dNdM

dN/dm (read as “d N divided by d m”) denotes a differential quantity describing how the number of objects changes with mass. If N(m) is a cumulative count of objects up to mass m, then dN/dm represents the density of objects at mass m and is typically nonnegative. In many practical contexts, especially in astrophysics, the related form dN/dM is used to describe the number of objects within a mass interval [M, M + dM], which is approximately equal to (dN/dM) dM.

Conventions differ in sign and definition. Some authors define N(m) as the number with mass greater than m, in which case dN/dm is nonpositive. To avoid ambiguity, it is common to specify whether one is using a cumulative distribution N(m) or a differential mass function dN/dm, and to note whether N counts objects up to m or above m.

Applications and examples. In astrophysics, dN/dM is central to the initial mass function (IMF), which describes how many stars form at each mass. The IMF is often modeled as a power law or a broken power law (for example, Salpeter’s slope for certain mass ranges), or as a log-normal distribution in other ranges. The quantity dN/dM is used to convert observed counts into a mass distribution, correct for incompleteness, and normalize to a total population or total mass.

Relation to totals. The cumulative distribution N(m) is obtained by integrating dN/dm over mass, N(m) = ∫ (dN/dm) dm, and the total number or total mass over a specified range follows from the appropriate integral limits.