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courses:ast402:interstellar-medium [2026/05/18 21:05] – [Jeans instability] shuvocourses:ast402:interstellar-medium [2026/05/18 21:06] (current) – [Jeans instability] shuvo
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 It describes the critical conditions under which a molecular cloud will spontaneously collapse under its own gravity to form a protostar. This concept, first investigated by Sir James Jeans in 1902, is based on the virial theorem, which assesses stability by balancing the total internal kinetic energy ($K$) and the absolute value of the gravitational potential energy ($|U|$) of a system. If twice the internal kinetic energy is less than the absolute value of the gravitational potential energy ($2K < |U|$), the inward pull of gravity will overwhelm the outward push of gas pressure, initiating a collapse. It describes the critical conditions under which a molecular cloud will spontaneously collapse under its own gravity to form a protostar. This concept, first investigated by Sir James Jeans in 1902, is based on the virial theorem, which assesses stability by balancing the total internal kinetic energy ($K$) and the absolute value of the gravitational potential energy ($|U|$) of a system. If twice the internal kinetic energy is less than the absolute value of the gravitational potential energy ($2K < |U|$), the inward pull of gravity will overwhelm the outward push of gas pressure, initiating a collapse.
  
-**The Mathematical Condition for Collapse**+**The Mathematical Condition for Collapse**\\
 For a spherical cloud of constant initial density $\rho_0$, mass $M_c$, and radius $R_c$, the gravitational potential energy is approximately: For a spherical cloud of constant initial density $\rho_0$, mass $M_c$, and radius $R_c$, the gravitational potential energy is approximately:
 $U \sim -\frac{3}{5}\frac{GM_c^2}{R_c}$.  $U \sim -\frac{3}{5}\frac{GM_c^2}{R_c}$. 
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 $\frac{3M_c k T}{\mu m_H} < \frac{3}{5}\frac{GM_c^2}{R_c}$. $\frac{3M_c k T}{\mu m_H} < \frac{3}{5}\frac{GM_c^2}{R_c}$.
  
-**The Jeans Mass and Jeans Length**+**The Jeans Mass and Jeans Length**\\
 The radius of the cloud can be expressed in terms of its mass and initial density as $R_c = \left( \frac{3M_c}{4\pi\rho_0} \right)^{1/3}$. Substituting this radius into the inequality allows us to solve for the minimum mass necessary to initiate a spontaneous collapse, known as the **Jeans mass** ($M_J$). The Jeans criterion states that collapse will occur if the cloud's mass exceeds the Jeans mass: The radius of the cloud can be expressed in terms of its mass and initial density as $R_c = \left( \frac{3M_c}{4\pi\rho_0} \right)^{1/3}$. Substituting this radius into the inequality allows us to solve for the minimum mass necessary to initiate a spontaneous collapse, known as the **Jeans mass** ($M_J$). The Jeans criterion states that collapse will occur if the cloud's mass exceeds the Jeans mass:
 $$M_c > M_J \approx \left(\frac{5kT}{G\mu m_H}\right)^{3/2} \left(\frac{3}{4\pi\rho_0}\right)^{1/2}$$. $$M_c > M_J \approx \left(\frac{5kT}{G\mu m_H}\right)^{3/2} \left(\frac{3}{4\pi\rho_0}\right)^{1/2}$$.
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 $R_c > R_J \approx \left(\frac{15kT}{4\pi G\mu m_H\rho_0}\right)^{1/2}$. $R_c > R_J \approx \left(\frac{15kT}{4\pi G\mu m_H\rho_0}\right)^{1/2}$.
  
-**Fragmentation During Collapse**+**Fragmentation During Collapse**\\
 As a molecular cloud collapses isothermally (where released gravitational energy is efficiently radiated away), its density increases while its temperature remains relatively constant. This increase in density progressively lowers the Jeans mass. As a result, localized regions of enhanced density within the cloud begin to independently satisfy the Jeans criterion and collapse on their own, leading to a cascading fragmentation of the original cloud into smaller structures.  As a molecular cloud collapses isothermally (where released gravitational energy is efficiently radiated away), its density increases while its temperature remains relatively constant. This increase in density progressively lowers the Jeans mass. As a result, localized regions of enhanced density within the cloud begin to independently satisfy the Jeans criterion and collapse on their own, leading to a cascading fragmentation of the original cloud into smaller structures. 
  
 This fragmentation process eventually halts when the collapsing fragments become opaque to radiation, trapping the heat and causing the collapse to transition from isothermal to adiabatic. During an adiabatic collapse, the temperature of the gas rises following the relation $T = K'\rho^{\gamma-1}$. Under these adiabatic conditions, the Jeans mass begins to increase according to the relation $M_J \propto \rho^{(3\gamma-4)/2}$, which sets a minimum limit on the mass of the fragments produced. This fragmentation process eventually halts when the collapsing fragments become opaque to radiation, trapping the heat and causing the collapse to transition from isothermal to adiabatic. During an adiabatic collapse, the temperature of the gas rises following the relation $T = K'\rho^{\gamma-1}$. Under these adiabatic conditions, the Jeans mass begins to increase according to the relation $M_J \propto \rho^{(3\gamma-4)/2}$, which sets a minimum limit on the mass of the fragments produced.
courses/ast402/interstellar-medium.txt · Last modified: by shuvo
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