# I Heat Energy Intercepted by the Earth

#### Dinoduck94

Summary
Am I missing anything with this calculation, and thought process?
I've recently been interested in how much energy the Earth intercepts from the Sun; the answer, unsurprisingly was an astronomical amount, measuring quite easily into the ZetaWatts.
However, the maths that got me that answer got me thinking... you can use the same method to determine the amount of energy the Earth intercepts, from any star in the universe.

The maths is very straight forward, and you only need 3 bits of information about the star:

Where 'P(i)' is the intercepted Power measured in Watts; 'σ' is the Stefan Boltzmann Constant; 'r' is the radius of the star; 'T' is the surface temperature of the star (in Kelvin); 'd' is the distance from the star (in the same units as 'r'); and 'A(i)' is the intercepting area of the Earth (in metres).

What I was shocked to discover, is that if the Earth was in a magical sweet spot where it sat with direct view of all the stars within 20 light years (except LP 145-131 as I couldn't confirm it's radius), a staggering 21,346,503 Watts of Heat energy is intercepted by the Earth.
Now obviously in the grand scale of things, this is a minute fraction of what Sol provides (with us being so close), in fact it is equivalent to dotting 0.167 Watt heaters, every kilometer, around the world (wouldn't exactly keep us warm if the sun disappeared!)- But it's still an awe inspiring amount of energy provided from minute little freckles of light in the night sky.

The reason I am posting this thread, though, is to gain the knowledge from those more knowing than me; is there anything that I have missed with this?
Am I correct in saying that radiated infrared energy will travel at the speed of light, and will not dissipate until it hits something i.e. the Earth?

I have attached an excel sheet showing the findings.

From doing this I also discovered that VY Canis Majoris (one of the largest known stars), sitting 3900 light years from Earth, bombards us with 803,083.71 Watts on its own!

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#### DrClaude

Mentor
Am I correct in saying that radiated infrared energy will travel at the speed of light, and will not dissipate until it hits something i.e. the Earth?
IR radiation is light, so it travels at the speed of light. Some of the energy will be absorbed by cosmic gas and dust on the way, but not that much.

The calculations appear correct.

#### Anand Sivaram

May I suggest you to find the total energy from Betelgeuse and Rigel. Betelgeuse is the brightest start in Near Infra Red.

#### davenn

Gold Member
Summary: Am I missing anything with this calculation, and thought process?

Now obviously in the grand scale of things, this is a minute fraction of what Sol provides (with us being so close), in fact it is equivalent to dotting 0.167 Watt heaters, every kilometer, around the world (wouldn't exactly keep us warm if the sun disappeared!)-

not sure how you got that figure ?? seems VERY small

from wiki
"Of the ~340 W/m² of solar radiation received by the Earth (at the top of the atmosphere), an average of ~77 W/m² is reflected back to space by clouds and the atmosphere and ~23 W/m² is reflected by the surface albedo, leaving ~240 W/m² of solar energy input to the Earth's energy budget ( at the surface)."

may clarification - (at the top of the atmosphere) and (at the surface)

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#### Drakkith

Staff Emeritus
2018 Award
not sure how you got that figure ?? seems VERY small

from wiki
"Of the ~340 W/m² of solar radiation received by the Earth (at the top of the atmosphere), an average of ~77 W/m² is reflected back to space by clouds and the atmosphere and ~23 W/m² is reflected by the surface albedo, leaving ~240 W/m² of solar energy input to the Earth's energy budget ( at the surface)."

may clarification - (at the top of the atmosphere) and (at the surface)
He's talking about light from other stars, not the Sun.

#### davenn

Gold Member
He's talking about light from other stars, not the Sun.
hmmm maybe ....... I was going by his opening statement that may have thrown me a curve ball

Summary: Am I missing anything with this calculation, and thought process?

I've recently been interested in how much energy the Earth intercepts from the Sun; the answer, unsurprisingly was an astronomical amount, measuring quite easily into the ZetaWatts.

then he went into stars .... I have to admit defeat on that one haha

#### Dinoduck94

May I suggest you to find the total energy from Betelgeuse and Rigel. Betelgeuse is the brightest start in Near Infra Red.
Interesting! With Betelguese we receive 11,182,492.24 Watts of Heat Energy, despite it being 642.5ly away; and Rigel hits us with 4,770,125.17 Watts of Heat Energy. These 2 stars together equal 75% of the heat energy we receive from all the stars within 20 light years. Truly the mind boggles with thinking of the entire system at work... If only we knew the required information about all the stars in the Galaxy - that would be an interesting calculation.

#### Dinoduck94

IR radiation is light, so it travels at the speed of light. Some of the energy will be absorbed by cosmic gas and dust on the way, but not that much.

The calculations appear correct.
Thank you! Do we know what the average reflection/absorption rate is of cosmic gas and dust - and how common it is throughout the Galaxy?
In other words, could we add a coefficient to the above to make it more accurate based on reflection/absorption of energy during it's travels?

Thanks

#### Anand Sivaram

My suggestion is to reconsider the list of stars you have in the excel sheet. You have already taken stars within 20 ly. Along with that consider many of the first and second order magnitude stars also. For example, Canopus, Vega, Arcturus etc. are not there in your list because they are more than 20 ly away.

I also thought that Betelguese would give very good numbers as it is the brightest star in IR wavelengths. At 640 lightyears and 140000 Solar luminance, it is one of the most luminous stars you could see with Naked Eye.

The reason I mentioned about first magnitude stars because, visual magnitude is a measure of received W/m2 in the visible spectrum. Considering it, there is no wonder that flux from Sirius has the maximum of 15MW as it has a magnitude of -1.46

Whereas stars like Proxima Centauri and Bernard's star are 12 magnitude stars, you require a good telescope to view them.

Flux density becomes 100 times, when the magnitude is changed by 5. That means, a star that has a visual magnitude of 0 has got 100 times more energy / sq meter received on the Earth compared to a star of magnitude 5.

Interesting! With Betelguese we receive 11,182,492.24 Watts of Heat Energy, despite it being 642.5ly away; and Rigel hits us with 4,770,125.17 Watts of Heat Energy. These 2 stars together equal 75% of the heat energy we receive from all the stars within 20 light years. Truly the mind boggles with thinking of the entire system at work... If only we knew the required information about all the stars in the Galaxy - that would be an interesting calculation.

#### Anand Sivaram

Reflection/Absorption depends upon wavelength. Each wavelength has different coefficient. Also it depends upon material through which light passes, example air, hydrogen etc. So, it may not be a simple coefficient at all.

In other words, could we add a coefficient to the above to make it more accurate based on reflection/absorption of energy during it's travels?

Thanks

#### cosmik debris

Interesting! With Betelguese we receive 11,182,492.24 Watts of Heat Energy
Wow, to two decimal places.

Cheers

#### Nugatory

Mentor
Wow, to two decimal places.
And seeing as how we’re considering any absorption to be negligible, and we only have two or three significant digits in our distance and size measurements....

@Dinoduck94 you are using a spreadsheet. Bring up that “Format cells” dialog and set your spreadsheet to scientific notation with two significant digits. That is, the contribution from Betelgeuse is best written as $1.1\times{10}^7$ watts; writing it as 11,182,492.24 is both misleading and clutters the discussion up with meaningless numbers.
(Exercise: The difference between 11,182,492.24 and 11,182,492.25 is to $1.1\times{10}^7$ as what is to the size of the Pacific Ocean? A bacterium? a small insect? An elephant? A mountain? The north island of New Zealand?).

#### klimatos

Summary: Am I missing anything with this calculation, and thought process?

What I was shocked to discover, is that if the Earth was in a magical sweet spot where it sat with direct view of all the stars within 20 light years (except LP 145-131 as I couldn't confirm it's radius), a staggering 21,346,503 Watts of Heat energy is intercepted by the Earth.
What is the source of your data? Virtually every reputable source of data for the Earth's heat budget gives a Solar Constant of 1,366 watts, more or less. Energy from the stars is negligible, due to the fact that its intensity is divided by the square of the distance between us and the stars.

#### Drakkith

Staff Emeritus
2018 Award
What is the source of your data? Virtually every reputable source of data for the Earth's heat budget gives a Solar Constant of 1,366 watts, more or less. Energy from the stars is negligible, due to the fact that its intensity is divided by the square of the distance between us and the stars.
I think the solar constant is W/m^2, while Dinoduck's is the total flux for the surface of the Earth. But I haven't done the math personally to confirm.

#### Nugatory

Mentor
Virtually every reputable source of data for the Earth's heat budget gives a Solar Constant of 1,366 watts,
That’s per square meter of the earth’s surface. OP is looking at the amount delivered to the entire planet, which has a cross section area of about $10^{14}$square meters. So if the $10^7$ Watts for Betelgeuse is correct, that’s ten decimal orders of magnitude less than what the sun is delivering, what you’d expect from a sun-sized star 100,000 times farther away than the sun.

#### Anand Sivaram

Correct.
Dinoduck is looking at total flux received on Earth. Whereas klimatos is talking about Solar flux.

Just to understand from both angle, let us consider Sirius, the brightest star in the night. As per Dinoduck's calculation, Sirius has the maximum flux on Earth around 15MW totally on Earth. This 15MW seems to be a huge power, if we calculate received flux for the whole πr2 for the Earth where r = 6371km. Corresponding figure for Sun is around 173000TW.

If we look from per square metre flux, then Sun has a flux of 1366 W/m2 and Sirus has a flux of 1/(12 Billion) compared to the Sun.

Magnitude of Sun = -26.74, Sirus = -1.46
Difference d = 25.28
Then brighness difference = 100 ^(d / 5) = 12 Billion

That means the energy received from stars are negligible.

I think the solar constant is W/m^2, while Dinoduck's is the total flux for the surface of the Earth. But I haven't done the math personally to confirm.

#### OmCheeto

Gold Member
I'm not sure why I like this thread, but I do.
Here are my results so far:

#### Drakkith

Staff Emeritus
2018 Award
Why isn't Sirius top on the list? Shouldn't the brightest star also have the largest amount of energy falling on the Earth?

#### Anand Sivaram

I think this calculation has a mistake. As Drakkith said, I also feel that Sirius should be there on the top.

Consider this way
Total Intercepted Energy ∝ Received W/m2 ∝ L / d2

where L = Luminosity of the star, d = distance to the star

These are the values of some of the stars listed, based on Wikipedia
L = Luminosity compared to the Sun
d = distance in Light years

| L | d | l/d2
Sirius | 25.4 | 8.6 | 0.34342
Hadar Beta Centauri | 41700.0 | 390.0 | 0.27416
Betelgeuse | 126000.0 | 640.0 | 0.30761

#### Drakkith

Staff Emeritus
2018 Award
Hmmm. After thinking about it more, perhaps the key is that some stars are visibly dim, but emit a much larger amount of light in the IR band.

#### Anand Sivaram

Luminosity of a star is generally measured relative to L⊙ - Solar luminosity 3.828 x 10^26 W. It is normally the bolometric luminosity measured using a Bolometer.
https://en.wikipedia.org/wiki/Luminosity#Measuring_luminosity

Solar luminosity includes all EM energy including UV and IR. But, neutrino energy is not included. It is derived from the Solar constant. We could see that the Solar constant includes Solar energy in all spectrum.
https://en.wikipedia.org/wiki/Solar_luminosity

There is also a term named Bolometric correction to get bolometric magnitude from visible magnitude
https://en.wikipedia.org/wiki/Bolometric_correction

Just found an interesting paper on Basic Stellar Astrophysics
http://www.astro.sunysb.edu/lattimer/PHY521/intro.pdf

#### OmCheeto

Gold Member
Hmmm. After thinking about it more, perhaps the key is that some stars are visibly dim, but emit a much larger amount of light in the IR band.
Or in the UV band.
Hadar and Sirius both have peak wavelengths in the UV.
Betelgeuse and R Doradus peak in the IR.

#### Anand Sivaram

@OmCheeto : I understood your calculations, depending upon the temperature peak wavelength changes and the percentage of visible energy changes.

But, the star Luminosity value includes al of these. Correct?
Below reference I got from the UNL website shows how it is calculated using Stefan-Boltzmann Law.
https://astro.unl.edu/naap/hr/hr_background2.html

#### OmCheeto

Gold Member
@OmCheeto : I understood your calculations, depending upon the temperature peak wavelength changes and the percentage of visible energy changes.

But, the star Luminosity value includes al of these. Correct?
Maths: yes
Current scientific literature: no*

*It's complicated.

"Heat Energy Intercepted by the Earth"

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