The Sun has mass `2.0xx10^(30)` kg and radiates energy at the rate `3.9xx10^(26)W` (A) At what rate is it mass changing? (b) What fraction of its original mass has it lost in this way since it began to burn hydrogen, about `4.5xx10^(9)` y ago?

Answer

To solve the problem, we will break it down into two parts as stated in the question.

### Part (A): Rate of Mass Change

1. **Given Data**:
- Power (P) = \(3.9 \times 10^{26} \, \text{W}\)
- Speed of light (c) = \(3 \times 10^8 \, \text{m/s}\)

2. **Using the Mass-Energy Equivalence**:
According to Einstein's mass-energy equivalence principle, the energy radiated can be expressed as:
\[ E = mc^2 \]
The power (rate of energy emission) can be expressed as:
\[ P = \frac{dE}{dt} = \frac{d(mc^2)}{dt} \]
Since \(c^2\) is a constant, we can simplify this to:
\[ P = c^2 \frac{dm}{dt} \]

3. **Rearranging for Rate of Mass Change**:
Rearranging the equation gives us:
\[ \frac{dm}{dt} = \frac{P}{c^2} \]

4. **Substituting the Values**:
\[ \frac{dm}{dt} = \frac{3.9 \times 10^{26}}{(3 \times 10^8)^2} \]

5. **Calculating**:
\[ c^2 = (3 \times 10^8)^2 = 9 \times 10^{16} \]
\[ \frac{dm}{dt} = \frac{3.9 \times 10^{26}}{9 \times 10^{16}} = 4.33 \times 10^{10} \, \text{kg/s} \]

### Part (B): Fraction of Mass Lost

1. **Time Duration**:
- Time (t) = \(4.5 \times 10^9 \, \text{years}\)
- Convert years to seconds:
\[ t = 4.5 \times 10^9 \times 3.154 \times 10^7 \, \text{s/year} \approx 1.42 \times 10^{17} \, \text{s} \]

2. **Total Mass Lost**:
The total mass lost over this time can be calculated as:
\[ \text{Mass lost} = \frac{dm}{dt} \times t \]
\[ \text{Mass lost} = 4.33 \times 10^{10} \, \text{kg/s} \times 1.42 \times 10^{17} \, \text{s} \approx 6.15 \times 10^{27} \, \text{kg} \]

3. **Original Mass of the Sun**:
- Original mass \(M_0 = 2.0 \times 10^{30} \, \text{kg}\)

4. **Calculating the Fraction of Mass Lost**:
The fraction of mass lost is given by:
\[ \text{Fraction lost} = \frac{\text{Mass lost}}{M_0} \]
\[ \text{Fraction lost} = \frac{6.15 \times 10^{27}}{2.0 \times 10^{30}} \approx 3.08 \times 10^{-3} \]

### Final Answers:
- (A) The rate at which the Sun is losing mass is approximately \(4.33 \times 10^{10} \, \text{kg/s}\).
- (B) The fraction of its original mass that the Sun has lost is approximately \(3.08 \times 10^{-3}\).