Mars (6800 km diameter) is viewed through a telescope on a night when it is 1.1 x 10⁸ km from the earth. Its angular size as seen through the eyepiece is 0.50°, the same size as the full moon seen by the naked eye. If the eyepiece focal length is 25 mm, how long is the telescope?

검증된 단계별 안내

Step 1: Understand the problem. The telescope's length refers to the focal length of the objective lens. To find this, we need to use the magnification formula for telescopes, which relates the angular size of the object, the focal length of the eyepiece, and the focal length of the objective lens.

Step 2: Recall the magnification formula for telescopes: \( M = \frac{f_{\text{objective}}}{f_{\text{eyepiece}}} \), where \( M \) is the magnification, \( f_{\text{objective}} \) is the focal length of the objective lens, and \( f_{\text{eyepiece}} \) is the focal length of the eyepiece.

Step 3: Calculate the magnification \( M \). The angular size of Mars as seen through the telescope is given as 0.50°, and the angular size of Mars without magnification can be calculated using the formula \( \theta = \frac{d}{D} \), where \( d \) is the diameter of Mars (6800 km) and \( D \) is the distance to Mars (1.1 \(\times\) 10^8 \(\text{ km}\)).

Step 4: Substitute the values into \( \theta = \frac{d}{D} \) to find the angular size of Mars without magnification. Then, use \( M = \frac{\text{angular size through telescope}}{\text{angular size without magnification}} \) to find the magnification.

Step 5: Rearrange the magnification formula \( M = \frac{f_{\text{objective}}}{f_{\text{eyepiece}}} \) to solve for \( f_{\text{objective}} \): \( f_{\text{objective}} = M \cdot f_{\text{eyepiece}} \). Substitute the calculated magnification and the given eyepiece focal length (25 mm) to find the telescope's length.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Angular Size

Angular size refers to the apparent size of an object as seen from a specific point, measured in degrees. It is determined by the actual size of the object and its distance from the observer. In this case, Mars has an angular size of 0.50°, which is crucial for calculating how the telescope will magnify the planet's image.

Telescope Focal Length

The focal length of a telescope is the distance from the lens or mirror to the point where light converges to form a clear image. It plays a vital role in determining the magnification of the telescope, which is calculated by the ratio of the focal length of the telescope to the focal length of the eyepiece. A longer focal length results in higher magnification.

Magnification

Magnification in telescopes is the factor by which the telescope enlarges the image of an object. It is calculated using the formula: Magnification = Focal Length of Telescope / Focal Length of Eyepiece. Understanding magnification is essential for determining how the telescope will present Mars compared to its actual size and distance.