How is the Temperature of Gas in a Cluster Calculated?

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SUMMARY

The temperature of gas in a cluster is calculated using the formula T = (mp * v^2) / k, where mp is the mass of a proton, v is the velocity dispersion, and k is the Boltzmann constant. A typical velocity dispersion of 700 km/s results in a temperature of approximately 6 x 10^7 K. The discrepancy in calculations arises from not converting the velocity from km/s to m/s, which is essential for accurate results. Proper unit conversion is crucial for achieving the correct temperature value.

PREREQUISITES
  • Understanding of the ideal gas law
  • Familiarity with the Boltzmann constant (k = 1.38 x 10^-23 J/K)
  • Knowledge of proton mass (mp = 1.67 x 10^-27 kg)
  • Ability to convert units from km/s to m/s
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  • Research the implications of temperature in astrophysical clusters
  • Learn about the ideal gas law in astrophysics
  • Explore the relationship between velocity dispersion and temperature
  • Study the role of the Boltzmann constant in thermodynamics
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Astronomers, astrophysicists, and students studying thermodynamics in astrophysical contexts will benefit from this discussion.

physgeek2033
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TL;DR
I am just trying to parse the following together:
The following link states that: "since kT ~ mpv2. A typical velocity dispersion 700 km/s implies T ~ 6 x 107 K from this source alone."

How did they get 6*10^7K ?

When I try this, using mp = 1.67*10^-27 , k =1.38*10^-23 I end up getting 84 as a final answer, nowhere near 6*10^7. Can anyone clarify this? https://pages.astronomy.ua.edu/keel...o the temperature associated with the cluster
 
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Check your work. Plugging in ##\frac{m_p v^2}{k}## gives me about ##6*10^7##.
Remember to convert 700 km/s to m/s.
 

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