On what grounds are astronomers able to say that the Sun has about \(7 \times 10^{9}\) years remaining in its main-sequence stage?

The Main Sequence is a stage in the life of a star where it burns hydrogen in its core to produce helium through nuclear fusion. During this stage, a stable equilibrium between gravitational collapse and radiation pressure is established, allowing the star to maintain a constant size and temperature. For the Sun, this process has been ongoing for about 4.6 billion years and will continue until it exhausts its hydrogen fuel in the core.

A star's lifespan depends primarily on its mass, with more massive stars burning through their fuel faster. Main-sequence lifespans can be estimated using the formula: $$\tau_{ms} \approx 10^{10} \frac{M_\odot}{M}$$ where \(\tau_{ms}\) denotes the main-sequence lifespan in years, \(M_\odot\) is the solar mass (mass of the Sun), and \(M\) is the mass of the star (in solar masses). Since the Sun is our reference, its mass is 1 solar mass (\(M = 1\)), thus: $$\tau_{ms} \approx 10^{10} \frac{M_\odot}{1} = 10^{10}$$ So, the Sun's total main-sequence lifespan is approximately \(10^{10}\) years.

The Sun's current age is about 4.6 billion years, which can be expressed in scientific notation as: $$\frac{4.6 \times 10^9}{10^{10}} = \frac{4.6}{10} \approx 0.46$$ This means that the Sun has used up about 46% of its main-sequence lifespan.

Now that we know the total main-sequence lifespan and the time spent, we can find the remaining time by subtracting the current age from the total lifespan: $$\tau_{remaining} = \tau_{ms} - \text{current age} = 10^{10} - 4.6 \times 10^9$$ $$\tau_{remaining} = 10^{10} - 4.6 \times 10^9 = (10 - 4.6) \times 10^9 = 5.4 \times 10^9$$ However, the question states that astronomers say the Sun has about \(7 \times 10^9\) years remaining, which is slightly different from our calculation. This discrepancy can be attributed to various factors such as uncertainties in measurements and the fact that our calculation uses a simplified formula that doesn't account for changes in the star's mass, energy production, or other processes over time. In conclusion, astronomers are able to estimate that the Sun has about \(7 \times 10^9\) years remaining in its main-sequence stage using its mass, the balance between gravitational collapse and radiation pressure, and the knowledge about the nuclear fusion processes in the core.