Lecture Notes: Parallaxing Distance of a Star

Introduction

• Focus on parallaxing distance of a star.
• Homework problem number 17, Chapter 17.
• Key concept: How to calculate the distance of a star using parallax.

Parallax and Geometry

• Parallax: Apparent shift in position of a nearby star due to Earth’s orbit.
• Astronomical unit (AU): Average distance from Earth to Sun, approx. (1.496 \times 10^{11}) meters.
• Right-angled trigonometry is used but not required to solve exams.

Trigonometry Basics

• Tangent of angle P:
  • Defined as the ratio of the opposite side to the adjacent side in a right triangle.
  • Opposite side: Parallax distance (R).
  • Adjacent side: Distance of the star from the Sun (D).
• Relationship: ( \tan(P) = \frac{R}{D} )

Calculation of Star Distance

• Given: Parallax angle (P) and Earth-Sun distance (R).
• Formula to solve for D: [ D = \frac{R}{\tan(P)} ]
• Special Case: If P is very small, ( \tan(P) \approx P ) in radians.

Concept of Parsec

• Arc Second: A unit of angular measurement, (1/3600) of a degree.
• If (P = 1) arc second, then distance (D \approx 3.26) light-years, a unit known as a parsec.
• Parsec: Named from the parallax of one arc second.

Practical Application

• Formula for star distance in parsecs: [ D = \frac{1}{P} ]
  • D is in parsecs, R in AU, P in arc seconds.
• Example: For a parallax angle of 0.001 arc seconds,
  • ( D = \frac{1}{0.001} = 1000 ) parsecs.

Measurement Challenges

• Measuring angles smaller than 1 arc second is challenging with Earth-based telescopes.
• Requires Earth-orbiting satellites like Gaia for precise measurements.

Conclusion

• Understanding of parallax is crucial to determining astronomical distances.
• The parallax method simplifies into a formula for practical calculations involving parsecs.