To estimate the number of stellar protons in the universe, we will follow these steps:
### Step 1: Calculate the Volume of a Typical Star
The volume \( V \) of a star can be calculated using the formula for the volume of a sphere:
\[
V = \frac{4}{3} \pi r^3
\]
Given that the radius \( r \) of the sun is \( 700,000 \) km, we first convert this to centimeters:
\[
r = 700,000 \, \text{km} = 700,000 \times 10^5 \, \text{cm} = 7 \times 10^8 \, \text{cm}
\]
Now, substituting this value into the volume formula:
\[
V = \frac{4}{3} \pi (7 \times 10^8)^3
\]
### Step 2: Calculate the Mass of a Typical Star
The mass \( M \) of the star can be calculated using the formula:
\[
M = \text{Density} \times \text{Volume}
\]
Given that the density of the star is \( 1.4 \, \text{g/cm}^3 \):
\[
M = 1.4 \, \text{g/cm}^3 \times V
\]
### Step 3: Estimate the Mass of Protons in a Typical Star
The mass of protons in a star can be estimated based on the composition of the star. We know that the star is composed of \( \frac{3}{4} \) hydrogen and \( \frac{1}{4} \) helium by mass. The mass of hydrogen contributes one proton per atom, while helium contributes two protons per atom. Therefore, the total mass of protons can be calculated as:
\[
\text{Mass of protons} = M \times \left(\frac{3}{4} + \frac{1}{4} \times \frac{1}{2}\right)
\]
### Step 4: Calculate the Total Number of Protons in the Universe
To find the total number of protons in the universe, we multiply the number of protons in one star by the total number of stars in the universe:
\[
\text{Total protons} = \text{Mass of protons} \times \text{Number of stars}
\]
Given that there are approximately \( 1 \times 10^{23} \) stars in the universe.
### Step 5: Final Calculation
After calculating the mass of a typical star and the mass of protons in that star, we can estimate the total number of protons in the universe.
1. Calculate the volume:
\[
V \approx \frac{4}{3} \times 3.14 \times (7 \times 10^8)^3 \approx 1.44 \times 10^{27} \, \text{cm}^3
\]
2. Calculate the mass:
\[
M \approx 1.4 \times 1.44 \times 10^{27} \approx 2.016 \times 10^{27} \, \text{g}
\]
3. Calculate the mass of protons:
\[
\text{Mass of protons} \approx 2.016 \times 10^{27} \times \left(\frac{3}{4} + \frac{1}{8}\right) \approx 1.8 \times 10^{27} \, \text{g}
\]
4. Total protons in the universe:
\[
\text{Total protons} \approx 1.8 \times 10^{27} \times 1 \times 10^{23} \approx 1.8 \times 10^{50}
\]
### Conclusion
Thus, the estimated number of stellar protons in the universe is approximately \( 1 \times 10^{80} \).
