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Ch. 44 - Astrophysics and Cosmology
Giancoli Douglas - Physics for Scientists and Engineers 5th edition
Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook
All textbooksGiancoli Douglas 5th editionCh. 44 - Astrophysics and CosmologyProblem 52
Chapter 39, Problem 52

(a) In order to measure distances with parallax at 100 ly, what minimum angular resolution (in degrees) is needed?
(b) What diameter mirror or lens would be needed?

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1
To solve part (a), recall that parallax is the apparent shift in the position of a nearby star against the background of distant stars due to Earth's orbit around the Sun. The parallax angle \( \theta \) (in radians) is related to the distance \( d \) (in parsecs) by the formula \( \theta = \frac{1}{d} \). Convert the distance from light-years to parsecs using the conversion factor: \( 1 \, \text{parsec} = 3.26 \, \text{light-years} \).
Next, calculate the parallax angle \( \theta \) in radians for a distance of 100 light-years. Use the formula \( \theta = \frac{1}{d} \), where \( d \) is the distance in parsecs. Then, convert the angle from radians to degrees using the conversion factor \( 1 \, \text{radian} = 57.3 \, \text{degrees} \). This gives the minimum angular resolution needed in degrees.
For part (b), use the Rayleigh criterion to determine the diameter of the mirror or lens. The angular resolution \( \theta \) (in radians) is related to the wavelength \( \lambda \) of light and the diameter \( D \) of the aperture by the formula \( \theta = 1.22 \frac{\lambda}{D} \). Rearrange this formula to solve for \( D \): \( D = 1.22 \frac{\lambda}{\theta} \).
Assume a typical wavelength of visible light, such as \( \lambda = 550 \, \text{nm} \) (nanometers). Convert this wavelength to meters: \( 1 \, \text{nm} = 10^{-9} \, \text{m} \). Substitute the value of \( \lambda \) and the angular resolution \( \theta \) (in radians, calculated in part (a)) into the formula for \( D \).
Finally, calculate the required diameter \( D \) of the mirror or lens in meters. Ensure all units are consistent throughout the calculation, and express the result in meters or another appropriate unit of length.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Parallax

Parallax is the apparent shift in position of an object when viewed from different angles. In astronomy, it is used to measure distances to nearby stars by observing their position against distant background stars at different times of the year. The parallax angle is the angle subtended at the star by the baseline of the Earth's orbit, and it is inversely proportional to the distance to the star.

Angular Resolution

Angular resolution refers to the smallest angle over which details can be distinguished in an image. It is crucial in astronomy for determining how closely two objects can be positioned before they appear as a single point of light. The minimum angular resolution required for a given distance can be calculated using the formula θ = d / D, where θ is the angular resolution in radians, d is the distance to the object, and D is the diameter of the lens or mirror.
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Diameter of Lens or Mirror

The diameter of a lens or mirror directly affects its ability to resolve fine details and gather light. A larger diameter allows for better angular resolution and greater light-gathering power, which is essential for observing faint objects in the universe. The relationship between diameter and resolution can be quantified using the Rayleigh criterion, which states that the minimum resolvable angle is inversely proportional to the diameter of the aperture.
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Related Practice
Textbook Question

Use special relativity and Newton’s law of gravitation to show that a photon of mass m = E/c² just grazing the Sun will be deflected by an angle ∆θ given by ∆θ = 2GM/c²R, where G is the gravitational constant, R and M are the radius and mass of the Sun, and c is the speed of light. Put in values and show ∆θ = 0.87". (General Relativity predicts an angle twice as large, 1.74".)

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