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Problem 58 Easy Difficulty

It has been proposed to use an array of infrared telescopes spread over thousands of kilometers of space to observe planets orbiting other stars. Consider such an array that has an effective diameter of 6000 km and observes infrared radiation at a wavelength of 10 $\mu$m. If it is used to observe a planet orbiting the star 70 Virginis, which is 59 light-years from our solar system, what is the size of the smallest details that the array might resolve on the planet? How does this compare to the diameter of the planet, which is assumed to be similar to that of Jupiter (1.40 $\times$ 10$^{5}$ km)? (Although the planet of 70 Virginis is thought to be at least 6.6 times more massive than Jupiter, its radius is probably not too different from that of Jupiter. Such large planets are thought to be composed primarily of gases, not rocky material, and hence can be greatly compressed by the mutual gravitational attraction of different parts of the planet.)

Answer

So $\Delta x$ is less than $1 \%$ of the diameter of the planet.

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Video Transcript

in this problem, we have to find the size of the smallest details that the area might resolve. And then we have to compare the size with the damage off the planet as we need to find this I dente X. And then we have to compare this Delta X with damage off the planet that is a d. P. In order to calculate the saddle eggs off you, The question Science theater is equals to 1.22 Lambda Divide by D here this language, the revenant off the light turned it is the damage off the length. So for smaller angle, uh, where theater is very, very small. This sine theta is approximately equals to theater, and then his squaring can be written as a theater is equals to 1.22 Lambda divided by D. Let's call it the question number one. Similarly, we can Why these three ties Theater is equals to Delta X, divided by oh now, by comparing these two equations and then solving the equation for that X, we can write, Delta X is equal to 1.22 Lambda are divided by deep, but throughout the values in tow this question this data exist equals to 1.22 into Lambda, which is equals to 1.0, multiplied by terrorists par minus 5 m into our which is equals toe 59 light years so we can watch it in tow meters. Eso we multiply 9.41 Multiply bitterness Part 15 meter Paarl idea divided by 6.0 multiplied by terrorists Paul 6 m From here we can wire the value for this eggs as their taxes equals two. 1.1 multiply Waiter is part 6 m or it can be written as Delta X is equal to 1 1000 and 100 kilometers. We can write the ratio as a delta x d y by D. P. So this will be 1102 kilometer divide by 1.4 multiplied by terrorists par five kilometers. So this is equals to 7.9. Multiply we turn is par minus three. So we can say that this Delta X is a smaller than this dp and off the question Thank you