2.8 Measuring the Astronomical Unit

Pre-Lecture Reading 2.8

Video Lecture

•
Measuring the Astronomical Unit (15:12)

Supplementary Notes

Parallax

A blue planet lies along the larger of a pair of two concentric circles. The center of the circles is marked with a dot. The diameter of this planet is labeled baseline, and a line is drawn from each edge of the planet through the common center of the circles. The angle formed by the intersection of these two lines is labeled theta, which is equal to the angular shift. The inner circle is labeled 360 degrees. The image has this equation: theta divided by 360 degrees is approximately equal to baseline divided by circumference (of big circle).

Figure 1: Earth-baseline parallax

A star lies along the left edge of the larger of two concentric circles. The star is orbited by a planet, which is directly above and below the star in the image. A line drawn between these two positions is labeled baseline, and a line is drawn from each of these positions through the common center of the circles (marked with a dot). The angle formed by the intersection of these two lines is labeled theta, which is equal to angular shift. The inner circle is labeled 360 degrees. The image has this equation: theta / 360 degrees is approximately equal to baseline / circumference (of big circle).

Figure 2: Stellar parallax

•
In both cases:

( 9 )

angular shift

360°

baseline

(2π × distance)

- •
  angular shift = apparent shift in angular position of object when viewed from different observing points
- •
  baseline = distance between observing points
- •
  distance = distance to object

•
If you know the baseline and the angular shift, solving for the distance yields:

( 10 )

distance =

baseline

2π

360°

angular shift

Note: Angular shift needs to be in degrees when using this equation.

•
If you know the baseline and the distance, solving for the angular shift yields:

( 11 )

angular shift =

360°

2π

baseline

distance

Note: Baseline and distance need to be in the same units when using this equation.

Standard astronomical baselines

•
Earth-baseline parallax
- •
  baseline = diameter of Earth = 12,756 km
- •
  This is used to measure distances to objects within our solar system.
•
Stellar parallax
- •
  baseline = diameter of Earth's orbit = 2 astronomical units (or AU)
- •
  1 AU is the average distance between Earth and the sun.
- •
  This is used to measure distances to nearby stars.

Radar Ranging

( 12 )

2 × distance = c × time

•
distance = distance to object
•
2 × distance = total distance that radio waves travel
•
c = speed of light = speed of radio waves
•
time = time that it takes for radio waves to travel to object, reflect, and travel back

Measuring the Astronomical Unit

Step 1

Venus is often the closest planet to Earth, making it a natural target for both Earth-baseline parallax and radar ranging measurements, which yield the distance to Venus in physically meaningful units, such as kilometers.

•
For example, when Venus is at closest approach to Earth, Earth-baseline parallax and radar ranging both measure the distance to Venus to be approximately 4.5 × 10⁷ km.

Step 2

Set the distance to Venus in kilometers equal to the distance to Venus in AU.

•
For example, when Venus is at closest approach to Earth, the distance to Venus is approximately 1 AU – 0.7 AU = 0.3 AU.

•
Hence: 0.3 AU = 4.5 × 10⁷ km

A diagram of the orbits of Venus and Earth around the sun. The orbital radius of Venus is labeled 0.7 AU. The orbital radius of Earth is 1 AU. The distance between Earth and Venus is 0.3 AU.

Figure 3

Step 3

Solve for 1 AU.

•
For example, when Venus is at closest approach to Earth:

0.3 AU

4.5 × 10⁷ km

0.3 AU

0.3

4.5 × 10⁷ km

0.3

1 AU

1.5 × 10⁸ km

Lab Link

Material presented in this unit is related to material presented in Lab 4 of Astronomy 101 Laboratory: Our Place in Space. In Lab 4: Cosmic Distance Ladder I: Parallax, we:

•
Use parallax to measure distances to objects on Earth.
•
Use parallax and Earth's diameter to measure distances to objects within our solar system.
•
Use parallax measurements of objects within our solar system to measure the astronomical unit (AU).
•
Use parallax and the AU to measure distances to nearby stars.

Video Lab Summary

•
Cosmic Distance Ladder I: Parallax (29:27)

Assignment 2

•
Do Questions 7 and 8.