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Chapter 6: Problem 52

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
About 0.271 meters.

Step by step solution

01

Convert angular resolution to radians

First, convert the given angular resolution from arcseconds to radians. Since 1 arcsecond = \(\frac{1}{3600}\) degrees, and 1 degree = \(\frac{\pi}{180}\) radians, calculate the angular resolution in radians as follows: \[ 0.2 \, \text{arcsec} = 0.2 \, \times \, \frac{1}{3600} \, \text{degrees} = 0.2 \, \times \, \frac{1}{3600} \, \times \, \frac{\pi}{180} \, \text{radians} \approx 9.696\times10^{-7} \text{radians} \]
02

Use the formula for linear size from angular resolution

The linear size of the smallest object the MRO can detect can be determined using the following formula: \[ \Delta l = r \, \theta \] where \( r \) is the altitude and \( \theta \) is the angular resolution. Here, \( r = 280 \, \text{km} = 280 \, \times \, 10^3 \, \text{m}\) and \( \theta = 9.696 \times 10^{-7} \, \text{radians} \).
03

Calculate the smallest detectable object size

Multiply the altitude by the angular resolution in radians: \[ \Delta l = 280 \, \times \, 10^3 \, \times \, 9.696 \, \times \, 10^{-7} \, \text{m} \ \approx 0.271 \, \text{m} \] This is the size of the smallest objects that the MRO can detect on the martian surface.

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Key Concepts

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

angular resolution
When observing distant objects, such as the surface of Mars, the clarity of the image is limited by the angular resolution of the imaging device. Angular resolution is the smallest angle between two points that can be distinguished by the instrument. A smaller angular resolution means finer detail can be seen.
To convert angular resolution from arcseconds to radians, remember:
  • 1 arcsecond = \frac{1}{3600}\ degree
  • 1 degree = \frac{\frac{\pi}{180}} radians

Thus, 0.2 arcseconds is converted as follows:
\[ \theta = 0.2 \times \frac{1}{3600} \times \frac{\pi}{180}\ radians \approx 9.696 \times 10^{-7} \text{ radians} \]
. Utilizing this, we can proceed to assess the smallest detectable objects using the MRO.
linear size calculation
To determine the smallest object size that the Mars Reconnaissance Orbiter can detect, we use the formula that relates linear size to angular resolution. The formula is: \[ \Delta l = r \cdot \theta \]
where:
  • \

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