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1) a) Calculate the Jeans length for the dense core of a giant molecular cloud with T=10 K, n = 1010/m³, and µ=2. b) Estimate the adiabatic sound speed for this core, using y=5/3. c) Use this speed to find the amount of time required for a sound wave to cross the cloud t;=2R/Vs. d) Compare your answer with the free-fall scale and comment your results.

1) a) Calculate the Jeans length for the dense core of a giant molecular cloud with T=10 K, n = 1010/m³, and µ=2. b) Estimate the adiabatic sound speed for this core, using y=5/3. c) Use this speed to find the amount of time required for a sound wave to cross the cloud t;=2R/Vs. d) Compare your answer with the free-fall scale and comment your results.

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i have attached some of the formulas that will be helpful. Thank you. 

**Helpful Equations** **Part a:** Jeans length = \( \left( \frac{15 \, kT}{4 \pi \, p \, G \, \mu \, m_m} \right)^{1/2} \) **Part b:** I'm not sure what equations to use. I think you can use this: \( MJ_{d \, p^{3/2}}^{~r-2} \) **Part c:** \[ T_{freefall} = \sqrt{\frac{3\pi}{32}} \cdot \frac{1}{G \rho} \] Given \( \gamma = 5/3 \)
Transcribed Image Text:**Helpful Equations** **Part a:** Jeans length = \( \left( \frac{15 \, kT}{4 \pi \, p \, G \, \mu \, m_m} \right)^{1/2} \) **Part b:** I'm not sure what equations to use. I think you can use this: \( MJ_{d \, p^{3/2}}^{~r-2} \) **Part c:** \[ T_{freefall} = \sqrt{\frac{3\pi}{32}} \cdot \frac{1}{G \rho} \] Given \( \gamma = 5/3 \)
### Problem Set: Star Formation in Giant Molecular Clouds **Problem 1:** **a)** *Calculate the Jeans length for the dense core of a giant molecular cloud with the following parameters:* - Temperature \( T = 10 \, \text{K} \) - Number density \( n = 10^{10} \, \text{m}^{-3} \) - Mean molecular weight \( \mu = 2 \) **b)** *Estimate the adiabatic sound speed for this core, using \( \gamma = 5/3 \).* **c)** *Use this speed to find the amount of time required for a sound wave to cross the cloud, given by the formula \( t_s = \frac{2R_J}{v_s} \).* **d)** *Compare your answer with the free-fall timescale and comment on your results.*
Transcribed Image Text:### Problem Set: Star Formation in Giant Molecular Clouds **Problem 1:** **a)** *Calculate the Jeans length for the dense core of a giant molecular cloud with the following parameters:* - Temperature \( T = 10 \, \text{K} \) - Number density \( n = 10^{10} \, \text{m}^{-3} \) - Mean molecular weight \( \mu = 2 \) **b)** *Estimate the adiabatic sound speed for this core, using \( \gamma = 5/3 \).* **c)** *Use this speed to find the amount of time required for a sound wave to cross the cloud, given by the formula \( t_s = \frac{2R_J}{v_s} \).* **d)** *Compare your answer with the free-fall timescale and comment on your results.*
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