[Paper Review] Crucial aspects of the initial mass function (I): The statistical correlation between the total mass of an ensemble of stars and its most massive star

M. Cerviño, Carlos Romann-Zuniga+4 more|DIGITAL.CSIC (Spanish National Research Council (CSIC))|Mar 28, 2013

Stellar, planetary, and galactic studies44 references16 citations

TL;DR

This paper establishes that the initial mass function (IMF) must be interpreted probabilistically, where stellar masses are independently and identically distributed random variables. It demonstrates that the correlation between a cluster's total mass and its most massive star arises naturally from this probabilistic framework, but only when the distribution of cluster masses and stellar counts is properly accounted for, invalidating simplistic IMF algebraic scaling approaches.

ABSTRACT

Our understanding of stellar systems depends on the adopted interpretation of the IMF, phi(m). Unfortunately, there is not a common interpretation of the IMF, which leads to different methodologies and diverging analysis of observational data.We study the correlation between the most massive star that a cluster would host, mmax, and its total mass into stars, M, as an example where different views of the IMF lead to different results. We assume that the IMF is a probability distribution function and analyze the mmax-M correlation within this context. We also examine the meaning of the equation used to derive a theoretical M-char_mmax relationship, N x int[Char_mmax-mup] phi(m) dm = 1 with N the total number of stars in the system, according to different interpretations of the IMF. We find that only a probabilistic interpretation of the IMF, where stellar masses are identically independent distributed random variables, provides a self-consistent result. Neither M nor N, can be used as IMF scaling factors. In addition, Char_mmax is a characteristic maximum stellar mass in the cluster, but not the actual maximum stellar mass. A -Char_mmax correlation is a natural result of a probabilistic interpretation of the IMF; however, the distribution of observational data in the N (or M)-cmmax plane includes a dependence on the distribution of the total number of stars, N (and M), in the system, Phi(N), which is not usually taken into consideration. We conclude that a random sampling IMF is not in contradiction to a possible mmax-M physical law. However, such a law cannot be obtained from IMF algebraic manipulation or included analytically in the IMF functional form. The possible physical information that would be obtained from the N (or M)-mmax correlation is closely linked with the Phi(M) and Phi(N) distributions; hence it depends on the star formation process and the assumed.

Motivation & Objective

• To resolve inconsistencies in interpreting the initial mass function (IMF) across different scientific contexts.
• To clarify why using total mass or star count as IMF scaling factors leads to self-inconsistent results.
• To establish that the m_max–M correlation is a statistical consequence of probabilistic IMF sampling, not a physical law.
• To demonstrate that the theoretical m_max derived from IMF equations is a characteristic value, not the actual maximum mass.
• To emphasize that observational m_max–M data depend critically on the underlying distributions of N and M, which are often neglected.

Proposed method

• Model the IMF as a probability density function φ(m), treating stellar masses as independent and identically distributed (i.i.d.) random variables.
• Use the equation 𝒩 × ∫_{m̂_max}^{m_up} φ(m) dm = 1 to derive the expected maximum mass m̂_max for a cluster of 𝒩 stars.
• Introduce the intensity function μ(m_b) = φ(m_b) / (1 − F(m_b)) to assess the likelihood of a star being in a given mass range given it exceeds m_b.
• Analyze the sampling distribution Φ_{m_max}(m_max|𝒩) to distinguish between the expected maximum (m̂_max) and the actual observed maximum (m_max).
• Account for the distributions Φ_𝒩(𝒩) and Φ_𝒫(𝒫) of total stellar mass and number of stars in the system, which influence observational correlations.
• Use Gumbel’s extreme value theory framework to model the statistical behavior of the maximum stellar mass in a sample.

Experimental results

Research questions

• RQ1 Why does the m_max–𝒫 correlation observed in clusters not follow directly from IMF algebraic manipulation?
• RQ2 What is the correct interpretation of the theoretical m̂_max derived from the IMF equation 𝒩 × ∫_{m̂_max}^{m_up} φ(m) dm = 1?
• RQ3 How does the distribution of cluster mass and star count (𝒩 and 𝒫) affect the observed m_max–𝒫 correlation in real data?
• RQ4 Why is the m̂_max not the actual maximum stellar mass in a cluster, and what is its statistical meaning?
• RQ5 Can a physical m_max–𝒫 law be derived from the IMF, or is it purely a statistical outcome of sampling?

Key findings

• Only a probabilistic interpretation of the IMF, where stellar masses are i.i.d. random variables, yields a self-consistent description of the m_max–𝒫 correlation.
• Neither total mass 𝒫 nor total number of stars 𝒩 can be used as scaling factors for the IMF, as this leads to inconsistent results.
• The value m̂_max is a characteristic maximum mass, not the actual maximum mass in a cluster; the actual m_max is stochastically larger with high probability when m̂_max is not near m_up.
• The observed m_max–𝒫 correlation in data includes a dependence on the distributions Φ_𝒩(𝒩) and Φ_𝒫(𝒫), which are typically ignored in standard analyses.
• The intensity function μ(m_b) shows that for m_b ≥ 10 M⊙ and not near m_up, there is over a 90% chance that the true m_max exceeds m̂_max.
• When m_b is close to m_up, the probability that the most massive star has mass m_b approaches 1, confirming that m̂_max cannot be interpreted as the actual maximum mass unless near the upper limit.