How likely is Earth going to be engulfed by the Sun during the red giant stage?

 The engulfment of Earth by the Sun during its red giant stage is a likely outcome in the very distant future, but it is not an immediate concern. The process you are referring to is related to the evolution of stars, including our Sun, as they exhaust their nuclear fuel.

Currently, the Sun is in the phase of its life known as the main sequence, where it primarily fuses hydrogen into helium in its core. In about 5 billion years, the Sun will enter the red giant phase. During this stage, it will expand and become much larger, likely reaching a size that could extend beyond the current orbit of Earth.

The expansion of the Sun into a red giant is expected to be gradual, and it will lose a significant amount of mass during this process. The outer layers of the Sun will be expelled into space, forming a shell of gas and dust known as a planetary nebula.

While the exact fate of Earth during the red giant phase is not certain, it is generally believed that the increased size of the Sun will cause the inner planets, including Earth, to be engulfed or severely affected by the expanding solar envelope. However, the timescale for this process is on the order of billions of years.

Before the red giant phase, other changes in the Sun's behavior, such as an increase in luminosity, are likely to make Earth uninhabitable. The timeframe for these changes is extremely long, and human civilization will likely face other challenges and changes long before the Sun reaches its red giant stage.

In summary, while the eventual engulfment of Earth by the Sun is a plausible scenario in the very distant future, it is not a concern for humanity in the short or even medium term. It's important to note that our understanding of stellar evolution is based on current scientific knowledge and may be refined as our understanding of astrophysics advances.