太阳中心温度估计

Estimating the temperature at the center of the sun involves understanding the physical conditions and nuclear reactions that take place in its core. One of the ways to estimate this temperature is by using the concept of hydrostatic equilibrium and the ideal gas law.

Here’s a simplified version of the computation:

1. Hydrostatic Equilibrium: This states that the pressure gradient within the sun balances the gravitational force. Mathematically, this can be written as:

$$\frac{dP}{dr}=-\frac{GM(r)\rho (r)}{r^2}$$

where $P$ is the pressure, $r$ is the radial distance from the center, $G$ is the gravitational constant, $M(r)$ is the mass enclosed within radius $r$, and $\rho (r)$ is the density.

2. Ideal Gas Law: The core of the sun is composed mainly of hydrogen in the plasma state, so we can use the ideal gas law:

$$P=\frac{\rho {\kappa }_{B}T}{\mu m_H}$$

where ${\kappa }_B$ is the Boltzmann constant, $T$ is the temperature, $\mu$ is the mean molecular weight, and $m_H$ is the mass of a hydrogen atom.

3. Combining the Equations: We can combine the equations to solve for the temperature. First, rearrange the ideal gas law to solve for $T$:

$$T=\frac{P\mu m_H}{\rho {\kappa }_B}$$

4. Estimate Values:
   4.1 The pressure at the center of the sun,$P_c$, is estimated to be about 2.5 $\times$ $10^{16}{P}_{\alpha }$.
   4.2 The density at the center, ${\rho }_c$, is about 1.5 $\times$ $10^{5}kg/m^3$.
   4.3 The mean molecular weight $\mu$ is approximately 0.61 for a fully ionized gas composed mainly of hydrogen helium.
   4.4 The Boltzmann constant, ${\kappa }_B$, is 1.38 $\times$ $10^{-23}J/K$.
   4.5 The mass of a hydrogen atom, $m_H$, is 1.67 $\times$ $10^{-27}kg$.

Let’s compute the temperature $T$:
$$T=\frac{P_c\mu m_H}{{\rho }_c{\kappa }_B}$$

Pugging in the values:
$$T=\frac{(2.5\times 10^{16}P\alpha )(0.61)(1.67\times 10^{-27}kg)}{(1.5\times 10^{5}kg/m^3)(1.38\times 10^{-23}J/K)}$$

$$T\approx 1.23\times 10^{7}K$$

Therefore, the estimated temperature at the center of the sun is approximately $1.23\times 10^{7}K$ Kelvin, or about 12 million Kelvin.