To estimate the minimum amount of energy a rogue star would need to power an Earth-sized planet for billions or trillions of years, we can consider the energy requirements for sustaining life on such a planet and the energy output of a star.
Energy requirements for sustaining life: Earth receives energy from the Sun in the form of sunlight, which is crucial for supporting life processes, including photosynthesis and maintaining a habitable temperature range. The total energy received from the Sun is approximately 174 petawatts (1 petawatt = 10^15 watts). However, not all of this energy is absorbed or used efficiently by Earth's ecosystems.
Energy output of a star: Stars emit energy through a process called stellar nucleosynthesis, primarily through nuclear fusion in their cores. The amount of energy a star produces depends on its mass, luminosity, and lifespan. Massive stars have higher luminosities but shorter lifespans, while smaller stars have lower luminosities but longer lifespans.
Assuming we are looking for a conservative estimate of a rogue star's energy output, we can consider a low-mass star known as a red dwarf. Red dwarfs are the most abundant type of star in the universe and have relatively long lifespans. Their energy output is significantly lower than that of larger stars like our Sun.
A typical red dwarf may have a luminosity of about 0.01% to 10% of the Sun's luminosity. Assuming the lower end of this range, let's consider a red dwarf with a luminosity that is 0.01% of the Sun's luminosity. This would correspond to approximately 0.0001 times the energy received by Earth from the Sun.
Given these estimates, the rogue star would need to emit energy at a rate of approximately 0.0001 times the energy Earth receives from the Sun to sustain an Earth-sized planet for billions or trillions of years. However, it's important to note that sustaining life on a planet depends on numerous other factors beyond just energy, such as the star's radiation levels, habitable zone, and the presence of other necessary conditions.
Keep in mind that our current understanding of rogue stars and the potential for sustaining life under such conditions is limited. Further research and discoveries may provide more precise estimates in the future.