Abstract
The size of a telescope affects what its ideal observation targets are. Larger telescopes will have higher angular resolution for greater wavelengths of light, while telescopes with smaller apertures will have better angular resolution for shorter wavelengths. However, this dependence is only valid to an order of magnitude: increasing the order of magnitude for the wavelength but not the aperture will result in a worse angular resolution.
Introduction
J-band refers to a specific range of wavelengths, from 1.1 to 1.4 microns (a micrometer is equal to 10-6 meters). This value means that light in the J-band spectrum falls into the infrared. Considering CCAT and Keck, two telescopes with different apertures (25 meters and 10 meters respectively), how does the angular resolution compare? The CCAT will detect wavelengths up to 850 microns, while Keck observes in J-band (1.1-1.4 microns). This question elaborates on my earlier blog post concerning the differences between large and small telescopes: it is a practical example of why you might want a telescope with a smaller aperture in some cases.
Solution
The relationship between aperture, wavelength and angular resolution can be defined as follows:
sin q = (1.220)(l /D)
Taking care to convert units (using meters as the standard unit), we can calculate the values for each telescope.
For CCAT, sin q = (1.220)(850*10-6 /25) = 4.15*10-5 and using the small angle approximation, we can say that q = 4.15*10-5. This value is given in radians, so a quick conversion to degrees gives us .0024° as the angular resolution of CCAT at this wavelength. Since there are 3600 arcseconds in a degree, this is an angular resolution of 8.64 arcseconds.
Keck, on the other hand, is observing at a much smaller frequency, at about 1 micron, with a slightly smaller aperture as well. So, sin q = (1.220)(1*10-6 /10) = 1.22*10-7. This is equivalent to 0.000007°, which is equal to 0.03 arcseconds. This is two orders of magnitude smaller than the resolution for CCAT, which makes sense since the wavelength Keck is studying here is two orders of magnitude smaller as well.
Epilogue
The angular resolution for CCAT was better for 850 micrometer wavelengths than Keck’s resolution was for 1 micrometer waves: we can compare CCAT’s angular resolution of 8.64 arcseconds to Keck’s resolution of 0.03 arcseconds. However, if we were to study different wavelengths of light, then the angular resolution for each telescope would also change – this one measurement of angular resolution cannot be considered the standard by which we judge the telescope.
Questions for Further Study (next blog post?)
Is it true that the formula for angular resolution can only work for ground-based telescopes? That is, how broadly can the term ‘aperture’ be defined? Can the resolution of x-ray telescopes be defined this way – do x-ray telescopes even have an angular resolution?
Acknowledgements
Thanks to Iryna Butsky and Monica He for their help!
I'm glad you decided to do a post on the optics activity. It seems to have gotten lost in the shuffle on other students' blogs.
ReplyDeleteYou have the effect of wavelength and aperture size correct, but your conclusions are a bit off. Given a choice, astronomers would almost always opt for a larger telescope over a smaller one. The tradeoff is between diameter and cost. Keck uses painstakingly polished glass mirrors, while radio telescopes can be essentially stamped out of metal.
Also, I'm a bit confused about your first equation. Why does "q" appear on both sides?
Oops! That would be a transcription error.. thanks!! I've fixed that now...
ReplyDeleteThese are great! A few questions:
ReplyDelete-Do you remember where the relationship comes from that the diffraction limit q ~ lambda/D? If D is the diameter of the aperture, what is the intensity of the light at angle q off from the center?
-Did anything in deriving the diffraction limit depend on being on the surface of the Earth? Did anything depend on optical vs radio vs X-rays? Can you use this reasoning to answer your questions about whether we can apply the diffraction limit for X-rays? for space-based missions?
For an optical telescope, you have many pixels, so you can see multiple stars in the same image (just like if you used a really good digital camera to take a picture of the sky). Each of those stars is blurred to the size of its diffraction limit. Having a larger aperture (smaller diffraction limit) does not hurt us most of the time, but it can get very expensive both to pay for bigger mirrors and to pay for bigger detectors (CCDs) to capture our wonderful high-resolution images.
For a radio telescope, if you only have one telescope, then you only have one pixel. A radio telescope adds up everything that it sees within one resolution unit (ie within the diffraction limit) and within a small wavelength range (the "bandpass") and reports that number. Then the tradeoff between big and small diffraction limit becomes important. Matthew Stevenson, one of the grad students here, is working on a project where he looks at the radio emission from "cold cores," where the gas is cold and dense and there is a lot of dust. He wants to measure the total flux from each object, so for him it is useful to have a broad angular resolution so that the whole object fits in one resolution unit. On the other hand, he doesn't want to accidentally include other objects, so there's a limit to how large his resolution can be. By contrast, I am using the same telescope as him to map "extended objects" where I want to see the structure in them, so I scan the telescope across a dust cloud that is larger than the diffraction limit of the telescope, and then I piece together a map based on what flux the telescope sees when I point it at different parts of the dust cloud.