Angular Resolution 1. 1

Diffraction Pattern through a Circular Aperture The analysis for light passing through a circular aperture such as a telescope is similar although more sophisticated (and can be found by consulting optics textbooks). The solution was first obtained in 1835 by Sir George Airy, and the resulting diffraction pattern – concentric rings – is known as the Airy pattern. The central bright spot is known as the Airy disk. Airy Pattern 91.0% 93.8% Enclosed power 83.8% Half maximum Normalized intensity 0.0175 0.0042 2

Angular Resolution When two point sources are separated by a sufficiently large angle in the sky that their Airy disks do not overlap, the two sources are said to be well resolved. (Note: by well resolved, we mean that we can easily separate the two sources, not that the individual sources are resolved.) Two well resolved objects (stars) 3

Angular Resolution As the angular separation between the two sources get smaller, their Airy disks start to overlap and the two sources become more difficult to resolve (separate). At what stage should we consider the two sources to no longer be resolved (perceived as separate)? At which stage does it become difficult/impossible to say whether there are one or two point sources? Two barely resolved objects (stars) 4

Angular Resolution Lord Rayleigh introduced an empirical criterion – now known as the Rayleigh criterion – for resolving two point sources: the central maximum of one diffraction pattern should lie at or beyond the first minimum of the other. That is, two point sources are considered resolved if their minimum angular separation in the sky satisfies (measured in radians) where  is the received wavelength of light and D is the entrance diameter of the telescope. Lord Rayleigh, 1842-1919 5

Diffraction Pattern through a Circular Aperture The intensity profile of the Airy disk is close to that of a Gaussian function. We shall return to the relevance of this point when discussing the seeing disk. Angular separation between the central maximum and first minimum is almost exactly equal to that between the full-width half maximum, providing a convenient manner to measure the angular resolution. Airy Pattern Solid curve: Airy disk Dashed curve: Gaussian profile Half maximum 6

Angular Resolution 1. Rayleigh’s criterion for angular resolution, θmin = 1.22 λ/D. θmin = 27.7 arcsec. 7

Angular Resolution 1. Rayleigh’s criterion for angular resolution, θmin = 1.22 λ/D. θmin = 27.7 arcsec. Angular diameter of Moon = 31.07 arcmin, and angular diameter of Jupiter = 31.8 arcsec at conjunction and 46.9 arcsec at opposition. 8

Angular Resolution 9

Angular Resolution 1. Rayleigh’s criterion for angular resolution, θmin = 1.22 λ/D. θmin = 27.7 arcsec. Angular diameter of Moon = 31.07 arcmin, and angular diameter of Jupiter = 31.8 arcsec at conjunction and 46.9 arcsec at opposition. Easily resolved for Moon, in principle resolved also for Jupiter. 10

Angular Resolution 2. 11

Angular Resolution 2. Rayleigh’s criterion for angular resolution, θmin = 1.22 λ/D. θmin = 0.69 arcsec. 12

Angular Resolution 2. Rayleigh’s criterion for angular resolution, θmin = 1.22 λ/D. θmin = 0.69 arcsec. Minimum size of crater that can be resolved is Dcrater = 1.29 km. 13

Angular Resolution 2. Rayleigh’s criterion for angular resolution, θmin = 1.22 λ/D. θmin = 0.69 arcsec. Minimum size of crater that can be resolved is Dcrater = 1.29 km. Only achievable if the seeing is much better than 0.69 arcsec, which is unlikely. 14

Atmospheric Seeing This is generally not the case (depending on the wavelength of light and the size-scale of atmospheric turbulence) and therefore the incoming wavefront is distorted. 15

Atmospheric Seeing Because of atmospheric seeing, light from a point source (e.g., star) is spread into a “seeing disk” that has a radial intensity profile resembling a Gaussian function (normal distribution). One of the best sites in the world for astronomical observations is the summit of Mauna (elevation of 4200 m) in Hawaii. The atmospheric seeing at this site is typically 0.6″. 16

Interferometry 3. Get interference maximum when Δ = d sin θ = mλ, where m = 0, 1, 2, … d = 12742 km 17

Interferometry 3. Get interference maximum when Δ = d sin θ = mλ, where m = 0, 1, 2, … For small angles θ, d θ = λ so that θ = λ/d = 3.4 mas. d = 12742 km 18

Interferometry 19

Interferometry 20

Interferometry 4. Suppose you would like to construct an interferometer with an angular resolution equivalent to that provided by a single-dish telescope with a diameter of 1 km. (a) What would be the required maximum separation of the two most distant antennas (use the formal definition for the angular resolution of a single-dish telescope and an interferometer)? (b) If you would like the interferometer to have a collecting area equal to a single- dish telescope with a diameter of 200 m, how many antennas would you need to construct if each antenna has a diameter of 10 m? 21

Interferometry 4. Suppose you would like to construct an interferometer with an angular resolution equivalent to that provided by a single-dish telescope with a diameter of 1 km. (a) What would be the required maximum separation of the two most distant antennas (use the formal definition for the angular resolution of a single-dish telescope and an interferometer)? Angular resolution single dish telescope is θmin = 1.22 λ/D. Angular resolution of an interferometer is θmin = λ/D. Thus, need to separate the two antennas by 1/1.22 = 0.82 km. (b) If you would like the interferometer to have a collecting area equal to a single- dish telescope with a diameter of 200 m, how many antennas would you need to construct if each antenna has a diameter of 10 m? 22

Interferometry 4. Suppose you would like to construct an interferometer with an angular resolution equivalent to that provided by a single-dish telescope with a diameter of 1 km. (a) What would be the required maximum separation of the two most distant antennas (use the formal definition for the angular resolution of a single-dish telescope and an interferometer)? Angular resolution single dish telescope is θmin = 1.22 λ/D. Angular resolution of an interferometer is θmin = λ/D. Thus, need to separate the two antennas by 1/1.22 = 0.82 km. (b) If you would like the interferometer to have a collecting area equal to a single- dish telescope with a diameter of 200 m, how many antennas would you need to construct if each antenna has a diameter of 10 m? Collecting area of antenna = πr2, where r is the antenna radius. N = 400. 23

Michelson Interferometer 5. Consider a one-dimensional interferometer comprising two antennas spaced separated by a distance d = 10 km along the x-direction. Assume that the interferometer receives monochromatic radio waves of wavelength  = 21cm, and perform observations at or near transit so that the angle  between the source direction and the normal to the antenna separation is very small. The interference fringes measured by such an interferometer for a celestial object that is extended in the x-direction can be most easily envisioned as the superposition of interference fringes from individual point sources distributed along the length of that object in the x-direction. 10 km 24

Michelson Interferometer 5. Consider a one-dimensional interferometer comprising two antennas spaced … a) If the interferometer measures zero visibility for two points sources with equal intensities, what is the minimum angular separation of the two sources in the sky along the x-direction? Position angle from transit 25

Michelson Interferometer 5. Consider a one-dimensional interferometer comprising two antennas spaced … a) If the interferometer measures zero visibility for two points sources with equal intensities, what is the minimum angular separation of the two sources in the sky along the x-direction? Say that one source produces the fringe colored black. Say that another source at an angular separation θ from the first produces the fringe colored red. The superposition of these two fringe patterns is a constant (blue line); i.e., VM = 0. Time Position angle from transit 26

Michelson Interferometer 5. Consider a one-dimensional interferometer comprising two antennas spaced … a) If the interferometer measures zero visibility for two points sources with equal intensities, what is the minimum angular separation of the two sources in the sky along the x-direction? To obtain VM = 0, Δ = d sin θ i.e., θ = ½ λ/d = 2.2 θ Time Position angle from transit 27

Michelson Interferometer 5. Consider a one-dimensional interferometer comprising two antennas spaced … b) If the interferometer measures unity visibility, what is the minimum non-zero angular separation of these two sources in the sky along the x-direction? Say that one source produces the fringe colored black. Say that another source at an angular separation θ from the first produces the fringe colored red. The superposition of these two fringe patterns (blue fringe) results in VM = 1. Time Position angle from transit 28

Michelson Interferometer 5. Consider a one-dimensional interferometer comprising two antennas spaced … b) If the interferometer measures unity visibility, what is the minimum non-zero angular separation of these two sources in the sky along the x-direction? To obtain VM = 1, Δ = d sin θ ≅ d θ = λ i.e., θ = λ/d = 4.3 θ Time Position angle from transit 29

Michelson Interferometer 5. Consider a one-dimensional interferometer comprising two antennas spaced … c) Sketch the visibility amplitude (y-axis) as the separation of the two antennas is increased from zero to d (x-axis) for the angular separation between the two sources derived in parts (a) and (b) (in your sketch, only the trend, and not the exact shape, of the visibility profile is important). Can you now see how the angular resolution of an interferometer, defined as /d, relates to the visibilities of two point sources? 30

Michelson Interferometer 5. Consider a one-dimensional interferometer comprising two antennas spaced … c) Sketch the visibility amplitude (y-axis) as the separation of the two antennas is … For situation a), consider what happens as you decrease the separation of the two antennas from d = 10 km. Finge pattern when d = 10 km, giving VM = 0, requiring Δ = ½ λ; i.e., θ = ½ λ/d. Time Position angle from transit 31

Michelson Interferometer 5. Consider a one-dimensional interferometer comprising two antennas spaced … c) Sketch the visibility amplitude (y-axis) as the separation of the two antennas is … For situation a), consider what happens as you decrease the separation of the two antennas from d = 10 km. Finge pattern when d = 7.5 km, giving 0 < VM < 1. Time Position angle from transit 32

Michelson Interferometer 5. Consider a one-dimensional interferometer comprising two antennas spaced … c) Sketch the visibility amplitude (y-axis) as the separation of the two antennas is … For situation a), consider what happens as you decrease the separation of the two antennas from d = 10 km. Finge pattern when d = 5.0 km, giving a larger VM than for d = 7.5 km. Time Position angle from transit 33

Michelson Interferometer 5. Consider a one-dimensional interferometer comprising two antennas spaced … c) Sketch the visibility amplitude (y-axis) as the separation of the two antennas is … For situation a), consider what happens as you decrease the separation of the two antennas from d = 10 km. Finge pattern when d = 2.5 km, giving a larger VM than for d = 5.0 km. Time Position angle from transit 34

Michelson Interferometer 5. Consider a one-dimensional interferometer comprising two antennas spaced … c) Sketch the visibility amplitude (y-axis) as the separation of the two antennas is … For situation a), consider what happens as you decrease the separation of the two antennas from d = 10 km. As d  0 km, VM  1. Time Position angle from transit 35

Michelson Interferometer 5. Consider a one-dimensional interferometer comprising two antennas spaced … c) Sketch the visibility amplitude (y-axis) as the separation of the two antennas is … For situation a), consider what happens as you decrease the separation of the two antennas from d = 10 km. As d  0 km, VM  1. 1 VM Time d 10 km 36

Michelson Interferometer 5. Consider a one-dimensional interferometer comprising two antennas spaced … c) Sketch the visibility amplitude (y-axis) as the separation of the two antennas is … For situation b), consider what happens as you decrease the separation of the two antennas from d = 10 km. Finge pattern when d = 10 km, giving VM = 1, requiring Δ = d sin θ ≅ d θ = λ; i.e., θ = λ/d (definition for angular resolution of an interferometer). Time Position angle from transit 37

Michelson Interferometer 5. Consider a one-dimensional interferometer comprising two antennas spaced … c) Sketch the visibility amplitude (y-axis) as the separation of the two antennas is … For situation b), consider what happens as you decrease the separation of the two antennas from d = 10 km. Finge pattern when d = 7.5 km, giving 0< VM < 1. Time Position angle from transit 38

Michelson Interferometer 5. Consider a one-dimensional interferometer comprising two antennas spaced … c) Sketch the visibility amplitude (y-axis) as the separation of the two antennas is … For situation b), consider what happens as you decrease the separation of the two antennas from d = 10 km. Finge pattern when d = 5.0 km, giving VM = 0. Time Position angle from transit 39

Michelson Interferometer 5. Consider a one-dimensional interferometer comprising two antennas spaced … c) Sketch the visibility amplitude (y-axis) as the separation of the two antennas is … For situation b), consider what happens as you decrease the separation of the two antennas from d = 10 km. Finge pattern when d = 2.5 km, giving 0< VM < 1. Time Position angle from transit 40

Michelson Interferometer 5. Consider a one-dimensional interferometer comprising two antennas spaced … c) Sketch the visibility amplitude (y-axis) as the separation of the two antennas is … For situation b), consider what happens as you decrease the separation of the two antennas from d = 10 km. As d  0 km, VM  1. Time Position angle from transit 41

Michelson Interferometer 5. Consider a one-dimensional interferometer comprising two antennas spaced … c) Sketch the visibility amplitude (y-axis) as the separation of the two antennas is … For situation b), consider what happens as you decrease the separation of the two antennas from d = 10 km. As d  0 km, VM  1. For two point sources, angular resolution of an interferometer corresponds to the situation where d increases to the point where VM = 1 after going through a null. 1 VM Time d 10 km 42

Michelson Interferometer 5. Consider a one-dimensional interferometer comprising two antennas spaced … d) If the interferometer measures zero visibility for an extended source with uniform brightness, what is the minimum angular size of this source in the sky along the x-direction? Consider two sources separated such that Δ = d sin θ ≅ d θ = ½ λ, resulting in VM = 0. Time Position angle from transit 43

Michelson Interferometer 5. Consider a one-dimensional interferometer comprising two antennas spaced … d) If the interferometer measures zero visibility for an extended source with uniform brightness, what is the minimum angular size of this source in the sky along the x-direction? Consider two sources adjacent to the previous two sources, also separated such that Δ = ½ λ, resulting in VM = 0. Time Position angle from transit 44

Michelson Interferometer 5. Consider a one-dimensional interferometer comprising two antennas spaced … d) If the interferometer measures zero visibility for an extended source with uniform brightness, what is the minimum angular size of this source in the sky along the x-direction? Consider adding two more adjacent sources such that Δ = ½ λ, resulting in VM = 0. Time Position angle from transit 45

Michelson Interferometer 5. Consider a one-dimensional interferometer comprising two antennas spaced … d) If the interferometer measures zero visibility for an extended source with uniform brightness, what is the minimum angular size of this source in the sky along the x-direction? Consider adding sets of two adjacent sources such that Δ = d sin θ ≅ d θ = ½ λ, resulting in VM = 0. The “first” and “last” sources are required to have Δ = d sin θ ≅ d θ  λ; i.e., minimum angular size  4.3. Time Position angle from transit 46

Michelson Interferometer 5. Consider a one-dimensional interferometer comprising two antennas spaced … e) Sketch the visibility amplitude (along the vertical axis) as the separation of the two antennas is decreased from d to zero (along the horizontal axis) for the angular size of the source derived in part (d) (in your sketch, only the trend, and not the exact shape, of the visibility profile is important). Can you now see how the angular resolution of an interferometer, defined as /d, relates to the visibilities of an extended source? 47

Michelson Interferometer 5. Consider a one-dimensional interferometer comprising two antennas spaced … e) Sketch the visibility amplitude (along the vertical axis) as the separation of the two antennas is decreased from d to zero (along the horizontal axis) for the angular size of the source derived in part (d) (in your sketch, only the trend, and not the exact shape, of the visibility profile is important). Can you now see how the angular resolution of an interferometer, defined as /d, relates to the visibilities of an extended source? As d decreases, Δ = d sin θ ≅ d θ < ½ λ, resulting in VM > 0. Time Position angle from transit 48

Michelson Interferometer 5. Consider a one-dimensional interferometer comprising two antennas spaced … e) Sketch the visibility amplitude (along the vertical axis) as the separation of the two antennas is decreased from d to zero (along the horizontal axis) for the angular size of the source derived in part (d) (in your sketch, only the trend, and not the exact shape, of the visibility profile is important). Can you now see how the angular resolution of an interferometer, defined as /d, relates to the visibilities of an extended source? As d decreases, Δ = d sin θ ≅ d θ < ½ λ, resulting in VM > 0. Time Position angle from transit 49

Michelson Interferometer 5. Consider a one-dimensional interferometer comprising two antennas spaced … e) Sketch the visibility amplitude (along the vertical axis) as the separation of the two antennas is decreased from d to zero (along the horizontal axis) for the angular size of the source derived in part (d) (in your sketch, only the trend, and not the exact shape, of the visibility profile is important). Can you now see how the angular resolution of an interferometer, defined as /d, relates to the visibilities of an extended source? As d decreases, Δ = d sin θ ≅ d θ < ½ λ, resulting in VM > 0. Time Position angle from transit 50

Michelson Interferometer 5. Consider a one-dimensional interferometer comprising two antennas spaced … e) Sketch the visibility amplitude (along the vertical axis) as the separation of the two antennas is decreased from d to zero (along the horizontal axis) for the angular size of the source derived in part (d) (in your sketch, only the trend, and not the exact shape, of the visibility profile is important). Can you now see how the angular resolution of an interferometer, defined as /d, relates to the visibilities of an extended source? As d decreases, Δ = d sin θ ≅ d θ  0, resulting in VM  1. Time Position angle from transit 51

Michelson Interferometer 5. Consider a one-dimensional interferometer comprising two antennas spaced … e) Sketch the visibility amplitude (along the vertical axis) as the separation of the two antennas is decreased from d to zero (along the horizontal axis) for the angular size of the source derived in part (d) (in your sketch, only the trend, and not the exact shape, of the visibility profile is important). Can you now see how the angular resolution of an interferometer, defined as /d, relates to the visibilities of an extended source? For an extended source, angular resolution corresponds to the first null in the visibility function. 1 VM d 10 km Position angle from transit 52