Imagine that our Sun was somehow replaced by a \(1-\mathrm{M}_{\odot}\) white dwarf star, and that our Earth continued in an orbit of semimajor axis \(1 \mathrm{AU}\) around this star. Discuss what effects this would have on our planet. What would the white dwarf look like as seen from Earth? Could you look at it safely with the unaided eye? Would the Earth's surface temperature remain the same as it is now?

White dwarfs are fascinating celestial bodies, remnants of stars like our own Sun after they have expended most of their nuclear fuel. Despite having a mass similar to the Sun's, usually around \(1-\text{M}_{\odot}\), they are incredibly small, roughly the size of Earth. This vast difference in volume compared to normal stars like the Sun leads to extreme density, where their atoms are packed together tightly.

Unlike the raging infernos of their previous stages, white dwarfs emit energy that is a remnant of their past thermonuclear processes. This leads to a luminosity far less than that of our Sun and a cooler temperature. It is this lower luminosity and innate coolness that significantly influences their impact on surrounding celestial bodies.

Should our Sun be replaced by a white dwarf, the consequences for Earth would be dire. The decreased luminosity would mean less heat and light reaching our planet, creating a much darker and cooler environment. This could initiate a global ice age, eradicating many life forms accustomed to Earth's current climate.

The white dwarf's gravity would still maintain Earth's orbital path due to their similar masses. However, without adequate warmth and light, Earth's biosphere would undergo drastic changes, potentially collapsing ecosystems and hindering the process of photosynthesis, which is essential for life as we know it.

A white dwarf's luminosity is significantly lower than that of the Sun. Luminosity, the total amount of energy a star emits per second, is a crucial factor affecting the climate of planets in its orbit. For a white dwarf, the compressed nature of its physical structure leads to reduced luminosity, despite its surface temperature being relatively high.

This reduced emission of energy and light means that, were a white dwarf to replace the Sun, the Earth would receive considerably less radiation, insufficient to maintain our current atmospheric and surface temperatures. For example, the white dwarf's output might not sustain the present-day hydrological cycle, leading to alterations in weather patterns and potentially the loss of surface water through freezing.

Contrary to what you might expect, the safety of viewing a white dwarf with the unaided eye is not guaranteed. Despite their lower overall luminosity, white dwarfs can exhibit a high surface brightness due to the concentrated emission of light over a smaller area.

Therefore, looking directly at a white dwarf could pose risks similar to staring at the Sun and should be done with caution. Proper solar filtration or indirect viewing techniques, such as pinhole projectors or eclipse glasses, should be utilized to observe these dense remnants of stars safely.

The surface temperature of Earth is inextricably linked to the amount of solar radiation it receives. If a white dwarf were to take the Sun's place, the substantial drop in luminosity would lead to a significant decrease in Earth's surface temperature.

The cooler, less intense radiation would be insufficient to sustain our current temperate climate. This could plunge our world into a prolonged state of winter, drastically transforming the environment. Areas that are currently habitable might become inhospitable due to the extreme cold, affecting not only human life but the entire flora and fauna on the planet.