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--- courses:ast402:star-formation [2026/06/02 06:03] – created shuvo
+++ courses:ast402:star-formation [2026/06/04 01:12] (current) – [Jeans Criterion] shuvo
@@ Line 1: / Line 1: @@
 ====== Star Formation ======
-### **Jeans Criterion**
+=====Jeans Criterion =====
 The **Jeans criterion** defines the critical conditions required for an interstellar cloud to undergo spontaneous gravitational collapse. According to the **virial theorem**, collapse occurs when twice the internal kinetic energy ($2K$) is less than the absolute value of the gravitational potential energy ($|U|$). For a spherical cloud of constant density $\rho_0$ and temperature $T$, the **Jeans mass** ($M_J$) represents the minimum mass required to initiate collapse:
 $$M_J \approx \left(\frac{5kT}{G\mu m_H}\right)^{3/2} \left(\frac{3}{4\pi\rho_0}\right)^{1/2}$$
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 Diffuse HI clouds are generally stable against collapse because their masses are much lower than their Jeans mass, whereas the **dense cores** of giant molecular clouds (GMCs) often satisfy this criterion.
-### **Homologous Collapse**
+The following derivation starts from the **virial theorem** and identifies the point where gravitational forces overwhelm internal gas pressure.
+**1. The Virial Theorem Starting Point**
+For a stable, gravitationally bound system in equilibrium, the virial theorem states that twice the total internal kinetic energy ($2K$) plus the gravitational potential energy ($U$) must equal zero:
+$$2K + U = 0$$
+where $K$ and $U$ are typically time-averaged values.
+To determine if a cloud will collapse, we compare these two energy terms. If **$2K < |U|$**, the gravitational force dominates, and the cloud will collapse. The critical boundary for stability occurs when **$2K = |U|$**.
+**2. Formulating Kinetic and Potential Energy**
+To apply this to a molecular cloud, we use the following approximations:
+**Internal Kinetic Energy ($K$):** Assuming an ideal monatomic gas, the total kinetic energy for a cloud containing $N$ particles at temperature $T$ is:
+$$K = \frac{3}{2} NkT$$
+where $k$ is Boltzmann’s constant.
+The total number of particles is the cloud mass ($M_c$) divided by the average mass per particle ($m = \mu m_H$):
+$$N = \frac{M_c}{\mu m_H}$$
+where $\mu$ is the mean molecular weight and $m_H$ is the mass of a hydrogen atom. Substituting this gives:
+$$K = \frac{3}{2} \frac{M_c kT}{\mu m_H}$$
+**Gravitational Potential Energy ($U$):** For a spherical cloud of constant density $\rho_0$, the potential energy is approximately:
+$$U \approx -\frac{3}{5} \frac{GM_c^2}{R_c}$$
+where $R_c$ is the cloud radius.
+**3. Deriving the Jeans Mass ($M_J$)**
+Substituting these expressions into the collapse condition ($2K < |U|$):
+$$2 \left( \frac{3}{2} \frac{M_c kT}{\mu m_H} \right) < \frac{3}{5} \frac{GM_c^2}{R_c}$$
+Simplifying the left side:
+$$\frac{3 M_c kT}{\mu m_H} < \frac{3}{5} \frac{GM_c^2}{R_c}$$
+To express this in terms of mass and density, we replace the radius $R_c$ using the volume formula for a sphere ($M_c = \frac{4}{3} \pi R_c^3 \rho_0$):
+$$R_c = \left( \frac{3M_c}{4\pi\rho_0} \right)^{1/3}$$
+Substituting $R_c$ into the simplified inequality:
+$$\frac{3 M_c kT}{\mu m_H} < \frac{3}{5} GM_c^2 \left( \frac{4\pi\rho_0}{3M_c} \right)^{1/3}$$
+Isolating $M_c$ reveals the minimum mass required to initiate collapse, known as the **Jeans mass ($M_J$)**:
+**$$M_J \approx \left( \frac{5kT}{G\mu m_H} \right)^{3/2} \left( \frac{3}{4\pi\rho_0} \right)^{1/2}$$**
+**4. Deriving the Jeans Length ($R_J$)**
+The Jeans criterion can also be expressed as the minimum radius needed for a cloud of a given density to collapse. By substituting the mass expression ($M_c = \frac{4}{3} \pi R_c^3 \rho_0$) back into the primary inequality:
+$$\frac{3 (\frac{4}{3}\pi R_c^3 \rho_0) kT}{\mu m_H} < \frac{3}{5} \frac{G(\frac{4}{3}\pi R_c^3 \rho_0)^2}{R_c}$$
+Solving for $R_c$ gives the **Jeans length ($R_J$)**:
+**$$R_J \approx \left( \frac{15kT}{4\pi G\mu m_H \rho_0} \right)^{1/2}$$**
+**Summary of Results**
+If a cloud's mass **$M_c > M_J$**, or its radius **$R_c > R_J$**, it is unstable and will undergo gravitational collapse. \\
+Diffuse HI clouds are generally stable because their masses are much lower than their Jeans mass, whereas the **dense cores** of giant molecular clouds often satisfy this criterion and collapse to form stars.
+=====Homologous Collapse =====
 Once the Jeans criterion is satisfied, if pressure gradients are initially negligible, the cloud undergoes **free-fall collapse**. This collapse is **isothermal** as long as the cloud remains optically thin and can radiate away the released gravitational potential energy. The equation of motion for a mass shell at radius $r$ is:
 $$\frac{d^2r}{dt^2} = -G \frac{M_r}{r^2}$$
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 Because $t_{ff}$ is independent of the initial radius $r_0$, all parts of a uniform cloud collapse in the same amount of time, a process known as **homologous collapse**. If the cloud is centrally condensed, it undergoes an **inside-out collapse** where the core collapses faster than the outer layers.
-### **Fragmentation of Collapsing Clouds**
+**Fragmentation of Collapsing Clouds:**
 As an isothermal collapse proceeds, the density $\rho$ increases while $T$ remains constant, which causes the Jeans mass to decrease ($M_J \propto \rho^{-1/2}$). Initial density inhomogeneities then allow smaller sections of the cloud to satisfy the Jeans criterion independently and collapse locally, leading to **fragmentation**. This cascading process segments the cloud into many smaller objects, explaining why stars often form in groups. Fragmentation ceases when the cloud becomes **optically thick** and the collapse becomes **adiabatic**. In an adiabatic collapse, the temperature rises ($T \propto \rho^{\gamma-1}$), causing $M_J$ to increase with density ($M_J \propto \rho^{1/2}$ for atomic hydrogen), which establishes a **minimum fragment mass**:
 $$M_{J,min} \approx 0.03 \left( \frac{T^{1/4}}{e^{1/2}\mu^{9/4}} \right) M_\odot$$
 For typical parameters, this limit is approximately **$0.01$ to $0.5$ $M_\odot$**.
-### **Formation of Protostars**
+=====Formation of Protostars=====
 A **protostar** forms when the central region of a collapsing fragment becomes optically thick, causing internal pressure to rise and slowing the collapse into a state of **near-hydrostatic equilibrium**. This hydrostatic core typically begins with a radius of about **$5$ AU**. The protostar is initially powered by a **shock wave** at its surface where infalling material from the envelope arrives at supersonic speeds, converting kinetic energy into heat. A second, more rapid collapse occurs when central temperatures reach ~2000 K, causing **molecular hydrogen to dissociate**, which removes the pressure support needed for equilibrium. Hydrostatic equilibrium is re-established once the core radius shrinks to about **1.3 times the size of the Sun**.
-### **Ambipolar Diffusion**
+**Ambipolar Diffusion:**
 Magnetic fields can inhibit cloud collapse by providing magnetic pressure. While charged ions and electrons are "frozen-in" to magnetic field lines, **neutral particles** (atoms and molecules) are not. Under the influence of gravity, neutrals slowly migrate toward the center of the cloud by drifting across magnetic field lines, a process called **ambipolar diffusion**. The neutrals collide with ions during this drift, which transfers some gravitational force to the magnetic field. The characteristic timescale for this diffusion is:
 $$t_{AD} \approx \frac{2R}{v_{drift}} \approx 10 \text{ Gyr} \left(\frac{n_{H_2}}{10^{10} \text{ m}^{-3}}\right) \left(\frac{B}{1 \text{ nT}}\right)^{-2} \left(\frac{R}{1 \text{ pc}}\right)^2$$
 This long timescale explains why dense cores can remain stable for millions of years before free-fall collapse begins.
-### **Hayashi Track**
+**Hayashi Track:**
-The **Hayashi track** is a nearly vertical evolutionary path on the **Hertzsprung-Russell (H-R) diagram** followed by a protostar with a **deeply convective envelope**. Because convection is highly efficient at transporting luminosity, the star's luminosity drops rapidly while its effective temperature increases only slightly as it contracts. The Hayashi track serves as a **boundary**; no stable hydrostatic stars can exist to its right (lower temperatures) because no mechanism can transport the required luminosity at those temperatures.
+The Hayashi track is a nearly vertical evolutionary path on the **Hertzsprung-Russell (H-R) diagram** followed by a protostar with a **deeply convective envelope**. Because convection is highly efficient at transporting luminosity, the star's luminosity drops rapidly while its effective temperature increases only slightly as it contracts. The Hayashi track serves as a **boundary**; no stable hydrostatic stars can exist to its right (lower temperatures) because no mechanism can transport the required luminosity at those temperatures.
-### **Pre-Main-Sequence Evolution**
+**Pre-Main-Sequence Evolution:**
-The **pre-main-sequence (PMS) phase** begins once the protostar settles into a quasi-static contraction. The evolution is now controlled by the **Kelvin-Helmholtz timescale** ($t_{KH}$), as the star radiates away gravitational energy released by its slow collapse. For a $1 M_\odot$ star, this takes approximately **40 Myr**. During this phase, **deuterium burning** may occur, temporarily slowing the contraction. As the core heats up, a **radiative zone** develops and expands, causing the evolutionary track to turn away from the Hayashi track and move horizontally toward higher temperatures.
+The pre-main-sequence (PMS) phase begins once the protostar settles into a quasi-static contraction. The evolution is now controlled by the **Kelvin-Helmholtz timescale** ($t_{KH}$), as the star radiates away gravitational energy released by its slow collapse. For a $1 M_\odot$ star, this takes approximately **40 Myr**. During this phase, **deuterium burning** may occur, temporarily slowing the contraction. As the core heats up, a **radiative zone** develops and expands, causing the evolutionary track to turn away from the Hayashi track and move horizontally toward higher temperatures.
-### **Brown Dwarfs**
+**Brown Dwarfs:**
 Objects with masses below approximately **$0.072 M_\odot$** (for solar composition) are known as **brown dwarfs**. Their cores never reach temperatures high enough to sustain stable hydrogen fusion to counteract gravitational collapse. They may temporarily burn **deuterium** (if $M > 0.013 M_\odot$) or **lithium** (if $M > 0.06 M_\odot$), but they eventually cool and fade. They are characterized by cool **L and T spectral types**.
-### **Birth of Massive Stars**
+**Birth of Massive Stars:**
 Massive stars ($M > 10 M_\odot$) evolve much faster than low-mass stars, reaching the main sequence in as little as **28,000 years** for a $60 M_\odot$ star. Because their central temperatures are high, they ignite hydrogen via the **CNO cycle**, which is highly temperature-dependent and maintains a **convective core** even after reaching the main sequence. Their PMS tracks are **nearly horizontal**, as they leave the Hayashi track almost immediately. Their intense ionizing radiation may disperse the surrounding cloud, potentially limiting the formation of nearby low-mass stars.
-### **Zero-Age Main-Sequence (ZAMS)**
+**Zero-Age Main-Sequence (ZAMS):**
-The **Zero-Age Main Sequence (ZAMS)** is the diagonal line on the H-R diagram where stars of various masses first reach a state of **equilibrium hydrogen burning**. At this point, nuclear energy production exactly balances the star's luminosity, and gravitational contraction stops.
+The Zero-Age Main Sequence (ZAMS) is the diagonal line on the H-R diagram where stars of various masses first reach a state of **equilibrium hydrogen burning**. At this point, nuclear energy production exactly balances the star's luminosity, and gravitational contraction stops.
+=====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.
-### **Initial Mass Function (IMF)**
+The relationship between the number of stars ($N$) and their mass ($M$) is expressed using the function $\xi$:
-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.
+$$\xi(\log_{10} M) = \frac{dN}{d(\log_{10} M)}$$
+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**
+=====HII Regions=====
 **HII regions** are large clouds of **ionized hydrogen** surrounding hot, young O and B stars. These stars emit intense **ultraviolet radiation** ($\lambda < 91.2$ nm) that ionizes the surrounding gas. Recombination of electrons and protons leads to a cascade of photons, including the red **$H\alpha$ Balmer line**, which causes these regions to fluoresce. The **Strömgren radius** ($r_S$) defines the equilibrium size of an HII region, where the ionization rate equals the recombination rate:
 $$r_S \approx \left(\frac{3N}{4\pi\alpha}\right)^{1/3} n_H^{-2/3}$$
 where $N$ is the rate of ionizing photons and $\alpha$ is the recombination coefficient.
-### **OB Associations**
+**OB Associations:**
-**OB associations** are loose groups of young, massive **O and B stars**. These clusters are typically **gravitationally unbound** because the intense radiation and stellar winds from the massive stars quickly disperse the remaining gas cloud that provided the binding mass. Consequently, the member stars tend to drift apart over time.
+OB associations are loose groups of young, massive **O and B stars**. These clusters are typically **gravitationally unbound** because the intense radiation and stellar winds from the massive stars quickly disperse the remaining gas cloud that provided the binding mass. Consequently, the member stars tend to drift apart over time.
-### **T Tauri Stars**
+**T Tauri Stars:**
-**T Tauri stars** are low-mass ($0.5$ to $2 M_\odot$) pre-main-sequence stars that have emerged from their dust cocoons but have not yet reached the ZAMS. They are characterized by **irregular luminosity variations**, strong **lithium absorption**, and **emission lines** from hydrogen, calcium, and iron. Many exhibit a **P Cygni profile** in their $H\alpha$ lines—a broad emission peak with a blueshifted absorption trough—indicating significant **mass loss** via strong stellar winds.
+T Tauri stars are low-mass ($0.5$ to $2 M_\odot$) pre-main-sequence stars that have emerged from their dust cocoons but have not yet reached the ZAMS. They are characterized by **irregular luminosity variations**, strong **lithium absorption**, and **emission lines** from hydrogen, calcium, and iron. Many exhibit a **P Cygni profile** in their $H\alpha$ lines—a broad emission peak with a blueshifted absorption trough—indicating significant **mass loss** via strong stellar winds.
-### **Herbig-Haro Objects**
+**Herbig-Haro Objects:**
 **Herbig-Haro (HH) objects** are small, bright nebulae found near young stars, created by high-speed **jets of gas** ejected from the protostar or its accretion disk. As these jets collide with the interstellar medium at supersonic speeds, the resulting **shocks** excite and ionize the gas, producing characteristic emission-line spectra.
-### **Circumstellar Disk Formation**
+**Circumstellar Disk Formation:**
 As a protostellar cloud collapses, it **spins up** to conserve **angular momentum** ($L = I\omega = \text{constant}$). The resulting centripetal acceleration halts the collapse in the plane perpendicular to the rotation axis, while collapse along the axis continues. This leads to the formation of a flattened **circumstellar accretion disk**. The **Hill radius** ($R_H$) defines the region around a growing protoplanet within the disk where its gravity dominates:
 $$R_H = a \left(\frac{M}{M_\odot}\right)^{1/3}$$
 where $a$ is the orbital radius and $M$ is the protoplanet mass. Most main-sequence stars rotate much slower than expected from simple collapse, implying that angular momentum is transferred away, likely by **magnetic braking** and stellar winds.