Incorporating Stellar Age into the Current Luminosity-Temperature Hertzsprung-Russell Diagram to Develop a More Comprehensive Model of Stellar Evolution

Word Count: 4153 

Abstract

The Hertzsprung-Russell (HR) diagram is one of the most important tools for studying the evolution of stars. However, the HR diagram is limited in that it is only able to predict a star’s phase based on its stellar luminosity and temperature, and is therefore inaccurate when attempting to predict other properties of a star (e.g. mass or age). This study serves to develop a new method that incorporates the parameter of stellar age to the HR diagram to more accurately predict the life of a star. Using Python programming code, a 2-dimensional multivariable polynomial regression model was developed to accomplish this goal, which yielded a R2 score of approximately 53.5%. Thus, around 53.5% of the variation in the predictions made by the model is accounted for by the natural model structure, which is relatively high for predicting such chaotic entities such as stars. This new simple yet accurate model for predicting the age of stars given their luminosity and temperature serves to provide a new way to improve the HR diagram and thus our understanding of stellar evolution.

Keywords: Hertzsprung-Russell diagram, Luminosity, Temperature, Age, Polynomial Regression

Introduction

The Hertzsprung-Russell (HR) diagram is one of the most important tools in the study of stellar evolution. As evident in its name, the HR diagram is the (indirect) combined effort of two astronomers: Ejnar Hertzsprung and Henry Norris Russell. Developed in the early 1890s by Danish astronomer Ejnar Hertzsprung and American astronomer Henry Norris Russell, the HR diagram plots the temperature (a physical quantity that measures the level of thermal energy in a body) or spectral type (the classification of stars based on their spectral properties such as temperature and color) of stars against their luminosity (the total quantity of light energy produced per unit of time by a celestial object) or absolute magnitude (the measure of the luminosity of a celestial object based on an inverse logarithmic astronomical magnitude scale) (Devorkin, 1991). Astronomers tend to use the HR diagram to summarise the evolution of stars or investigate the properties of stellar clusters. Every star goes through a certain set of evolutionary stages determined by its internal structure and its production of energy—whether the star’s fuel-burning process (e.g. PP Chain—the proton-proton chain reaction is the set of nuclear fusion reactions by which stars with mass less than or equal to that of the Sun convert hydrogen to helium—or CNO Cycle—the carbon-nitrogen-oxygen cycle is the set of nuclear fusion reactions by which stars with mass greater than 1.3 solar masses convert hydrogen to helium) or metallicity (the abundance of all elements more massive than helium in a star). The power of the HR diagram lies within its ability to outline the life of a star through its four main evolutionary phases: main sequence, giant, supergiant, and dwarf (Swinburne, 2020).

Description of the HR Diagram

The HR diagram contains four main regions: the main sequence branch, the giants branch, the supergiants branch, and the white dwarfs branch (Swinburne, 2020).

Figure 1: Hertzsprung-Russell Diagram. The x-axis represents the surface temperature of a star, and the y-axis represents the luminosity of a star with respect to the sun (units are in solar luminosities)

As seen in Figure 1 above, the main sequence branch runs from the bottom right corner (location of cool, faint stars) to the top left corner (location of hot, luminous stars). The main sequence phase is where stars spend 90% of their lives, and at this point, the star is relatively small, with hydrogen fusing in its core and maintaining hydrostatic equilibrium (the state of a star at which the outward-directed internal pressure balances out the inward-directed gravitational pressure). (Dluzhnevskaya, 1967). 

Both the giants and supergiants branches are approximately located in the rightmost two-thirds of the top half of the HR diagram. Giant and Supergiant stars form once a main sequence star exhausts all the hydrogen fuel in its core, during which the core collapses because of the lack of burning fuel and the outer shells of the star rapidly expand outwards. When stars are in this phase of their evolution, they have low surface temperatures, high luminosities, and large radii. (Dluzhnevskaya, 1967)

The white dwarfs branch covers an arch at the bottom left of the diagram. White dwarves are mainly composed of electron degenerate matter, and form once a giant or supergiant star sheds all of its outer layers through fusion. These stars have high surface temperatures, low luminosities, and small radii. A dwarf star is considered to be essentially the final stage of the vast majority of stars. (Dluzhnevskaya, 1967)

Literature Review

Despite the HR diagram’s usefulness in modeling stellar evolution, it does not consider other factors such as a star’s mass or age. In order to improve the comprehensiveness of the HR diagram, many researchers have sought out to develop other models such as the spectroscopic Hertzsprung-Russell diagram (Langer & Kudritzki, 2014), multidimensionally colorized HR diagram (Eyer & Mowlavi 2008), or the BaSTI stellar evolution isochrones (Hidalgo et al. 2018).

One such model, the sHR (spectroscopic Hertzsprung-Russell) diagram, “shows the inverse of the flux-mean gravity versus the effective temperature. Observed stars whose spectra have been quantitatively analyzed can be entered in this diagram without the knowledge of the stellar distance of absolute brightness” (Langer & Kudritzki, 2014). Langer and Kudritzki (2014) therefore suggest that the star evolution of intermediate and low mass stars at constant mass can be accurately predicted without the use of the absolute magnitude of a star, and only its temperature. Therefore, the sHR diagram is a modification of the HR diagram involving removing a variable that allows researchers to simplify the process of predicting the life of stars.

Another group of useful variations to the HR diagram is one specializing in pulsating variable stars (stars whose brightness changes constantly because of their environment or innate rotation caused by magnetic fields), for which multiple extra-dimensional variations of the HR diagram were developed—as opposed to the sHR, a lesser-dimensional variation of the HR diagram. Furthermore, when interpreting the HR diagram, “the interstellar extinction, metallicity, duplicity, and rotational effects may also affect the morphology (a study of the forms of things) of the HR diagram” (Eyer & Mowlavi, 2008). Thus, these stellar components can also be added to the HR diagram to more deeply categorize and analyze the behaviors of stars. For example, when studying the properties of variable stars in the HR diagram, “a third dimension can be added… by using colors to scale the third parameter… [such as] the fraction of variable stars, the amplitude of variability, or its period” (Eyer & Mowlavi, 2008). Adding a color scale as a variable to the HR diagram can prove to be visually effective in describing stars according to their rotational properties; it would therefore be possible to predict a star’s age or stellar phase based on its speed of rotation. Furthermore, “[w]hen the absolute magnitude [, which corresponds to stellar luminosity] cannot be obtained, we can still work on a colour-colour diagram… [although they] are more degenerate than the classical HR diagram” (Eyer & Mowlavi, 2008).

A final and more recent approach to expanding upon HR diagrams are the updated 

BaSTI (a Bag of Stellar Tracks and Isochrones) stellar evolution models and isochrones (lines on a diagram or map connecting points at which something occurs or arrives at the same time) for the HR diagram. The researchers involved in this project have found a comprehensive array of new models that are able to discover several correlations among the properties of stars: temperature, absolute magnitude, metallicity, mass, radius, and luminosity. These models are able to cover all types of stars, from pre-main-sequence to white dwarves. The main uses of their very comprehensive stellar evolution models are to more accurately categorize new stars into more advanced evolutionary phases (e.g. early, middle, to late red giant phase), to discover a more accurate way to determine the ages of star clusters (large groups of stars), and to find a more comprehensive modification of the HR diagram that encompasses more properties of a star. The creators of this set of models “believe that this updated BaSTI release will be an important tool to investigate field, cluster, Galactic, and extragalactic stellar populations” (Hidalgo et al., 2018). However, the researchers also project that a new model for the HR diagram that involves the age of stars is needed.

Proposed Research Topic

Overall, the current 2-dimensional diagram is relatively limited in terms of representing the trends of stars of different ages, as little research has been done on the specific modification to the HR diagram (Eyer & Mowlavi, 2008; Chiosi et al., 2003). Adding the parameter of age (in addition to luminosity/absolute magnitude and temperature) to the HR diagram could prove very useful in categorizing stars in various stellar evolution phases because these parameters are the main determiners of the life of a star. Thus, the analysis of isochrones or stellar clusters while accounting for stellar age could be very helpful in developing a more complete stellar evolution model diagram, as suggested by Hidalgo et al (2018). 

Overall, the Hertzsprung-Russell diagram is an essential diagnostic diagram for stellar structure and evolution that has been used by astronomers for over a hundred years (Swinburne). The previously aligned models—sHR diagram, specialized HR diagram for pulsating stars, and the BaSTI stellar evolution models—all provide new ways to interpret the HR diagram, whether it be increasing or decreasing its complexity. However, there still lacks a detailed model of portraying the relationship between age, luminosity, and temperature of a star. A more concrete model of the age of stars as they transition through their stellar phases is needed to understand the way luminosity and temperature of a star variate throughout its life—thus developing a more comprehensive understanding of stellar evolution. Therefore, this research paper explores the research question: how can stellar age be incorporated into the current luminosity-temperature HR diagram model to develop a more comprehensive model of stellar evolution between main sequence and red giant stars?

Methods

Method Design

The purpose of this study was to determine whether a modified HR diagram and model with a third parameter—stellar age—would be more useful for astronomers than the current version. The units of analysis for this study are the main sequence dwarf stars and red giant stars, which all evolved from stars of similar mass to the sun (between 0.5 and 4 solar masses). The reason why only main sequence dwarf stars and red giant stars are being analyzed in this study is that there is too little data collected on white dwarf stars and black holes; similarly, the researcher lacks the necessary knowledge to be able to account for the chaotic behavior of those two stellar phases. The chosen research type is a secondary data analysis conducted on the temperature, luminosity, and age of the stars that are categorized within the unit of analysis.

The secondary data analysis method was chosen for three main reasons. Firstly, I was unable to accurately collect stellar data myself, as I lacked access to the instruments required to do so. Secondly, this method allowed me to analyze a higher amount of data and therefore obtain more comprehensive results. Finally, this method enabled more flexibility to choose the specific unit of analysis so that it was possible to analyze only Sun-like stars, accounting for both mass and metallicity.

Data Acquisition

The stellar data analyzed in this study was obtained from the Vizier database, which is the largest catalog used to perform research on celestial objects. It contains a myriad of public surveys on astronomical objects ranging from asteroids, stars, and galaxies. For this research paper, the data on main sequence dwarf stars were obtained from the Geneva-Copenhagen Survey of Neighbourhood III (Homberg et al., 2009). These main sequence stars have similar properties to the sun in terms of metallicity and mass. Furthermore, the data on red giant stars were acquired from the Seismology and Spectroscopy of CoRoGEE red giants (Anders et al., 2017). According to the survey specifications, these red giant stars were predicted to have evolved from stars that were of similar mass and metallicity to the sun. These two surveys were chosen because they both contained data of stars that are currently sun-like or used to be sun-like before evolving into a red giant. All the data collected and analyzed in this research study were of stars located in the Milky Way. Both datasets were downloaded from the website as CSV files and compiled into a single file containing all of the data on main sequence dwarf stars and red giant stars: temperature, luminosity, and age. 

Procedure

Searched for, found, and downloaded public data from the Vizier database.
During this process, the bubbling selection function on the database was utilized to filter the data into only observing the luminosity, temperature, and age of the stellar data.
Used Python programming (an interpreted, high-level and general-purpose programming language that is very easy to learn and is commonly used for machine learning tasks) to compile the data onto a single file.
Used Python programming to graph the data onto a three-dimensional space to visualize a possible new model for the HR diagram.
Within this research study, the data was separated from the main sequence dwarf stars and the red giant stars by color within the graph.
Implemented a machine learning algorithm—Multivariate Polynomial Regression (Multivariate Polynomial Regression: a supervised machine learning algorithm that involves multiple data variables to make predictions)—to generate a mathematical model that can accurately predict the age of a sun-like star, given its temperature and luminosity. The Scipy (a core package that provides many user-friendly and efficient numerical routines and techniques) and Scikit-learn (a core package that provides many efficient tools for machine learning and statistical modeling) Python libraries were utilized to build, tune, and test the model.
A 2-dimensional polynomial regression model was developed because of the computational power of the researcher’s computer, as well as to minimize the complexity of the model in order to increase the applicability of the model.
Used the Lasso method (a regression analysis method that performs both variable selection and regularization to improve the prediction accuracy of a machine learning or statistical model) for the regularization (a technique used to decrease the variance of the model, which thereby increases the overall prediction accuracy) of the machine learning model and to account for the vast difference between the nominal magnitudes of temperature and luminosity. The reason why regularization is needed is that the large difference in magnitudes between temperature and luminosity may develop bias within the model.
Evaluated the accuracy of the developed model by evaluating its R2 score (a statistic that provides some information about the goodness of fit of a model; how well the regression predictions approximate the real data points).
Limitations of the Research Method

The limitations to the research method performed within this research study are the time frame, the availability of the data, the researcher’s knowledge of the subject of stellar evolution, and the computational capacity of the researcher’s computer. Firstly, the researcher had less than a year to formulate, develop, and conduct the study, which may have limited its comprehensiveness. Secondly, the researcher’s access to professional telescopic instruments was limited, so they had to resort to searching for data on an expansive, albeit limited database. Thirdly, the researcher does not have especially extensive knowledge on the subject of stellar evolution and stellar processes, so not all stellar phases could be analyzed. Finally, the computation capacity of the researcher’s computer limited the potential accuracy and complexity of the developed age-determining stellar evolution model.

Results

Figure 2: Dot plot of the data on 1436 sun-like main sequence dwarf stars and 606 red giant stars based on their luminosity, surface temperature, and age. Temperatures are given in kelvin, luminosities are given in zero-point luminosities (the luminosity of a star whose absolute bolometric magnitude is 0; 3.0128 ⨉ 1028 Watts), and stellar ages are given in gigayears (billions of years).

Firstly, a dot plot of the stellar data was constructed to obtain an initial understanding of the data, as displayed above. The main sequence stars were indicated in the orange color, while the red giant stars were indicated in the red color. When considering the main sequence dwarf stars and the red giant stars separately, red giant stars tend to have lower temperatures than main sequence stars. Furthermore, the majority of stars in either group have a stellar age of 10 billion years or younger, especially within the red giant group.

Overall, a trend within the overall data seemed to stand out: as luminosity decreases and temperature decreases, the age of stars increases. To further substantiate this observation, a Multivariate Polynomial Regression model method was utilized. 

Figure 3: Surface plot generated by the Multivariate Polynomial Regression model, along with the raw data in Figure 1.

Using this machine learning method, a model for the transition between the main sequence star and the red giant star was developed, extending off of the HR diagram by using age as an external parameter. The model is represented by the following equation:

A=max(0, 9.72773101-(2.5049210-2) T-(3.48992101) L -(4.3842910-3) T2

      +(3.07793) L2),

where A is the age in billions of years, T is the temperature in kelvin, and L is the luminosity in zero-point luminosities. When adjusted for scaling, the luminosity of a star is observed to be a more significant factor in predicting the age of a star. 

Furthermore, the score of this model was R2=0.535076, meaning that 53.5% of the variability between temperature, luminosity, and age was accounted for in the model described above. Considering the variability of the data used within the study and the unpredictability of stellar evolution, this R2value was substantial, and suggests that the model is relatively accurate.

Discussion

Raw Data

Firstly, the raw data itself accurately aligned with previous research done on the different phases of the HR diagram—specifically, the main sequence track and the red giant track. The pattern of main-sequence dwarf stars tending to have higher temperatures than red giant stars was consistent with DeVorkin’s (1991) study on the HR diagram and the differentiation of the different phases of a star, which asserted that the increased density within main sequence stars causes them to have higher temperatures than red giant stars. Furthermore, the higher luminosity within red giant stars supported the research done by Basu (2020), which investigated the dynamics of red giants by analyzing their asteroseismology (the study of oscillations in stars). Their research found that the luminosities produced by red giants are intrinsically luminous from their inner convection, which causes them to be more luminous than main-sequence stars. Finally, the approximate ages of either star types within this research study corroborated with Hasselquist’s study on the Stellar Age Distribution of the Milky Way, which suggested that the vast majority of stars in the Milky way are between the ages of 2 and 10 billion years old. Thus, overall, the data collected supported previous research on stellar life and more specifically, the HR diagram, because it is able to accurately predict both the temperature and luminosity, given the phase of the star. The overall validity of the data in this study was therefore established because it aligned with previous research done in the area.

Analysis of Research Model

After establishing the validity of the data, the research model was analyzed and evaluated. In order to answer the research question, a mathematical model of stellar age was produced to obtain a better understanding of stars. The predictions of the age of stars are consistent with similar previous studies on stellar age determination such as Mowlavi’s (2008) study and McBride’s (2020) study. Mowlavi’s research on stellar age used Geneva models (a set of stellar evolution models specialized in predicting the age of high-rotation stars), which are very computationally advanced. Similarly, McBride’s proposed machine learning models on predicting stellar age contain multiple layers of a neural network and are therefore very computationally intensive. However, unlike those studies, the model proposed in this study only scales up to the second degree, or a quadratic polynomial. The multivariate polynomial regression model, with Lasso regularization, aligned above in the results section, therefore, has relatively lower complexity than previous solutions, with little caveats in terms of error because this model was able to obtain a R2 score similar to the models developed by Mowlavi and McBride (53.5% vs 55.46% and 58.34%).

Evaluation of the R2Score

The score of the machine learning model, or the correlation coefficient squared value, is relatively high, at 0.535076, which suggests that the overall model is relatively aligned with the data. Specifically, the score indicates that 53.5076% of the variance in the relationship between stellar temperature, luminosity, and age is accounted for within the model. The variability in the data can be explained with the deviations of metallicity and stellar mass, aligning with Lorenzo-Oliveira’s (2016) research that developed a model relating a star’s age, mass, and metallicity; their research suggests that stellar age and metallicity are positively correlated, as well as stellar age and mass. Just as the model developed in this study largely agrees with Lorenzo-Oliveira’s model, the multivariate polynomial regression model developed here is slightly more accurate. Thus, the model proposed within this study is relatively precise and has strong potential for future applications, the most prominent one being providing the third parameter to the HR diagram. 

Overall, the model developed within this research paper can act as a direct extension of the HR diagram. Instead of a two-dimensional diagram, represented by the HR diagram, a three-dimensional model can be used to help better predict the evolution of stars by accounting for not only temperature and luminosity, but also stellar age. Directly aligning with the BaSTI stellar evolution models and isochrones stacking for HR diagram, the stellar evolution model proposed within this paper offers a modified and simplified model for estimating the phases of stars: specifically, the main sequence or red giant phases.

Beyond expanding upon the HR diagram, the proposed stellar age prediction model in this research paper has several other potential applications. One of the other prominent ones is using stellar age as a current scientific clock. Specifically, it can help predict the ages of their host galaxies and globular clusters through aging red giant and white dwarf stars (aligning with the use of the HR diagram), and even the age of the universe by faint white dwarves. There is potential within the model developed within this research study, and it may be able to help not only develop a more comprehensive HR diagram, but also a better understanding of our universe.

Conclusions

Research Topic and Method

The overall purpose of this research paper was to expand upon the Hertzsprung-Russell diagram using extra parameters to improve stellar evolution predictions. Similarly, this research paper was meant to help develop a more comprehensive understanding of the stellar evolution phase between main sequence stars and red giants. Therefore, this research addressed the following research question: how can stellar age be incorporated into the current luminosity-temperature HR diagram model to develop a more comprehensive model of stellar evolution between main sequence and red giant stars?

A secondary data analysis study was conducted to develop a new model to predict the age of stars that are currently in the phases of the main sequence, red giant, or in transition between. The algorithmic method of determining a new model of stellar evolution involved using a Multivariate Polynomial Regression algorithm of the second degree to predict the age of the star, given its temperature and luminosity as parameters.

Results and Discussion Overview

This research paper made several connections to current literature, as well as corroborated with previously established ideas. Firstly, the raw data exudes the following trends: main sequence dwarf stars tend to have higher temperatures than red giant stars; red giants generally have higher luminosities than main-sequence stars; and the vast majority of stars have stellar ages between 2 and 10 billion years old. Secondly, the multivariate polynomial regression algorithm presented in the paper was able to obtain a relatively high score (R^2 value), suggesting that it is capable of making relatively accurate predictions of stellar age. Furthermore, upon being extrapolated to 9879 main-sequence and red giant stars surveyed by Piskunov (1980), the model was capable of estimating their ages within an average R2score of 50.4379%. Finally, the overall results accurately align with previously established stellar evolution isochrone models as well as other stellar age predictors.

Beyond other models and research, this algorithm developed in this paper offers a more accurate and novel way to estimate the age of a star with similar mass and metallicity as the sun. At the same time, this research’s use of computational analysis and machine learning models is not without weaknesses and indicates that more research is needed within the area of stellar evolution modeling.

Limitations

Despite the strong results, there were several limitations involved within this study. Firstly, the model itself could most likely be made more accurate with increased regularization and more advanced regression and/or machine learning methods. Secondly, only the two phases of main-sequence stars and red giant stars were analyzed, meaning that the stellar nebula, pre-main sequence, white dwarf, neutron star, and black hole (among others) phases were not included within the research study. Third and finally, this research study only analyzed one explorative (independent) parameter, stellar age, and was not able to explore other stellar components. 

Future Research

In order to address the limitations of the stellar age model developed previously in this paper, further exploration of other components of stellar evolution is needed. Multiple studies conducted by researchers such as Chiavassa (2018), Berger (2020), and Hasselquist (2020) have already shown other ways that stellar age can be used to help develop a better understanding of stellar evolution: simulating the dynamics of stars based on the HR diagram, developing a model that relates stellar age and radius, and incorporating stellar mass into the HR diagram. Ultimately, a similar approach in using machine learning to model the components of stars, especially age, can help us obtain not only a better understanding of overall stellar life, but also a better understanding of the universe and its fundamental properties.

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