Context and Why This Matters
Lecture 02 fixed how sharp an image a telescope can in principle deliver. This lecture answers the practical question every instrument designer must solve next: how large, in millimeters, does that sharp image actually appear at the focal plane, and how small must a detector’s pixels be to record it without throwing resolution away?
Learning Goals
- Derive the plate-scale relation linking angular separation on the sky to physical separation at the focal plane.
- Compute the plate scale and pixel scale of a real telescope-plus-instrument combination.
- Apply the Nyquist sampling criterion to determine whether a given pixel size adequately samples a telescope’s diffraction-limited or seeing-limited image.
Opening Phenomenon
The Dark Energy Camera (DECam), a 570-megapixel imager built from 62 CCDs on the 4 m Blanco telescope, uses 15 μm pixels and a 11.28 m effective focal length to achieve a 0.274 arcsec/pixel image scale -- a deliberate engineering choice, not an accident, that this lecture’s plate-scale relation shows was necessary to properly sample the site’s seeing disk.
Terms Developed in Context
These terms form the vocabulary of the lecture's reasoning arc; each is tied to the model, the working equation, and the evidence used to test the instrument.
- plate scale
Definition and use: Plate scale denotes the physical quantity, process, or instrument component captured by this lecture-specific statement: Multiplying the plate scale by a detector’s physical pixel size (in mm) gives the pixel scale in arcsec/pixel -- the practical number that determines both a camera’s field of view (pixel scale × detector width) and its ability to resolve fine structure.
Development in this lecture: A small angle θ (radians) subtended on the sky is imaged, by simple similar-triangles geometry through a telescope of effective focal length f, onto a physical size x = fθ at the focal plane; converting θ to arcseconds and f to millimeters gives the standard plate-scale relation s = 206265″/f[mm], in arcsec per mm.
Connections: It is used with pixel scale to interpret the lecture model. The quantitative connection is the working relation \(s = \dfrac{206265^{\prime\prime}}{f\,[\mathrm{mm}]}, \qquad \text{pixel scale} = s \times p\,[\mathrm{mm}], \qquad \text{Nyquist: pixel scale} \le \dfrac{\mathrm{FWHM}}{2}\).
Boundary or common confusion: Assuming a smaller pixel is always better. Undersampling wastes optical resolution, but excessively small pixels (oversampling) spread the same total signal across more pixels, each carrying its own read-noise contribution (Lecture 05), which can lower the effective signal-to-noise ratio for faint sources without any corresponding gain in usable resolution...
- pixel scale
Definition and use: Pixel scale denotes the physical quantity, process, or instrument component captured by this lecture-specific statement: Multiplying the plate scale by a detector’s physical pixel size (in mm) gives the pixel scale in arcsec/pixel -- the practical number that determines both a camera’s field of view (pixel scale × detector width) and its ability to resolve fine structure.
Development in this lecture: Every published instrument specification sheet for a large telescope’s imager (e.g., DECam, the Keck imagers, HST’s WFC3) states a plate scale in arcsec/mm or arcsec/pixel, computed directly from the telescope’s focal length via the same relation derived in this lecture, confirming the formula is the actual engineering tool used, not a textbook simplification.
Connections: It is used with plate scale, field of view to interpret the lecture model. The quantitative connection is the working relation \(s = \dfrac{206265^{\prime\prime}}{f\,[\mathrm{mm}]}, \qquad \text{pixel scale} = s \times p\,[\mathrm{mm}], \qquad \text{Nyquist: pixel scale} \le \dfrac{\mathrm{FWHM}}{2}\).
Boundary or common confusion: Assuming a smaller pixel is always better. Undersampling wastes optical resolution, but excessively small pixels (oversampling) spread the same total signal across more pixels, each carrying its own read-noise contribution (Lecture 05), which can lower the effective signal-to-noise ratio for faint sources without any corresponding gain in usable resolution...
- field of view
Definition and use: Field of view denotes the physical quantity, process, or instrument component captured by this lecture-specific statement: Multiplying the plate scale by a detector’s physical pixel size (in mm) gives the pixel scale in arcsec/pixel -- the practical number that determines both a camera’s field of view (pixel scale × detector width) and its ability to resolve fine structure.
Development in this lecture: The same telescope focal length produces a much smaller plate scale (arcsec/mm) than a smaller telescope of the same aperture ratio, exactly the inverse-focal-length scaling this lecture derives, which is why long-focal-length Cassegrain and coudé foci are chosen specifically for high-plate-scale spectroscopy (Lecture 09) while short-focal-length prime foci are chosen for wide-field imaging.
Connections: It is used with pixel scale, focal plane to interpret the lecture model. The quantitative connection is the working relation \(s = \dfrac{206265^{\prime\prime}}{f\,[\mathrm{mm}]}, \qquad \text{pixel scale} = s \times p\,[\mathrm{mm}], \qquad \text{Nyquist: pixel scale} \le \dfrac{\mathrm{FWHM}}{2}\).
Boundary or common confusion: Assuming a smaller pixel is always better. Undersampling wastes optical resolution, but excessively small pixels (oversampling) spread the same total signal across more pixels, each carrying its own read-noise contribution (Lecture 05), which can lower the effective signal-to-noise ratio for faint sources without any corresponding gain in usable resolution...
- focal plane
Definition and use: Focal plane denotes the physical quantity, process, or instrument component captured by this lecture-specific statement: A small angle θ (radians) subtended on the sky is imaged, by simple similar-triangles geometry through a telescope of effective focal length f, onto a physical size x = fθ at the focal plane; converting θ to arcseconds and f to millimeters gives the standard plate-scale relation s =...
Development in this lecture: This lecture answers the practical question every instrument designer must solve next: how large, in millimeters, does that sharp image actually appear at the focal plane, and how small must a detector’s pixels be to record it without throwing resolution away?
Connections: It is used with field of view, Nyquist sampling to interpret the lecture model. The quantitative connection is the working relation \(s = \dfrac{206265^{\prime\prime}}{f\,[\mathrm{mm}]}, \qquad \text{pixel scale} = s \times p\,[\mathrm{mm}], \qquad \text{Nyquist: pixel scale} \le \dfrac{\mathrm{FWHM}}{2}\).
Boundary or common confusion: Assuming a smaller pixel is always better. Undersampling wastes optical resolution, but excessively small pixels (oversampling) spread the same total signal across more pixels, each carrying its own read-noise contribution (Lecture 05), which can lower the effective signal-to-noise ratio for faint sources without any corresponding gain in usable resolution...
- Nyquist sampling
Definition and use: Nyquist sampling denotes the physical quantity, process, or instrument component captured by this lecture-specific statement: The Nyquist sampling criterion, borrowed directly from signal processing, requires at least two pixels across the FWHM of the smallest resolvable feature (the diffraction-limited or seeing-limited PSF) to avoid throwing away resolution the optics already deliver (undersampling), while...
Development in this lecture: Imagers deliberately designed with pixels close to half the expected seeing-disk or diffraction-limited FWHM (the Nyquist criterion) are found, in on-sky commissioning tests, to recover the full resolution the optics deliver, while imagers with coarser pixels are observed to lose resolving power regardless of how sharp the optical image actually is -- direct evidence that pixel sampling, not just optical quality, sets a real...
Connections: It is used with focal plane, undersampling to interpret the lecture model. The quantitative connection is the working relation \(s = \dfrac{206265^{\prime\prime}}{f\,[\mathrm{mm}]}, \qquad \text{pixel scale} = s \times p\,[\mathrm{mm}], \qquad \text{Nyquist: pixel scale} \le \dfrac{\mathrm{FWHM}}{2}\).
Boundary or common confusion: Assuming a smaller pixel is always better. Undersampling wastes optical resolution, but excessively small pixels (oversampling) spread the same total signal across more pixels, each carrying its own read-noise contribution (Lecture 05), which can lower the effective signal-to-noise ratio for faint sources without any corresponding gain in usable resolution...
- undersampling
Definition and use: Undersampling denotes the physical quantity, process, or instrument component captured by this lecture-specific statement: The Nyquist sampling criterion, borrowed directly from signal processing, requires at least two pixels across the FWHM of the smallest resolvable feature (the diffraction-limited or seeing-limited PSF) to avoid throwing away resolution the optics already deliver (undersampling), while...
Development in this lecture: Multiplying the plate scale by a detector’s physical pixel size (in mm) gives the pixel scale in arcsec/pixel -- the practical number that determines both a camera’s field of view (pixel scale × detector width) and its ability to resolve fine structure.
Connections: It is used with Nyquist sampling, oversampling to interpret the lecture model. The quantitative connection is the working relation \(s = \dfrac{206265^{\prime\prime}}{f\,[\mathrm{mm}]}, \qquad \text{pixel scale} = s \times p\,[\mathrm{mm}], \qquad \text{Nyquist: pixel scale} \le \dfrac{\mathrm{FWHM}}{2}\).
Boundary or common confusion: Assuming a smaller pixel is always better. Undersampling wastes optical resolution, but excessively small pixels (oversampling) spread the same total signal across more pixels, each carrying its own read-noise contribution (Lecture 05), which can lower the effective signal-to-noise ratio for faint sources without any corresponding gain in usable resolution...
- oversampling
Definition and use: Oversampling denotes the physical quantity, process, or instrument component captured by this lecture-specific statement: The Nyquist sampling criterion, borrowed directly from signal processing, requires at least two pixels across the FWHM of the smallest resolvable feature (the diffraction-limited or seeing-limited PSF) to avoid throwing away resolution the optics already deliver (undersampling), while...
Development in this lecture: Multiplying the plate scale by a detector’s physical pixel size (in mm) gives the pixel scale in arcsec/pixel -- the practical number that determines both a camera’s field of view (pixel scale × detector width) and its ability to resolve fine structure.
Connections: It is used with undersampling to interpret the lecture model. The quantitative connection is the working relation \(s = \dfrac{206265^{\prime\prime}}{f\,[\mathrm{mm}]}, \qquad \text{pixel scale} = s \times p\,[\mathrm{mm}], \qquad \text{Nyquist: pixel scale} \le \dfrac{\mathrm{FWHM}}{2}\).
Boundary or common confusion: Assuming a smaller pixel is always better. Undersampling wastes optical resolution, but excessively small pixels (oversampling) spread the same total signal across more pixels, each carrying its own read-noise contribution (Lecture 05), which can lower the effective signal-to-noise ratio for faint sources without any corresponding gain in usable resolution...
Evidence
- Every published instrument specification sheet for a large telescope’s imager (e.g., DECam, the Keck imagers, HST’s WFC3) states a plate scale in arcsec/mm or arcsec/pixel, computed directly from the telescope’s focal length via the same relation derived in this lecture, confirming the formula is the actual engineering tool used, not a textbook simplification.
- Imagers deliberately designed with pixels close to half the expected seeing-disk or diffraction-limited FWHM (the Nyquist criterion) are found, in on-sky commissioning tests, to recover the full resolution the optics deliver, while imagers with coarser pixels are observed to lose resolving power regardless of how sharp the optical image actually is -- direct evidence that pixel sampling, not just optical quality, sets a real imager’s effective resolution.
- The same telescope focal length produces a much smaller plate scale (arcsec/mm) than a smaller telescope of the same aperture ratio, exactly the inverse-focal-length scaling this lecture derives, which is why long-focal-length Cassegrain and coudé foci are chosen specifically for high-plate-scale spectroscopy (Lecture 09) while short-focal-length prime foci are chosen for wide-field imaging.
Derivation and Model
- A small angle θ (radians) subtended on the sky is imaged, by simple similar-triangles geometry through a telescope of effective focal length f, onto a physical size x = fθ at the focal plane; converting θ to arcseconds and f to millimeters gives the standard plate-scale relation s = 206265″/f[mm], in arcsec per mm.
- Multiplying the plate scale by a detector’s physical pixel size (in mm) gives the pixel scale in arcsec/pixel -- the practical number that determines both a camera’s field of view (pixel scale × detector width) and its ability to resolve fine structure.
- The Nyquist sampling criterion, borrowed directly from signal processing, requires at least two pixels across the FWHM of the smallest resolvable feature (the diffraction-limited or seeing-limited PSF) to avoid throwing away resolution the optics already deliver (undersampling), while using many more than two pixels per FWHM (oversampling) needlessly divides the same signal across more pixels, each with its own noise floor (Lecture 05).
Working Equation
\[ s = \dfrac{206265^{\prime\prime}}{f\,[\mathrm{mm}]}, \qquad \text{pixel scale} = s \times p\,[\mathrm{mm}], \qquad \text{Nyquist: pixel scale} \le \dfrac{\mathrm{FWHM}}{2} \]
Worked Example
- Keck plate scale from f=17.5 m: s = 206265″/(1000×17.5) = 11.787 arcsec/mm, roughly 3.29× the plate scale of HST (f=57.6 m, s=3.581 arcsec/mm) despite Keck’s much larger aperture, because plate scale depends on focal length, not aperture, alone.
- DECam pixel scale from its published f=11.28 m and p=15 μm pixels: pixel scale = 18.286 arcsec/mm × 0.015 mm = 0.274 arcsec/pixel, matching the camera’s own published specification.
- Nyquist check against Keck’s diffraction limit: half the V-band Rayleigh resolution is 6.9 milliarcsec/pixel -- far finer than DECam’s 0.274″/pixel, confirming (correctly) that DECam is designed to Nyquist-sample the site’s seeing disk, not Keck’s diffraction limit, since DECam sits on a different, seeing-limited 4 m telescope.
Common Misconception
Assuming a smaller pixel is always better. Undersampling wastes optical resolution, but excessively small pixels (oversampling) spread the same total signal across more pixels, each carrying its own read-noise contribution (Lecture 05), which can lower the effective signal-to-noise ratio for faint sources without any corresponding gain in usable resolution once the Nyquist criterion is already satisfied.
Active-Learning Guidance
Given DECam’s pixel scale and the Blanco 4 m telescope’s seeing-limited image quality (typically 0.9-1.1 arcsec FWHM at the site), estimate how many pixels sample the FWHM of a typical stellar image, and state whether this imager is undersampled, well-sampled, or oversampled relative to the Nyquist criterion.
Lab Connection
Lab 02 computes the plate scale and pixel scale for several real telescope-instrument pairs and checks each against the Nyquist criterion using the diffraction or seeing limit established in Lecture 02.
Synthesis Questions
- What was measured directly, or taken from a published instrument specification, in this lecture’s worked example, and what was derived from a physical law?
- Which assumption in the derivation would most change the interpretation if it were wrong?
- How does this lecture’s technique connect to the lab and problem set that follow it?
References
- OpenStax Astronomy 2e (local extracted reference copy: references/openstax-astronomy-2e-extracted.txt) covers observational instrumentation only descriptively (Chapter 6, astronomical instruments); this course derives the optics, detector, and error-budget physics explicitly, well beyond that introductory treatment.
- Every numeric result above is computed programmatically in
materials/ASTR340/src/generate_astr340_content.py, not hand-typed.