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--- courses:ast402:star-formation [2026/06/03 01:52] – shuvo
+++ courses:ast402:star-formation [2026/06/03 01:55] (current) – [Formation of Protostars] shuvo
@@ Line 45: / Line 45: @@
 **Initial Mass Function (IMF):**
 The initial mass function (IMF), denoted as $\xi$, describes the **relative number of stars** that form in different mass intervals from a fragmented cloud. Fragmentation typically produces a large abundance of **low-mass stars** and very few massive stars. While the function is well-modeled for higher masses, it is less certain for objects below $0.1 M_\odot$, where it may become relatively flat.
+The relationship between the number of stars ($N$) and their mass ($M$) is expressed using the function $\xi$:
+$$\xi(\log_{10} M) = \frac{dN}{d(\log_{10} M)}$$
+where $dN$ represents the number of stars in a specific mass interval. To find the number of stars in a linear mass range ($dN/dM$), the equation can be rewritten using the chain rule:
+$$dN = \xi(\log_{10} M) \cdot d(\log_{10} M) = \frac{\xi(\log_{10} M)}{M \ln 10} \, dM$$
+This shows that the distribution is **strongly mass-dependent**, typically resulting in a large abundance of low-mass stars and very few massive stars.
+**Key Characteristics of IMF are:\\
+**Mass-Dependency:** Fragmentation segments a cloud into many smaller objects, with the probability of formation heavily favoring lower masses.\\
+**Low-Mass Behavior:** The IMF is considered less certain for objects below approximately **$0.1 \, M_\odot$**. In this regime, the function may become **relatively flat**, suggesting a high population of low-mass stars and brown dwarfs.\\
+**Comparison to Planets:** For comparison, the source notes that the mass distribution of **extrasolar planets** in certain intervals follows a similar power-law behavior, where the number of planets $N$ varies as $\frac{dN}{dM} \propto M^{-1}$.
 =====HII Regions=====