Astro 2/27

Escape Velocity and Energy Conservation

• Escape velocity is the minimum speed required for an object to escape the gravitational field of a planet or other celestial body.

• Calculated using the formula: ( V = \sqrt{\frac{2GM}{r}} ) where:

  • G = gravitational constant

  • M = mass of the celestial body

  • r = radius from the center of the body to the point of escape.

• Depends on both the mass of the body and its radius.

• The concept derives from energy conservation principles.

Temperature and Average Kinetic Energy

• Temperature is a measure of the average kinetic energy of particles in a substance.

• The relationship is given by the formula:

  • ( KE_{avg} = \frac{1}{2}mv_{rms}^2 )

  • Where ( v_{rms} ) = root mean square velocity.

• Average molar speed for nitrogen (( N_2 )) at room temperature is approximately 500 m/s.

• Not all molecules have the same velocity, but ( v_{rms} ) indicates a characteristic average velocity.

Factors Influencing a Planet's Atmosphere

• Whether a planet can maintain an atmosphere depends on:

  • Distance from its star (e.g., Earth vs. Mars).

  • Solar energy absorption versus energy radiation.

• The rate of energy gain from the sun equals the rate of energy loss through radiation, represented by the equation:

  • ( T = \left( \frac{L(1-a)}{16\pi \epsilon \sigma d^2} \right)^{1/4} )

  • L = solar luminosity

  • a = albedo (reflectivity of the planet)

  • ( \sigma ) = Stefan-Boltzmann constant

  • d = distance from the sun.

Albedo and Energy Absorption

• Albedo (a) is the fraction of solar energy reflected by a planet:

  • If a = 1/3, then the planet absorbs 2/3 of the incident light.

• The incorporation of albedo into planetary temperature equations adjusts energy absorption estimates.

Examples of Planetary Calculations

• For Mars (albedo ~0.2):

  • Emits 80% solar energy, absorbs 20%.

  • Calculated temperature: 215 K, aligning closely with measured average (209 K).

• For Earth, predicted temperature is 255 K, indicating a lack of liquid water.

  • This emphasizes the role of atmospheric gases in maintaining higher temperatures (greenhouse effect).

Atmosphere Retention and Escape Velocity

• Atmosphere retention relates to RMS speed of atmospheric molecules compared to escape velocity:

  • Earth: Nitrogen molecules have ( v_{rms} \approx 500 m/s ) while escape velocity ( \approx 11,200 m/s ).

  • Thus, significant retention of nitrogen on Earth vs. potential for loss on planets like Mars.

Case Study: Pluto's Atmosphere

• Pluto's characteristics:

  • Mass: ( 1.3 \times 10^{22} ) kg

  • Radius: 1190 km

  • Measured Temperature: ~50 K.

• The escape velocity for Pluto is approximately 1200 m/s.

• Calculate RMS speed for nitrogen at 50 K, yielding about 210 m/s.

• Since RMS speed is not far lower than the escape velocity, Pluto’s atmosphere is on the edge of stability and could lose its atmosphere over time.

The Greenhouse Effect

• Discussion on how atmospheric gases trap heat, raising average planetary temperatures.

• Comparison between planets with various atmospheric compositions show different heat retention capabilities.

Exoplanet Detection Methods

• Radial Velocity Method: Measures star wobble due to an orbiting planet through Doppler shifts in spectral lines.

• Transit Method: Involves detecting light dips as planets transit in front of their stars, providing data on:

  • Orbital period (time between transits).

  • Size of planet (depth of light dip indicates size).

Statistical Overview of Exoplanets

• By 2023, about 5839 confirmed exoplanets exist, primarily detected through:

  • Radial velocity and transit methods.

• Observational biases exist preventing detection of small and distant planets.

• Kepler Observatory revolutionized exoplanet discovery rates.

Conclusion and Future Directions

• Continued exploration into properties affecting atmosphere retention, exoplanet characteristics, and habitability.

• Ongoing research into the dynamics of atmospheres on different celestial bodies allows for better understanding of planetary systems.