Resolution (DP IB Physics: HL)

A woman is rowing a boat towards two ancient statues situated on an island. The woman can just resolve the images of the two statues.

Describe whether the Rayleigh criterion applies to this scenario.

The pupils in the woman's eyes have a diameter of 2.3 mm. The average wavelength of the sunlight is 570 nm.

She can just resolve the images of the two statues when her distance from them is 10 km.

Around the other side of the island, there are two further statues. These are situated at a distance apart of 0.73 m. 

Determine how far the woman can row away from the island before she can no longer resolve these two statues.

The sun begins to set and the average wavelength of light is now 500 nm. The woman's pupils dilate to give a wider diameter of 3.1 mm.

Determine the new distance at which the woman can resolve the second set of statues.

Two distant stars are detected on earth using a telescope with a circular receiving dish.

Outline the criteria to enable light from the two stars to be just resolved.

The stars are both situated in a galaxy 5.4 × 10²⁵ m from the Earth and are 3.7 × 10²⁰ m apart.

The wavelength of light received from the stars is 9.3 µm.

Calculate the minimum size of the circular receiving dish required to resolve the image of the two stars.

The same telescope is used to observe a different galaxy at a distance of 3.8 × 10²⁷ m from Earth. The astronomer wishes to resolve two stars with a separation of 2.3 × 10²² m. 

Determine whether the telescope is able to resolve the light from the stars.

The astronomer working with the telescope wishes to determine if sodium is present in the stars. To do this they use a diffraction grating to split the light from the star. 

The two wavelengths of light they wish to resolve are 589.0 nm and 589.6 nm.

Determine the resolving power of the diffraction grating required to resolve these wavelengths.

Monochromatic light of wavelength 610 nm is incident normally on a single narrow slit. A diffraction pattern is formed on a screen.

Sketch a graph to show the variation of relative light intensity with angle, measured from the centre of the slit.

The single narrow slit is replaced by a diffraction grating.

Describe how the Rayleigh criterion allows the new diffraction pattern to be predicted.

A second monochromatic light source of 630 nm is placed alongside the original monochromatic light source of wavelength 610 nm. The width of the beam of light from these sources is 2.0 mm.

The diffraction grating is replaced with with 200 lines per mm. 

Calculate the resolving power of the new grating under the same conditions

A binary star system is observed using a telescope with a circular lens. The images of the two stars can just be resolved according to the Rayleigh criterion.

Outline what is meant by the statement “just resolved according to the Rayleigh criterion” in this context.

The circular lens has a diameter of 4 mm. The distance from the Earth to the binary star system is 6.2 × 10¹⁶ m, and the average wavelength of light emitted by the stars is 470 nm.

Determine the separation of the two stars.

The telescope is directed at a different area of the sky to detect further binary systems, using a larger diameter lens. A potential binary system with the same average wavelength of light, and the same separation between the stars is detected at 9.1 × 10¹⁸ m distance from the earth.

Determine the diameter of the new lens.

The light from the binary system is passed through a diffraction grating with 430 lines per mm. The difference in wavelength between the two stars is 1.3 nm. 

A man stands on an airstrip at night as a car approaches him. His pupils have a diameter of 3.2 mm and the wavelength of the light is 490 nm. He can just resolve the light from the headlights of the car into two distinct points.

The car is moving at 20 m s⁻¹ and the headlights are 1.6 m apart.

Draw a labelled sketch of the variation of intensity of light with angle, θ, as it falls on the man's retina at the point where the two headlights can just be resolved.

Hovering above the airstrip is a helicopter on which the man can just resolve two light sources. The helicopter is hovering at 25% of the distance at which the car lights were able to be resolved, and the separation between the lights is 0.45 m. The pupil diameter of the man remains unchanged.

Determine the wavelength of the light emitted by the helicopter's lights.

The man is holding blue lights in each hand to guide the helicopter to a safe landing. He switches to using red lights whilst maintaining the same separation.

Explain the effect on how the helicopter pilot observes the lights

There is a supermassive black hole at the centre of the Andromeda galaxy. It is difficult for astronomers using a telescope to resolve images of the region around this black hole directly. 

(i)

Sketch on the axis the variation in the intensity of the diffraction pattern produced when light from a point object passes through a circular aperture. 

[2]

Astronomers investigating the supermassive black hole at the centre of the Andromeda galaxy detect radio waves at a frequency of 240 GHz. By correlating the information from several radio telescopes, they can obtain the same resolution as a single radio telescope with a diameter of 4500 km. 

Calculate the angular separation of the black hole as detected by the telescope on Earth. 

The centre of the Andromeda galaxy is 27 000 light years away from the Earth and the black hole at its centre has a mass equal to 9 × 10³⁶ kg. The following equation can be used to calculate the radius of a black hole: 

The Hale telescope in Palomar California was designed and built by the California Institute of Technology in the 1940s. 

Draw a diagram to show the path of two rays emerging from stellar objects, parallel to the axis, through the Hale telescope, as far as the eyepiece. 

The Hale telescope was one of the first telescopes to use a diffraction grating. The diameter of the diffraction grating is 12.5 cm with 250 lines per mm. The wavelength of the incident light is 2.5 µm. 

Calculate the resolving power of the diffraction grating. 

To reduce atmospheric absorption problems the telescope was built in the Palomar Mountain Range. 

A binary star system in Alpha Centuri can be observed from Earth. Alpha Centuri A and B are 4.35 light years away from Earth and are 23 AU from each other. 

Calculate the angular separation of the two stars from Earth. Give your final answer to one significant figure. 

State and explain the changes that would be observed on the graph from your answer to (b) part (i) if Alpha Centuri A and B both moved further away from the Earth by an equal amount. 

The James Webb Space Telescope has a mirror diameter of 6.5 m and operates with a wavelength range of 0.6 µm to 28 µm. 

Determine whether it can resolve two binary stars 3.6 × 10¹⁰ m apart at a distance of 3.19 × 10¹⁷ m away. 

The James Webb Space Telescope (JWST) was launched with the criteria to "see further" than the Hubble Space Telescope (HST). The following information is available: 

Evaluate how each of the factors given in the table will enable the James Webb Space Telescope to be effective in meeting its criteria.