Astro 2/25

Introduction to Temperature and Atmospheric Effects

• Temperature is commonly understood as a measure of how hot or cold an object is.

• Definition: Temperature represents the average kinetic energy of molecules in a substance.

Kinetic Energy and Temperature

• Kinetic Energy Formula: KE = 1/2 m v²

  • m = mass of the molecule

  • v = velocity of the molecule

• The average kinetic energy is related to temperature by the equation:

  [ KE_{avg} = \frac{3}{2} k T ]

  • k is Boltzmann's constant (1.38 × 10⁻²³ J/K)

  • Kelvin Scale: 0 K equals -273 °C (absolute zero).

Air Molecule Movement

• Understanding the movement of air molecules helps gauge room temperature.

• Average speed of air molecules can be calculated using the above kinetic energy relation and data for air composition.

Typical RMS Speed Calculation

• Use mass of nitrogen (N₂) for calculation:

  • Nitrogen has a molecular mass of about 28 amu (atomic mass unit).

  • RMS speed estimated to be around 506 m/s, suggesting air molecules move relatively fast but circulate randomly.

  • Wind vs. Air Molecules: Wind moves air molecules in a single direction, whereas air molecules scatter in multiple directions, impacting pressure and perceived movement.

Sound Speed and Temperature Effect

• The speed of sound in air is approximately 343 m/s (depends on temperature):

  • Sound travels as pressure waves (density changes), illustrating how energy moves through a medium.

• As temperature increases, sound speed increases due to faster moving molecules.

Absence of Atmosphere: Lunar Example

• Moon's Atmosphere: Minimal atmosphere due to low gravity, leading to escape of gaseous molecules.

• The concept of Escape Velocity is critical to determine if a planet can retain its atmosphere.

Escape Velocity Explained

• Escape velocity is the minimum speed that an object must achieve to break free from a planet's gravitational influence.

  • Formula:[ v_{escape} = \sqrt{\frac{2GM_{planet}}{R_{planet}}} ]

  • G is the gravitational constant; M is planet mass, R is planet radius.

• Example for Earth gives an escape velocity around 11,200 m/s, significantly higher than the average speed of nitrogen molecules in the atmosphere.

Atmospheric Composition and Temperature Relationships

• Earth’s average temperature (290 K) allows for a compatible RMS speed of atmospheric molecules, keeping them bound to the planet.

• Key Factors Affecting Temperature:

  • Distance from the Sun

  • Atmospheric composition (clouds, greenhouse gases)

  • Planetary rotation and angle of tilt affecting sunlight exposure.

Energy Absorption and Radiation

• Power Absorption: Earth absorbs solar energy, calculated from luminosity spread over area.[ P_{in} = \text{Intensity} \times \text{Area}{Earth} = \frac{L{sun}}{4\pi d^2} \cdot \pi R^2 ]

• Energy Emission: Energy is re-radiated based on emissions from Earth's thermal balance.

• Key Relationship: [ P_{in} = P_{out} ]

  • Balance in energy absorbed vs. energy emitted determines equilibrium temperature.

Conclusion and Implications

• For Earth, complicating factors such as greenhouse gases contribute to higher average temperatures than predicted by simple models.

• Understanding planetary atmospheres and their temperature dynamics also informs the potential for life, drawing attention to the importance of maintaining suitable conditions in terms of pressure and temperature.