To solve this problem, we need to use the known proportionality relationships between a star’s mass, luminosity, and lifetime:<\/p>\n\n\n\n
- \n
- Luminosity (L)<\/strong> as a function of mass (M) for main-sequence stars: L\u221dM3.5L \\propto M^{3.5}L\u221dM3.5<\/li>\n\n\n\n
- Lifetime (T)<\/strong> as a function of mass: T\u221dMLT \\propto \\frac{M}{L}T\u221dLM\u200b<\/li>\n<\/ol>\n\n\n\n
Step 1: Combine the equations<\/h3>\n\n\n\n
Since L\u221dM3.5L \\propto M^{3.5}L\u221dM3.5, substitute this into the equation for lifetime:T\u221dMM3.5=M\u22122.5T \\propto \\frac{M}{M^{3.5}} = M^{-2.5}T\u221dM3.5M\u200b=M\u22122.5<\/p>\n\n\n\n
So the lifetime of a star<\/strong> is inversely proportional to the 2.5 power of its mass<\/strong>.T\u221dM\u22122.5T \\propto M^{-2.5}T\u221dM\u22122.5<\/p>\n\n\n\n
Step 2: Express mass in terms of luminosity<\/h3>\n\n\n\n
From L\u221dM3.5L \\propto M^{3.5}L\u221dM3.5, we solve for mass:M\u221dL1\/3.5=L2\/7M \\propto L^{1\/3.5} = L^{2\/7}M\u221dL1\/3.5=L2\/7<\/p>\n\n\n\n
Now substitute this into the expression for lifetime:T\u221d(L2\/7)\u22122.5=L\u22125\/7T \\propto (L^{2\/7})^{-2.5} = L^{-5\/7}T\u221d(L2\/7)\u22122.5=L\u22125\/7<\/p>\n\n\n\n
So we find:T\u221dL\u22125\/7T \\propto L^{-5\/7}T\u221dL\u22125\/7<\/p>\n\n\n\n
Step 3: Apply this to a star with twice the Sun\u2019s luminosity<\/h3>\n\n\n\n
Let L=2L\u2299L = 2L_{\\odot}L=2L\u2299\u200b, then:TT\u2299=(LL\u2299)\u22125\/7=2\u22125\/7\\frac{T}{T_{\\odot}} = \\left( \\frac{L}{L_{\\odot}} \\right)^{-5\/7} = 2^{-5\/7}T\u2299\u200bT\u200b=(L\u2299\u200bL\u200b)\u22125\/7=2\u22125\/7<\/p>\n\n\n\n
Now calculate:2\u22125\/7\u22480.659752^{-5\/7} \\approx 0.659752\u22125\/7\u22480.65975<\/p>\n\n\n\n
Final Answer:<\/h3>\n\n\n\n
0.66\\boxed{0.66}0.66\u200b<\/p>\n\n\n\n
\n\n\n\nExplanation <\/h3>\n\n\n\n
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.<\/p>\n\n\n\n
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\u221dM3.5L \\propto M^{3.5}L\u221dM3.5<\/p>\n\n\n\n
Meanwhile, a star\u2019s 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\u221dMLT \\propto \\frac{M}{L}T\u221dLM\u200b<\/p>\n\n\n\n
Substituting the first relation into the second gives:T\u221dMM3.5=M\u22122.5T \\propto \\frac{M}{M^{3.5}} = M^{-2.5}T\u221dM3.5M\u200b=M\u22122.5<\/p>\n\n\n\n
This shows that more massive stars live shorter lives. But we want a relationship between lifetime and luminosity<\/strong>, so we solve the first equation for mass in terms of luminosity:M\u221dL2\/7M \\propto L^{2\/7}M\u221dL2\/7<\/p>\n\n\n\n
Then,T\u221d(L2\/7)\u22122.5=L\u22125\/7T \\propto (L^{2\/7})^{-2.5} = L^{-5\/7}T\u221d(L2\/7)\u22122.5=L\u22125\/7<\/p>\n\n\n\n
This final expression tells us that stellar lifetime decreases as luminosity increases, but not as sharply as the “half-as-long” quote suggests.<\/p>\n\n\n\n
For a star with twice the Sun\u2019s luminosity,T\/T\u2299=2\u22125\/7\u22480.66T\/T_{\\odot} = 2^{-5\/7} \\approx 0.66T\/T\u2299\u200b=2\u22125\/7\u22480.66<\/p>\n\n\n\n
So, such a star lives about 66 percent<\/strong> as long as the Sun, not half. The poetic quote has a grain of truth, but stars burn less<\/strong> than twice as bright for half<\/strong> the time.<\/p>\n\n\n\n
![\"\"](\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-1328.jpeg\")
<\/figure>\n","protected":false},"excerpt":{"rendered":"\u201cThe light that burns twice as bright burns half as long\u2026\u201d The origin of the above quote (with \u201cflame\u201d or \u201ccandle\u201d sometimes substituted for \u201clight\u201d) 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 […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-41302","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/41302","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/comments?post=41302"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/41302\/revisions"}],"predecessor-version":[{"id":41313,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/41302\/revisions\/41313"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=41302"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=41302"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=41302"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
- Lifetime (T)<\/strong> as a function of mass: T\u221dMLT \\propto \\frac{M}{L}T\u221dLM\u200b<\/li>\n<\/ol>\n\n\n\n