--- title: "One converged mass scale does not establish an IMF" description: "Read the resolution figures in Guszejnov 2020 v1, separating mass-weighted medians, number distributions and claims about a stellar initial mass function." canonical_url: "https://sandboxphysics.com/stories/molecular-clouds/why-a-converged-mass-is-not-the-imf/" language: "en" page_type: "story" alternate_languages: zh-CN: "https://sandboxphysics.com/zh/stories/molecular-clouds/why-a-converged-mass-is-not-the-imf/" last_modified: "2026-11-04" --- [Molecular cloud collapse and magnetic support](https://sandboxphysics.com/stories/molecular-clouds/) Molecular clouds · L4 # One converged mass scale does not establish an IMF Sandbox Physics · 2026-11-04 Open Can magnetized turbulence set the mass scale of stars? by Guszejnov and colleagues, pinned to arXiv:2002.01421v1. Follow Figures 8, 9 and 10 in that order. A mass-weighted characteristic mass can settle with resolution even as the number of low-mass objects continues changing. ## Before you begin Read the Jeans-scale and resolution chapters and be comfortable with logarithmic axes. Mass weighting, sink particles and the IMF are explained below. Keep the pinned paper open beside the worked examples. ## Find out what counts as an object A stellar initial mass function, or IMF, describes how many stars have different initial masses. The paper evolves ideal isothermal magnetohydrodynamics and uses sink particles to follow dense collapse beyond the resolved gas scale. Sinks accrete surrounding gas and carry mass and motion. Relating them to stars retains physical and resolution assumptions. Start with section 2.2. The GIZMO meshless finite-mass method keeps gas-cell mass fixed. Sink creation requires high density, gravitational binding and contraction along three axes, with additional criteria. Accretion has radius, binding and angular-momentum conditions too. A dense image pixel alone does not define one of these objects. Sphere initial conditions place a cloud with a random velocity field inside diffuse surrounding gas. Periodic Box conditions are driven to saturated turbulence before gravity is enabled, and driving continues afterward. These are evolving three-dimensional velocity and magnetic fields. A gas patch seeded with two coherent cosines does not reproduce that turbulence. The paper also uses a normalized mass-to-flux ratio $\mu$. In equation 9, $\mu=1$ follows the authors’ homogeneous-sphere, uniform-field critical convention. Box fields undergo turbulent dynamo amplification, so initial field strength is not a fully independent final-state control. Keep geometry and initial conditions with the magnetic conclusion. Reading: Guszejnov et al. 2020, Can magnetized turbulence set the mass scale of stars? arXiv:2002.01421v1 · Federrath & Klessen 2012, arXiv:1209.2856v1 ## Calculate mass weighting with five specified masses Section 3.1 chooses $M_{50}$ as a characteristic mass. Sort objects by mass and accumulate the mass they contain. The halfway point defines a mass-weighted median. For the specified discrete exercise below, use the first object mass crossing that threshold. An ordinary median instead counts objects until it reaches the middle of the list. Use the specified list $(1,1,1,1,8)$ in any consistent mass unit. Its total is $12$ and its number median is $1$. The four small objects contain only $4$; reaching half the total, $6$, requires the object of mass $8$. Thus $M_{50}=8$. Replace one unit-mass object with one hundred objects of mass $0.01$. Total mass stays $12$, object count rises from $5$ to $104$, and the ordinary median becomes $0.01$. The mass-weighted median remains $8$. A large number-distribution change can leave a high-mass-dominated statistic unchanged. These are supplied exercise masses, not paper data. The standalone worked-example program recomputes the operation with exact rational numbers, preserving the equal-total-mass condition. $$ \sum_{m_i\le M_{50}} m_i\ \ge\ \frac12\sum_i m_i $$ Choose the first mass threshold reaching half the cumulative mass, with the stated convention for ties. This differs from reaching half the object count. Reading: Guszejnov et al. 2020, Can magnetized turbulence set the mass scale of stars? arXiv:2002.01421v1 · Molecular-cloud Jeans standalone scientific reproduction ## Figure 8 percentiles are different distribution locations The horizontal axis is logarithmic gas-element mass $\Delta m$. Moving left means finer mass resolution; the upper axis gives the corresponding element count. The vertical axis gives logarithmic sink-mass percentiles at $5\%$ star formation efficiency, meaning the fraction of the original cloud mass incorporated into sinks. The curves mark the zeroth, sixteenth, fiftieth, eighty-fourth and hundredth percentiles. Intermediate curves use cumulative mass, and the endpoints are the minimum and maximum sink masses. The sixteenth and eighty-fourth curves are not statistical error bars on a characteristic-mass estimate. With finer resolution the minimum continues decreasing, but higher mass percentiles begin flattening. This supports insensitivity of named mass statistics over the computed resolution range. Whether the low-mass object population has settled requires the later number-distribution plot. Pinned v1 has an unresolved model pairing here: the paragraph introducing Figure 8 names M2e5\_R30, whereas the caption and next sentence name M2e4\_R10. The numerical example is an approximate chain, $\Delta m\approx3\times10^{-8}M_0=0.01M_\odot$. Different model masses give different approximate dimensional values. Preserve that context without declaring a units error or choosing the intended model for the authors. Reading: Guszejnov et al. 2020, Can magnetized turbulence set the mass scale of stars? arXiv:2002.01421v1 ## Figure 9 compares with a highest-resolution fit The horizontal variable divides mass resolution by $M_0\mathcal M^{-4}$, where $M_0$ is initial cloud mass and $\mathcal M$ is three-dimensional turbulent RMS speed relative to sound speed. The vertical variable is the logarithm of $M_{50}/M_{50,\mathrm{fit}}$. This scaling puts different clouds on one comparison. Section 3.5 says explicitly that $M_{50,\mathrm{fit}}$ comes from equation 17 fitted to the highest available resolutions. It serves as a proxy for the infinite-resolution value. There is no independently known exact continuum solution on that axis. Approaching the fit supports resolution insensitivity over the computed range. Equation 22 gives the empirical condition $\Delta m\lesssim0.05M_0\mathcal M^{-4}$. The following proposed Eulerian criterion resolves the sonic length with several cells. This is the length at which turbulent velocity dispersion equals sound speed. After equation 23 the authors warn that the coefficient depends on the MHD and gravity solvers. It cannot simply become another method’s local Jeans stopping rule. $$ \Delta m\lesssim0.05M_0\mathcal M^{-4},\qquad \Delta x\lesssim0.2L_{\rm sonic}\approx0.2R_{\rm cloud}\mathcal M^{-2} $$ The mass-resolution condition is empirical. The spatial condition is a proposed estimate whose numerical coefficient is method dependent. Reading: Guszejnov et al. 2020, Can magnetized turbulence set the mass scale of stars? arXiv:2002.01421v1 ## Figure 10 restores the full number distribution The horizontal axis is logarithmic sink mass. The vertical axis is the logarithm of counts per logarithmic mass interval, $dN/d\log M$. Colors identify mass-resolution levels, again compared at $5\%$ formation efficiency. Low-mass object counts now have much more influence on the picture. As resolution improves, a low-mass peak near the resolution limit keeps moving and more low-mass sinks appear. A small amount of mass can be divided into many objects. The specified-list exercise explains why a mass-weighted median can respond weakly to that change. The plot marks a brown-dwarf boundary around $0.08M_\odot$ and shades a region where isothermality is expected to fail. The shading indicates a physical approximation, not an observational uncertainty band. Dense gas eventually has difficulty radiating heat, so keeping one sound speed misses an important change. The authors restrict their resolution conclusion to the relatively large objects containing most of the mass. Read that scope together with the continuing low-mass change. A characteristic mass and the full observed IMF ask different questions. Reading: Guszejnov et al. 2020, Can magnetized turbulence set the mass scale of stars? arXiv:2002.01421v1 ## Separate the roles of magnetic fields, flows and added physics Section 4.2 compares with observed IMFs and discusses the characteristic scale, environmental variation and low-mass distribution. It does not conclude that ideal isothermal MHD alone explains the full observed IMF. Radiation heating and outflows can alter fragmentation, accretion and the fate of gas in different ways. Federrath and Klessen 2012 v1 provides a separate comparison: it connects a star formation rate to density statistics and states limits on statistical magnetic-pressure approximations. A formation rate asks how much gas enters stars per time. An IMF asks how that mass is divided among objects. Sharing the phrase star formation does not make the conclusions interchangeable. An inward gathering flow also does not automatically supply outward support. A turbulence-support claim needs its actual velocity correlations, driving, dissipation and geometry. A magnetic-criticality claim needs the adopted mass-to-flux convention. Reading: Guszejnov et al. 2020, Can magnetized turbulence set the mass scale of stars? arXiv:2002.01421v1 · Federrath & Klessen 2012, arXiv:1209.2856v1 ## Carry the same questions to another resolution study Identify the statistic first: number weighting, mass weighting or a named percentile. Note what the horizontal refinement changes and whether initial conditions and evolutionary stage match. Then locate the reference: an exact solution, an independent algorithm or a highest-resolution fit. Find the part of the distribution still moving and connect it to the physical approximation. When a statistic flattens, state its name, conditions and tested range. A broader conclusion needs evidence addressing the broader quantity. Reading: Guszejnov et al. 2020, Can magnetized turbulence set the mass scale of stars? arXiv:2002.01421v1 · Molecular-cloud Jeans standalone scientific reproduction ## Exercises and explanations 1. Why does splitting one supplied unit-mass object into one hundred small objects change the ordinary median but not the mass-weighted median? Read the explanation Total mass remains $12$. The small objects and the three remaining unit masses sum to $4$, so crossing half the total still requires the object of mass $8$. By number, the hundred masses of $0.01$ dominate the list, giving ordinary median $0.01$. The statistics weight the same list differently. 2. Can the sixteenth and eighty-fourth curves in Figure 8 be read as a confidence interval on the characteristic mass? Read the explanation They are different locations in the cumulative mass distribution. A statistical interval for an estimator needs a specified sampling or inference procedure and uncertainty definition. Those are different objects from these percentile curves. 3. Does approaching the highest-resolution fit in Figure 9 provide a known exact infinite-resolution value? Read the explanation The authors explicitly use the fit as a proxy. The curves support decreasing sensitivity within the computed range. Read them together with the changing low-mass population in Figure 10 to identify which distribution properties have stabilized. ## Try the prediction ### [Watch pressure and gravity change the same gas](https://sandboxphysics.com/labs/molecular-cloud-jeans/) 1. Compare peak density and complete gas fields under grid refinement, defining each quantity. Do not count two-dimensional bright patches as stars. 2. Recompute the specified mass-list exercise. Predict both medians first, then inspect the output and the unchanged total mass. 3. Return to pinned v1 Figures 8–10. Record each axis, weighting, common evolutionary stage and reference source, then identify which conclusion needs added physics. [Previous: Replaying an official Jeans initial condition](https://sandboxphysics.com/stories/molecular-clouds/replay-the-official-jeans-input/) ## Sources and calculation scope 1. [Federrath & Klessen 2012, arXiv:1209.2856v1](https://arxiv.org/abs/1209.2856v1) Pinned v1, sections 2.2/2.4, 3.1–3.3 and 7: isothermal driven three-dimensional MHD, sink criteria, the statistical magnetic-pressure approximation and its limitations. Their driving, sinks and numerical dynamo requirement are not part of the gas-patch calculation. 2. [Guszejnov et al. 2020, Can magnetized turbulence set the mass scale of stars? arXiv:2002.01421v1](https://arxiv.org/abs/2002.01421v1) Pinned v1, sections 2–4, Figures 8–10 and equations 8–11/17/22–23. Read the mass-weighted percentiles and the number distribution separately. The Figure 8 model-name pairing and approximate dimensional example are preserved in context; no author correction is assumed. 3. [Molecular-cloud Jeans standalone scientific reproduction](https://sandboxphysics.com/stories/molecular-clouds/materials/molecular-cloud-jeans-science.zip) Independent finite-volume and oversampled pseudospectral producers; analytic Jeans and full linear MHD references; all ten support cases, a fixed longer case, original official inputs and retained failures. Default replay actually recomputes its scientific steps offline after ordinary Python dependencies are installed.