“The light that burns twice as bright burns half as long…” The origin of the above quote (with “flame” or “candle” sometimes substituted for “light”) is unclear. It is often attributed to either Lao Tzu or to the character Eldon Tyrell from the 1982 movie Blade Runner. Stars follow a similar law, although the factor isn’t precisely 1/2. In this problem, you will figure out the precise factor that the quote should have to apply to stars. Using the proportionality relationships for stellar luminosity as a function of mass and stellar lifetime as a function of mass, combine the two equations to arrive at a proportionality for stellar lifetime as a function of luminosity. Consider a star with luminosity twice that of the Sun’s. Compute the star’s main sequence lifetime as a multiple of the Sun’s main sequence lifetime. Enter your result below as a decimal. For example, if you found
, enter “0.3”. (Here T is the star’s lifetime and
is the Sun’s main sequence lifetime.
The Correct Answer and Explanation is:
To solve this problem, we need to use the known proportionality relationships between a star’s mass, luminosity, and lifetime:
- Luminosity (L) as a function of mass (M) for main-sequence stars: L∝M3.5L \propto M^{3.5}L∝M3.5
- Lifetime (T) as a function of mass: T∝MLT \propto \frac{M}{L}T∝LM
Step 1: Combine the equations
Since L∝M3.5L \propto M^{3.5}L∝M3.5, substitute this into the equation for lifetime:T∝MM3.5=M−2.5T \propto \frac{M}{M^{3.5}} = M^{-2.5}T∝M3.5M=M−2.5
So the lifetime of a star is inversely proportional to the 2.5 power of its mass.T∝M−2.5T \propto M^{-2.5}T∝M−2.5
Step 2: Express mass in terms of luminosity
From L∝M3.5L \propto M^{3.5}L∝M3.5, we solve for mass:M∝L1/3.5=L2/7M \propto L^{1/3.5} = L^{2/7}M∝L1/3.5=L2/7
Now substitute this into the expression for lifetime:T∝(L2/7)−2.5=L−5/7T \propto (L^{2/7})^{-2.5} = L^{-5/7}T∝(L2/7)−2.5=L−5/7
So we find:T∝L−5/7T \propto L^{-5/7}T∝L−5/7
Step 3: Apply this to a star with twice the Sun’s luminosity
Let L=2L⊙L = 2L_{\odot}L=2L⊙, then:TT⊙=(LL⊙)−5/7=2−5/7\frac{T}{T_{\odot}} = \left( \frac{L}{L_{\odot}} \right)^{-5/7} = 2^{-5/7}T⊙T=(L⊙L)−5/7=2−5/7
Now calculate:2−5/7≈0.659752^{-5/7} \approx 0.659752−5/7≈0.65975
Final Answer:
0.66\boxed{0.66}0.66
Explanation
The well-known quote “The light that burns twice as bright burns half as long” poetically captures the idea that more luminous objects exhaust their fuel more quickly. For stars, we can examine how close this saying comes to physical truth using astrophysical relationships.
Stellar luminosity increases steeply with mass. Specifically, the luminosity LLL of a main-sequence star scales roughly as the 3.5 power of its mass MMM:L∝M3.5L \propto M^{3.5}L∝M3.5
Meanwhile, a star’s main-sequence lifetime TTT is proportional to the amount of fuel it has divided by the rate at which it uses it. Since fuel is proportional to mass and usage rate is luminosity,T∝MLT \propto \frac{M}{L}T∝LM
Substituting the first relation into the second gives:T∝MM3.5=M−2.5T \propto \frac{M}{M^{3.5}} = M^{-2.5}T∝M3.5M=M−2.5
This shows that more massive stars live shorter lives. But we want a relationship between lifetime and luminosity, so we solve the first equation for mass in terms of luminosity:M∝L2/7M \propto L^{2/7}M∝L2/7
Then,T∝(L2/7)−2.5=L−5/7T \propto (L^{2/7})^{-2.5} = L^{-5/7}T∝(L2/7)−2.5=L−5/7
This final expression tells us that stellar lifetime decreases as luminosity increases, but not as sharply as the “half-as-long” quote suggests.
For a star with twice the Sun’s luminosity,T/T⊙=2−5/7≈0.66T/T_{\odot} = 2^{-5/7} \approx 0.66T/T⊙=2−5/7≈0.66
So, such a star lives about 66 percent as long as the Sun, not half. The poetic quote has a grain of truth, but stars burn less than twice as bright for half the time.
