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Chapter 6: Problem 42

The Mars Reconnaissance Orbiter (MRO) flies at an average altitude of \(280 \mathrm{km}\) above the martian surface. If its cameras have an angular resolution of 0.2 arcsec, what is the size of the smallest objects that the \(M R O\) can detect on the martian surface?

Short Answer

Expert verified
The smallest objects that MRO can detect are 2.71 meters in size.

Step by step solution

01

- Convert Angular Resolution to Radians

First, convert the angular resolution from arcseconds to radians. Use the conversion factor where 1 arcsec equals \(4.848 \times 10^{-6} \) radians. Multiply the given angular resolution by this factor: \[0.2 \, \text{arcsec} \times 4.848 \, \times 10^{-6} \frac{\text{radians}}{\text{arcsec}} = 9.696 \times 10^{-7} \text{radians} \]
02

- Use Small-Angle Approximation Formula

Use the small-angle approximation, which states that for very small angles, the linear size \(L\) of the smallest object that can be resolved is given by \[ L = \theta d \] where \(\theta\) is the angular resolution in radians and \(d\) is the distance from the observer to the object. Here, \(d = 280 \text{km} = 280 \times 10^{3} \text{m} \).
03

- Calculate the Linear Size

Plug the values into the formula. The smallest object size \ L \ is given by: \[ L = 9.696 \times 10^{-7} \times 280 \times 10^{3} = 2.71 \text{m} \]

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Key Concepts

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

angular resolution
Angular resolution describes a camera's ability to distinguish between two separate points. It is often measured in arcseconds, a unit of angular measurement.
One arcsecond is 1/3600 of a degree, and it helps in determining the detail an imaging system can capture.
An important point to remember is that a higher angular resolution means the camera can see finer details.
In this exercise, the Mars Reconnaissance Orbiter (MRO) has an angular resolution of 0.2 arcseconds. It's crucial to convert this unit into radians for further calculations.
The conversion factor is where 1 arcsecond equals (

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Most popular questions from this chapter

The advantage of an interferometer is that a. the resolution is dramatically improved. b. the focal length is dramatically increased. c. the light-gathering power is dramatically increased. d. diffraction effects are dramatically decreased. e. chromatic aberration is dramatically decreased.

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