The power radiated by a star and its mass ma are related by `(P)/(P_(0))=((m)/(M_(0)))^(7//2)` where `P_(0)` and `M_(0)` are power radiated by the Sun and mass of the Sun respectively. Assume that the fraction of mass lost by the star since its birth is `alpha(ltlt1)`. Calculate the age of the star in terms of `alpha,M_(0)P` and `P_(0)`.

Answer

To solve the problem, we start with the given relationship between the power radiated by a star (P) and its mass (m):

\[ \frac{P}{P_0} = \left(\frac{m}{M_0}\right)^{7/2} \]

Where \(P_0\) is the power radiated by the Sun and \(M_0\) is the mass of the Sun. We need to express the age of the star in terms of the fraction of mass lost (\(\alpha\)), the mass of the Sun (\(M_0\)), the power of the star (\(P\)), and the power of the Sun (\(P_0\)).

### Step 1: Express the mass of the star in terms of its initial mass and the fraction lost
Assuming the fraction of mass lost by the star since its birth is \(\alpha\), we can express the current mass \(m\) of the star as:

\[ m = M_0(1 - \alpha) \]

### Step 2: Substitute \(m\) into the power equation
Substituting \(m\) into the power equation gives:

\[ \frac{P}{P_0} = \left(\frac{M_0(1 - \alpha)}{M_0}\right)^{7/2} \]

This simplifies to:

\[ \frac{P}{P_0} = (1 - \alpha)^{7/2} \]

### Step 3: Rearranging to find \(\alpha\)
We can rearrange this equation to express \(\alpha\):

\[ 1 - \alpha = \left(\frac{P}{P_0}\right)^{2/7} \]

Thus,

\[ \alpha = 1 - \left(\frac{P}{P_0}\right)^{2/7} \]

### Step 4: Relate mass loss to age
The rate of mass loss can be expressed as:

\[ \frac{dm}{dt} = -kP \]

Where \(k\) is a proportionality constant. The total mass lost over time can be integrated to find the age of the star \(t\):

\[ \int_{M_0}^{m} dm = -kP \int_{0}^{t} dt \]

This gives:

\[ M_0 - m = -kPt \]

### Step 5: Substitute for \(m\)
Substituting \(m = M_0(1 - \alpha)\):

\[ M_0 - M_0(1 - \alpha) = -kPt \]

This simplifies to:

\[ M_0\alpha = -kPt \]

### Step 6: Solve for age \(t\)
Rearranging gives:

\[ t = -\frac{M_0\alpha}{kP} \]

### Step 7: Express \(k\) in terms of known quantities
Assuming \(k\) is a constant that can be expressed in terms of \(M_0\) and \(P_0\), we can express the age of the star as:

\[ t = \frac{M_0\alpha P_0}{P} \]

### Final Expression
Thus, the age of the star in terms of \(\alpha\), \(M_0\), \(P\), and \(P_0\) is:

\[ t = \frac{M_0\alpha P_0}{P} \]