Hotter than the sun

Which gives off more heat, you, or the Sun? If we're talking about net heat, it's obviously the Sun, but if we're talking about heat output per unit mass, it's not so clear. Calculate the ratio of your body's heat output per unit mass to that of the Sun.

Details and Assumptions

  • The mass of the Sun is 2 × 1 0 30 2\times10^{30} kg
  • The radius of the Sun is 7 × 1 0 8 7\times 10^8 m
  • The surface temperature of the sun is 5780 5780 K, and it is a perfect blackbody.
  • Your daily consumption of food is 2,000 kilocalories, and you radiate it all as heat.
  • You weigh 75 kg.


The answer is 6637.41.

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2 solutions

Michael Mendrin
Jun 24, 2015

Okay, that was a little weird.

Edit: Here's how it's worked out

M = 2 × 10 30 × k g M=2\times { 10 }^{ 30 } \times kg
R = 7 × 10 8 × m e t e r R=7\times { 10 }^{ 8 } \times meter
T = 5780 × k e l v i n T=5780 \times kelvin
C a l o r i e = 1000 ( 0.00116222 ) × w a t t × h o u r Calorie=1000(0.00116222) \times watt \times hour
σ = 5.670373 × 10 8 w a t t × m e t e r 2 × k e l v i n 4 \sigma =5.670373\times { 10 }^{ -8 }watt \times{ meter }^{ -2 } \times{ kelvin }^{ -4 }


σ \sigma is the Stefan-Boltzman constant

Total radiated energy from sun in 24 24 hour period divided by sun’s mass

S u n = 24 × h o u r M ( 4 π R 2 σ T 4 ) Sun=\dfrac { 24 \times hour }{ M } \left( 4\pi { R }^{ 2 }\sigma { T }^{ 4 } \right)

Total radiated energy from body in 24 24 hour period divided by body’s mass

B o d y = 2000 × C a l o r i e 75 × k g Body=\dfrac { 2000 \times Calorie }{ 75 \times kg }

Ratio of Body to Sun

B o d y S u n = 6627.46... \dfrac { Body }{ Sun } =6627.46...

which is a dimensionless ratio, i.e., all the physical dimensions drop out

Note: Calorie, with capital "C", is 1000 1000 times greater than calorie, with lower case "c". Calories is commonly used for foods, while calories is used for stuff like locomotive steam engine thermodynamic analysis. Maybe that explains why that family is glowing so much.

I did not know the difference but got right answer by using both values one by one.Thanks

Ayush Verma - 5 years, 11 months ago

Dear Michael Mendrin, Nothing can stop your intelligence to know what people meant to you. I confess that there was a subconscious meaning but it was also a semi-intention. Please forget it and do accept my semi-apology. Thanks! Lu Chee Ket

Lu Chee Ket - 4 years ago
Lu Chee Ket
Dec 11, 2015

P = A σ T 4 P = A \sigma T^4

A A = 4 π R 2 4 \pi R^2 = 4 π ( 7 × 1 0 8 ) 2 = 6.15752160103599 × 1 0 18 m 2 4 \pi (7 \times 10^8)^2 = 6.15752160103599 \times 10^{18} m^2

T 4 T^4 = 578 0 4 5780^4 = 1116121190560000 K 4 1116121190560000 K^4

P = 6.15752160103599 × 1 0 18 × 5.7 × 1 0 8 × 1116121190560000 P = 6.15752160103599 \times 10^{18}\times 5.7 \times 10^{-8} \times 1116121190560000 = 3.89698671867486 × 1 0 26 W . 3.89698671867486 \times 10^{26} W.

3.89698671867486 × 1 0 26 W 2 × 1 0 30 k g \displaystyle \frac{3.89698671867486 \times 10^{26} W}{2 \times 10^{30} kg} = 1.94849335933743 × 1 0 4 W / k g 1.94849335933743 \times 10^{-4} W/ kg

Since 2000 Calories = 2000 × \times 4184 Joules = 8368000 J,

8368000 24 × 3600 = 96.8518518518519 W ; \displaystyle \frac{8368000}{24 \times 3600} = 96.8518518518519 W;

96.8518518518519 W 75 k g = 1.29135802469136 W / k g \displaystyle \frac{96.8518518518519 W}{75 kg} = 1.29135802469136 W/ kg

Ratio = 1.29135802469136 1.94849335933743 × 1 0 4 \displaystyle \frac{1.29135802469136}{1.94849335933743 \times 10^{-4}} = 6.59302174169231 E + 3 6.59302174169231E+3

With 5.670373 5.670373 × 1 0 8 \times 10^{-8} instead of 5.7 5.7 × 1 0 8 \times 10^{-8} , 6.62746946764281 E + 3 6.62746946764281E+3 is obtainable.

This ratio may give reasonable fact of why the sun can last for a long period of time in space.

Answer: 6593 \boxed{6593} e l s e else 6627 \boxed{6627}

The question may mention significant figures wanted.

Lu Chee Ket - 5 years, 6 months ago

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