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Chapter 7: Problem 1

The angular displacement of an image (in radians) due to aberration is approximately \(v / c,\) as long as \(v \ll c .\) Use the fact that the Earth orbits the Sun once per year at a distance of \(1.50 \times 10^{8} \mathrm{km}\) to find the maximum displacement of a star's image due to the motion of the Earth. Express your answer in arc seconds.

Short Answer

Expert verified
The maximum displacement is approximately 20.49 arc seconds.

Step by step solution

01

Determine the Earth's orbital speed

The Earth orbits the Sun once per year in a nearly circular path. To find the speed, use the formula for the circumference of a circle and the duration of the orbit. Circumference of Earth's orbit: \[ 2 \times \pi \times (1.50 \times 10^{8} \text{ km}) \]Duration of one year in seconds: \[ 1 \text{ year} = 365.25 \times 24 \times 60 \times 60 \text{ seconds} \]Compute the speed: \[ v = \frac{2 \pi \times (1.50 \times 10^{8})}{365.25 \times 24 \times 60 \times 60} \]
02

Calculate the Earth's orbital speed

Perform the calculation to find the speed. Earth's speed: \[ v = \frac{2 \pi \times 1.50 \times 10^8}{3.15576 \times 10^7} \approx 29800 \text{ m/s} \]
03

Use the aberration formula

Using the formula for angular displacement due to aberration: \[ \theta = \frac{v}{c} \]where \( c \) is the speed of light (\( 3 \times 10^8 \text{ m/s} \)) and \( v \) is the Earth's orbital speed.
04

Compute the angular displacement in radians

Substitute the values into the formula: \[ \theta = \frac{29800}{3 \times 10^8} \approx 9.93 \times 10^{-5} \text{ radians} \]
05

Convert radians to arc seconds

There are \( 206265 \) arc seconds in a radian. Multiply the angular displacement in radians by this conversion factor: \[ \theta_{\text{arc seconds}} = 9.93 \times 10^{-5} \times 206265 \approx 20.49 \text{ arc seconds} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Angular Displacement
Angular displacement measures the angle through which an object has moved about a given point or axis. It is often used in the context of circular motion. For example, in Earth's orbit around the Sun, angular displacement shows how far Earth has traveled along its orbit.
Angular displacement is calculated using the formula: \( \theta = \frac{v}{c} \), where \(v\) is the velocity and \(c\) is the speed of light. This formula assumes that the speed \(v\) is much smaller than the speed of light \(c\).
Understanding this concept helps us relate motion dimensions (like distance and speed) to angles, which is crucial in fields such as astronomy and physics.
Aberration of Light
Aberration of light refers to the apparent shift in the position of a celestial object due to the Earth's motion. This effect is caused by the combination of the speed of light and the velocity of the observer.
An everyday analogy would be walking in the rain; the direction of raindrops seems to alter because of your movement, similar to light from a star changing its apparent position.
In calculations, aberration of light uses the formula \( \theta = \frac{v}{c} \). By knowing Earth's orbital speed and the speed of light, we can calculate the angular displacement to predict this shift accurately.
Earth's Orbital Speed
Earth's orbital speed is how fast Earth travels in its orbit around the Sun. To quantify this, we need to know the circumference of Earth's nearly circular orbit and the time it takes to complete one orbit (365.25 days).
The formula used is: \( v = \frac{2 \pi \times r}{T} \), where \(r\) is the radius of the Earth's orbit (1.50 × 10^8 km) and \(T\) is the time in seconds for one year.
Using this method, Earth's orbital speed appears to be approximately 29800 m/s. This speed is crucial for calculating phenomena like the aberration of light and understanding the dynamics of our solar system.
Arc Seconds
An arc second is a unit of angular measurement equivalent to 1/3600th of a degree. This unit is commonly used in astronomy for measuring small angles, such as the apparent sizes and separations of stars.
There are 206265 arc seconds in one radian. To convert an angular displacement from radians to arc seconds, we multiply the value in radians by this conversion factor.
In our example, an angular displacement of approximately 9.93 × 10^-5 radians converts to 20.49 arc seconds. This conversion helps us understand and express tiny angular movements in a more manageable and relatable way.

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Most popular questions from this chapter

Show that if tachyons exist, their rest mass must be an imaginary number if the energy is to be real for \(v > c\). (An imaginary number is the square root of a negative number.)

You and your friend carry identical clocks. Your friend passes by in a rapidly moving train. As your clock ticks off 1.00 s, you see your friend's clock tick off 0.50 s. How much time would your friend see your clock tick off in the time it takes your friends clock to tick off 1.00 s?

Some radioactive particles are traveling at \(0.999 c .\) If their lifetime is \(10^{-20}\) s when they are at rest, what is their lifetime at this speed? How far do they travel in that time (as viewed in the frame at which they are moving at \(0.999 c ?\)

We define the redshift, \(z\), as the shift in wavelength, \(\Delta \lambda\) divided by the rest wavelength \(\lambda_{0}\) On the assumption that only radial motions are involved, find an expression for \(z\) as a function of \(v / c\).

You and your friend carry identical meter sticks and identical clocks. Your friend goes by on a fast moving train, holding the meter stick parallel to the direction of motion of the train. If in the time it takes your clock to tick off 1.00 s you see 0.5 s tick off on your friend's clock, how long does your friend's stick appear to you? How long does your stick appear to your friend, assuming the sticks are parallel to each other.
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