If nuclear fusion in the sun's core suddenly stopped today, would the sky be dark in the daytime tomorrow? why

No, it would take millions of years before we noticed an effect. The reason is that the photons produced by nuclear reaction in the Sun's interor scatter very often. They travel only a very short distance inbetween collisions with the ions. After a collision they travel in some pretty much random direction compared to the direction before the collision. This is known as a random walk.

If the photons travel a distance d on average betwen collisions, then after N collisions they'll have travelled on average a distance of sqrt(N) d
(not N d)

d is about 10^(-6) meters. The radius of the Sun is of order 10^8 meters, so the number of collisions N needed for a photon to escape is of order 10^28. The total distance the photons will have travelled inbetween all the collsions is thus 10^22 meters, which is about a million lightyears.

If nuclear fusion in the sun's core suddenly stopped today, the sky would not be dark in the daytime tomorrow. The reason is that the sun's core is not directly responsible for producing the visible light we see during the day. Instead, it generates energy through nuclear fusion, primarily converting hydrogen into helium.

The energy generated from nuclear fusion in the sun's core takes a long time to reach the surface, as it goes through several processes like radiation and convection. The sunlight we observe is primarily produced by the sun's photosphere, which is the visible surface of the sun. The photosphere emits photons of light due to the intense heat generated from the fusion reactions happening below.

If nuclear fusion were to cease in the sun's core, it would lead to a gradual reduction in the energy output of the sun. However, the sunlight we receive on Earth is a result of the immense amount of energy that has already been emitted from the sun's interior over millions of years. It would take approximately 8 minutes and 20 seconds for the light to travel from the sun to the Earth, so even if nuclear fusion stopped, we would still receive sunlight for a few more minutes. Therefore, the sky would not go dark immediately; it would take some time for the effects to be noticeable.

If nuclear fusion in the sun's core were to suddenly stop, the sky would not be completely dark in the daytime the next day. This is because there are two sources of light that illuminate our planet during the day: direct sunlight and sunlight scattered by the Earth's atmosphere.

Direct sunlight is the primary source of illumination during the day. It is produced by nuclear fusion reactions in the sun's core, where hydrogen atoms combine to form helium, releasing an enormous amount of energy in the process. This energy is then emitted as light and other forms of electromagnetic radiation. If nuclear fusion were to stop, the sun's core would no longer produce this energy, resulting in a drastic decrease in the intensity of direct sunlight reaching the Earth.

However, even without direct sunlight, the sky would still not be completely dark. Sunlight interacts with the Earth's atmosphere, causing the atmosphere to scatter the light in different directions. This scattering phenomenon is responsible for the blue color of the sky. The molecules and particles in the atmosphere scatter shorter wavelength light (blue and violet) more effectively than longer wavelength light (red and orange). This scattering of light in all directions ensures that some light still reaches our eyes, even if the source of the light is not as bright or intense.

Therefore, while the lack of direct sunlight would significantly reduce the brightness and intensity of daylight, the scattering of light by the Earth's atmosphere would still result in some illumination during the day. The sky would appear noticeably darker, possibly with a bluish-gray hue, but not completely dark.