Imagine a giant ruler at the planet's distance that is 1 arcsecond across in your image. Given this angular length, and this distance to the ruler in AU, you can calculate the physical length of the ruler in AU. This then allow you to convert your measurement of the semi-major axis from arcseconds to AU. 0/ 360° - length / circumference (of big circle) 360° 0 = angular length = 1 arcsecond length
Imagine a giant ruler at the planet's distance that is 1 arcsecond across in your image. Given this angular length, and this distance to the ruler in AU, you can calculate the physical length of the ruler in AU. This then allow you to convert your measurement of the semi-major axis from arcseconds to AU. 0/ 360° - length / circumference (of big circle) 360° 0 = angular length = 1 arcsecond length
College Physics
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ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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Transcribed Image Text:**Transcription for Educational Website:**
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**Understanding Angular Measurements in Astronomy**
Imagine a giant ruler at the planet's distance that is 1 arcsecond across in your image. Given this angular length, and this distance to the ruler in Astronomical Units (AU), you can calculate the physical length of the ruler in AU. This will then allow you to convert your measurement of the semi-major axis from arcseconds to AU.
**Diagram Explanation:**
- The diagram is a visual representation of how angular measurements translate to physical distances.
- It features a large circle segment with a labeled angle (θ) at the center.
- θ (theta) is denoted as the angular length, equivalent to 1 arcsecond.
- The formula provided in the diagram is: θ / 360° ≈ length / circumference (of the big circle).
- The circle's circumference is used to relate angular measurements to linear measurements.
- The diagram shows the central angle θ subtending an arc on the circle's circumference.
- The area shaded in blue and the yellow rectangle represent conceptual aids related to measurement.
Understanding these relationships is crucial for translating astronomical observations into meaningful physical distances.
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![The physical length of the ruler as a fraction of the circumference of the big circle is the same as the angular length of the ruler as a fraction of 360°:
\[
\frac{\text{length}}{\text{circumference}} = \frac{\theta}{360^\circ}
\]
Since circumference = 2π × radius and the radius of the big circle is the distance to the ruler/planet:
\[
\frac{\text{length}}{2\pi \times \text{distance}} = \frac{\theta}{360^\circ}
\]
Solving for the physical length of the ruler yields:
\[
\text{length} = 2\pi \times \text{distance} \times \frac{\theta}{360^\circ}
\]
**Question:** Use this equation to determine a conversion factor from 1 arcsecond to AU at the planet’s distance. You will need to convert θ = 1 arcsecond to degrees first.
\[
\boxed{\phantom{Length Answer Box}} \text{ AU}
\]
Show your work for this calculation.
\[
\boxed{\phantom{Work Explanation Box}}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdc986032-179a-454d-ad20-2c76f1bd2607%2Fa9419bda-db74-4092-aec5-875aa9164e96%2F0zzx8qxt_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The physical length of the ruler as a fraction of the circumference of the big circle is the same as the angular length of the ruler as a fraction of 360°:
\[
\frac{\text{length}}{\text{circumference}} = \frac{\theta}{360^\circ}
\]
Since circumference = 2π × radius and the radius of the big circle is the distance to the ruler/planet:
\[
\frac{\text{length}}{2\pi \times \text{distance}} = \frac{\theta}{360^\circ}
\]
Solving for the physical length of the ruler yields:
\[
\text{length} = 2\pi \times \text{distance} \times \frac{\theta}{360^\circ}
\]
**Question:** Use this equation to determine a conversion factor from 1 arcsecond to AU at the planet’s distance. You will need to convert θ = 1 arcsecond to degrees first.
\[
\boxed{\phantom{Length Answer Box}} \text{ AU}
\]
Show your work for this calculation.
\[
\boxed{\phantom{Work Explanation Box}}
\]
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