Astro 120: Homework Chapter 17 - Problem 14

Key Concepts

• Flux: Amount of energy from a star received at a certain location.
• Luminosity: Total power output of a star.
• Distance (D): Distance between the source and the point where flux is received.

Master Equation for Flux

• Flux (F) received from a star is calculated using: [ F = \frac{L}{4\pi D^2} ]
• Where:
  • (L) is the luminosity (power) of the source.
  • (D) is the distance from the source.

Problem Overview

• Compare flux received by Jupiter from the Sun to the flux received by Earth.
• Important factors:
  • Both Earth and Jupiter receive power from the same source (the Sun).
  • Distance from the Sun differs between Earth and Jupiter.

Hypothesis

• Jupiter, being further from the Sun, receives less flux compared to Earth.
• Jupiter's distance from the Sun: 5 Astronomical Units (AU)
• Earth's distance from the Sun: 1 AU

Setting Up the Ratio

• Compare flux at Jupiter ((F_J)) to flux at Earth ((F_E)).
• Substitute values into the flux equation:
  • ( F_J = \frac{L}{4\pi D_{J-S}^2} )
  • ( F_E = \frac{L}{4\pi D_{E-S}^2} )
• Form the ratio: [ \frac{F_J}{F_E} = \frac{L}{4\pi D_{J-S}^2} \div \frac{L}{4\pi D_{E-S}^2} ]

Simplifying the Ratio

• Cancel out common terms (L and (4\pi)) in the ratio: [ \frac{1/D_{J-S}^2}{1/D_{E-S}^2} ]
• Simplify by flipping and multiplying: [ \frac{D_{E-S}^2}{D_{J-S}^2} ]

Calculation

• Substitute distance values (in AU):
  • (D_{E-S} = 1 AU)
  • (D_{J-S} = 5 AU)
• Calculate: [ \frac{1^2}{5^2} = \frac{1}{25} ]
• Jupiter receives (\frac{1}{25}) the flux compared to Earth.

Conclusion

• The amount of flux Jupiter receives is (\frac{1}{25}) of what Earth receives.
• Final answer: (F_J = \frac{1}{25} F_E)

Note

• The primary challenge often lies in setting up the right ratio and simplifying the equation.
• Units often cancel out, leaving a simple numeric ratio.