the Sun change. So, how do we find out?

The Solution: The "Solar Twin" Crowd

Since we can't wait, the author, Steven Spangler, decided to look at a crowd of Solar Twins.

Imagine you want to know if a specific person, "Mr. Smith," is naturally grumpy or if he's just having a bad day. You can't wait 50 years to see his whole life. Instead, you go to a high school reunion for people exactly his age, height, and background. If you look at the whole group and see that everyone is smiling, you can guess Mr. Smith is probably happy too. If the whole group is frowning, maybe it's just a bad day for everyone.

The "High School Reunion" in this paper is the Open Cluster M67.

M67 is a group of stars that were all born at the same time, from the same "cloud" of gas, and are roughly the same age as our Sun (about 4.5 billion years old).
Because they are all the same age and type, they should all shine with the exact same brightness.

The Experiment: Measuring the "Fuzziness"

If these stars were perfect, steady lightbulbs, they would all line up perfectly on a graph. They would form a razor-thin, straight line called the Main Sequence.

However, in reality, the line is a little bit fuzzy or thick. The author asked: Why is the line fuzzy?

There are three main reasons for the fuzziness:

The "Double Trouble" (Binaries): Some stars are actually two stars orbiting each other. When we look at them, their lights blend together, making them look brighter than they should be. It's like looking at two flashlights taped together; they look like one super-bright light.
The "Blurry Glasses" (Measurement Error): Our telescopes aren't perfect. Sometimes we measure a star's brightness slightly wrong, just like a blurry photo.
The "Mood Swings" (Intrinsic Variability): The stars themselves might actually be changing brightness over long periods.

The Detective Work

Spangler went through the data of 1,278 stars in M67 and did some serious "culing" (filtering):

He threw out the obvious "Double Troubles" (stars known to be binaries).
He threw out stars that were too young or too old.
He was left with 170 "Solar Twins" that looked like single, steady stars.

He then measured how "fuzzy" the line was. He used a mathematical model to figure out how much of that fuzziness was caused by:

Undetected Binaries: Stars that look single but are actually double (the "hidden couples").
Measurement Noise: The "blurry glasses."
Real Variability: The actual mood swings.

The Results: It's Mostly Just "Blurry Glasses"

Here is the punchline: The fuzziness was almost entirely explained by measurement errors and hidden binary stars.

The "Hidden Couples" (Binaries): The math suggested that about 25% to 45% of the "single" stars were actually hidden binary pairs. This makes sense because finding every single binary is very hard.
The "Mood Swings" (Variability): After accounting for the hidden couples and the blurry glasses, there was almost no room left for the stars to be changing their own brightness.

The author calculated that if the stars are changing brightness, it's a very small amount—less than 6% to 9% of their total light.

What Does This Mean for Earth?

The author compares this limit to the effect of human-caused climate change.

Doubling the amount of CO2 in our atmosphere creates a warming effect equivalent to the Sun getting about 1.7% brighter.
The study found that the Sun (and its twins) could theoretically vary by up to 6% to 9%.

The Verdict:
While the study didn't prove the Sun is wildly unstable, it set a "speed limit." It tells us that the Sun isn't changing its brightness by huge amounts (like 20% or 30%) over long periods. If the Sun is responsible for some of Earth's ancient climate shifts, it's likely doing so with a subtle "nudge" rather than a massive "push."

The Future: Better Glasses

The author admits that this study is limited by the quality of the old data (from the 1990s). It's like trying to measure a hair's width with a ruler.

He suggests that if we use data from the Gaia space telescope (which has much sharper "glasses" and better precision), we could shrink that "fuzzy line" down. This might allow us to finally detect if the Sun has those subtle, long-term mood swings or if it's as steady as a rock.

In short: We looked at a crowd of solar twins to see if they flicker. They seem pretty steady, but our old tools were a bit blurry. With better tools, we might finally know for sure.

1. Problem Statement

The paper addresses the potential for long-term variations in solar luminosity (timescales of centuries to millions of years) to drive terrestrial climate changes observed in the geological record (Phanerozoic Eon). While short-term solar variability is well-understood, long-term variability remains unconstrained. Direct measurements of the Sun are limited to the last few decades, and proxy records (e.g., isotopes) have significant uncertainties.

The core scientific question is whether solar-type stars exhibit intrinsic luminosity variations of 1–14% over long timescales, which could explain paleoclimatic events like the Paleocene-Eocene Thermal Maximum (PETM) without invoking extreme greenhouse gas forcing.

2. Methodology

Data Source and Selection

Target: The open star cluster M67, chosen for its age (~4.0 Gyr), which is comparable to the Sun, and its proximity (890 pc).
Dataset: Photometric data ( $V$ and $B-V$ magnitudes) from Geller, Latham, & Mathieu (2015), originally based on Montgomery, Marschall, & Janes (1993).
Sample Culling:
- Started with 420 stars identified as single members of M67 (spectroscopic binaries were removed).
- Restricted to solar-type stars: $0.60 \le (B-V) \le 0.90$ (Spectral classes G0–K1).
- Further filtered by magnitude difference $|\Delta m| \le 0.90$ relative to a theoretical isochrone.
- Final Sample: 170 stars.

Analytical Framework

Variable Definition: The study defines a variable $x = \Delta m = m_0 - m$ , where $m$ is the measured magnitude and $m_0$ is the theoretical magnitude from an isochrone for the same color.
Broadening Mechanisms: The width of the main sequence in the Color-Magnitude Diagram (CMD) is broadened by four factors:
1. Undetected binaries (blended light).
2. Photometric measurement errors.
3. Differential extinction.
4. Distance dispersion within the cluster.
Modeling Approach: The author fits an empirical probability density function (PDF), $p_x(x)$ $p_{x} (x)$ , derived from the data, against a theoretical model from Spangler (2025a). The theoretical model assumes:
- Intrinsic luminosity variations follow a Gaussian distribution with normalized width $\sigma$ .
- Undetected binaries follow a specific mass-ratio distribution.
- The observed distribution is a convolution of binary contamination and intrinsic variability (plus noise).

Fitting Process

The fit determines two primary parameters:
- $A$ : The fraction of "residual" (undetected) binaries in the cull sample.
- $\sigma$ : The Gaussian width representing the combined variability of primary and secondary stars.
A linear trend in $\Delta m$ vs. $(B-V)$ was investigated and removed in a secondary "detrended" analysis to ensure robustness.

3. Key Results

Parameter Retrieval

Binary Fraction ( $A$ ): The fit yields an acceptable range of $0.25 \le A \le 0.45$ (Best fit $A = 0.353$ ). This is higher than the known binary fraction in M67, suggesting a significant population of undetected binaries or limitations in the model, though it aligns with recent photometric binary studies of other clusters.
Variability Width ( $\sigma$ ): The acceptable range for the total Gaussian width is $0.10 \le \sigma \le 0.135$ (Best fit $\sigma = 0.117$ ).

Decomposition of Variance

The author calculates the expected width due solely to photometric noise ( $\sigma_\phi$ $σ_{ϕ}$ ) using error estimates from the source data.
- Calculated $\sigma_\phi \approx 0.101$ .
Comparison: The retrieved $\sigma$ (0.10–0.135) is statistically consistent with the noise-only value ( $\sigma_\phi = 0.101$ ).
Intrinsic Variability Limit ( $\sigma_{IV}$ ): By subtracting the noise variance from the total variance ( $\sigma_{IV} = \sqrt{\sigma^2 - \sigma_\phi^2}$ $σ_{I V} = σ^{2} - σ_{ϕ}^{2}$ ), the author derives upper limits for intrinsic stellar variability:
- $\sigma_{IV} \le 0.058$ (using best-fit $\sigma$ ).
- $\sigma_{IV} \le 0.089$ (using maximum acceptable $\sigma$ ).

4. Key Contributions

Methodological Refinement: The paper applies a rigorous statistical deconvolution technique to separate the effects of undetected binaries, photometric noise, and intrinsic variability in open cluster data.
Quantitative Constraints: It provides the first quantitative upper limits on long-term solar-type variability derived specifically from the width of the main sequence in an age-matched cluster (M67).
Noise Dominance Demonstration: The study demonstrates that current ground-based photometric data is dominated by measurement errors, which mimic intrinsic variability, thereby explaining previous ambiguities in cluster CMD widths.
Paleoclimatic Context: The derived limits ( $\sim 6-9\%$ ) are compared to the radiative forcing required to explain geological climate shifts (e.g., doubling of $CO_2$ requires $\sim 1.7\%$ luminosity change). The results suggest that while large variations are not ruled out, they are not definitively detected.

5. Significance and Future Outlook

Current Status: The results are not conclusive regarding the existence of 10–14% long-term solar variability. The data is consistent with the hypothesis that all observed broadening is due to measurement error and undetected binaries.
Paleoclimatic Relevance: The upper limits ( $\sigma_{IV} \le 0.089$ ) are of the same order of magnitude as the luminosity changes needed to explain major geological climate events, meaning the question remains open but constrained.
Future Improvements: The author argues that the method is highly promising but requires:
- Higher Precision Data: Utilizing Gaia DR3 photometry, which offers significantly lower noise and better binary discrimination (photometric binaries) than ground-based data.
- Advanced Modeling: Replacing analytic approximations with Monte Carlo simulations to better model binary trajectories in the CMD and differential extinction.

Conclusion: The paper establishes a framework for using open clusters to constrain solar variability but concludes that current data precision is insufficient to distinguish intrinsic variability from noise. Future application of Gaia data could potentially yield definitive detections or tighter upper limits.