The Solar Weight Loss Plan

Bad Astronomy
The entire universe in blog form
July 14 2014 7:30 AM

What’s the Quicker Solar Weight Loss Plan: Solar Wind, or Nuclear Fusion?

solar prominence
An eruptive prominence blew off the Sun in 2013. That blue dot next to it? That's how big the Earth is compared to it. Sweet dreams.

Photo by NASA/SDO

I know this may seem obvious, but … the Sun is big.

Phil Plait Phil Plait

Phil Plait writes Slate’s Bad Astronomy blog and is an astronomer, public speaker, science evangelizer, and author of Death From the Skies!  

Really, really big. It’s more than 100 times wider than the Earth, and more than a million Earths would fit inside it. If you could weigh them on a cosmic scale, you’d find the Sun is more than 300,000 times the mass of the Earth!

But that last number is dropping. Slowly, over time, the Sun is losing mass. It’s actually doing this in two ways: directly, via its solar wind, and indirectly, by converting mass into energy and shining brightly.

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This raises an obvious question: Which one is faster? Which method is better at making the Sun shed those kilograms?

The Solar Wind Diet

The solar wind is just that: a vast stream of subatomic particles blowing away from the Sun. It’s composed of various types of particles, including electrons, protons, and even things like helium nuclei. Shaped by the Sun’s powerful and complex magnetic field, these particles are flung into space at high speed, ranging from a few hundred kilometers per second to many thousands.

aurora
When the solar wind impacts Earth, the planet's magnetic field can channel it into our atmosphere to create beautiful aurorae.

Photo by David Cartier

The Sun blows off just as many electrons as protons, but protons are so much more massive we can ignore the electrons (and helium is only about 4 percent of the total, so we can ignore it too). Using satellites we can directly measure how many protons fly past the Earth in the solar wind. If you had a little net in space one centimeter on a side, you’d catch about 300 million protons every second at the Earth’s distance from the Sun.

But that means you’d miss a lot. The Sun is sending them out in all directions (though, to be fair, more along the Sun’s equator than at the poles, which we’ll ignore), and your little net is smaller than a postage stamp. To catch all the protons, you’d need to make a shell around the Sun at the Earth’s orbit. That shell would have an area of a staggering 3 x 1027 square centimeters—that’s 3 octillion little nets!

If one net catches 3 x 108 protons every second, then our big shell would catch 9 x 1035 of them. Each proton has a mass of 1.7 x 10-24 grams, so every second that means the Sun blows off about 1.5 trillion grams, or 1.5 million tons of material!

OK, so that’s Diet Plan 1, which loses the Sun 1.5 million tons per second. So what about Diet Plan 2?

The Nuclear Fusion Diet

fusion
Nuclear fusion in the Sun takes many steps, but in the end four protons fuse to become a single nucleus of helium, and a whole lot of energy is released.

Drawing by Borb via Wikimedia Commons

The Sun gives off energy, and that energy has to come from somewhere. Deep in its core, the Sun is busily converting mass into energy. That energy works its way out of the Sun and flies away into space in the form of light.

We know how bright the Sun is, that is, how much light it gives off. And we know—thanks to Einstein—how much mass it takes to create energy. So let’s see …

Doing the same sort of math as above, except measuring the light we see from the Sun per square centimeter as measured from the Earth, we find that the total energy emitted by the Sun is 4 x 1033 ergs per second (an erg is a teeny unit of energy).

But we also know that energy = mass x the speed of light squared. Rearranging to solve for the mass, and using the usual constant c for the speed of light (3 x 1010 cm/sec), we get

Mass = energy / c2  

Plugging and chugging: mass = 4 x 1033 / 9 x 1020 = 4.4 x 1012 grams per second, or more than 4 million tons per second.

So there you go. The Sun loses 4 million tons of mass per second due to fusion. Fusion wins over solar wind as a stellar diet plan, by about a factor of two or three.

To be honest, I find it surprising the numbers are that close. Without any prior knowledge, it seems like either one could be thousands or even millions of times the other. Yet they’re about the same.

Run Away!

An interesting thought: The Earth orbits the Sun, held sway by its gravity. But as the mass of the Sun goes down, the Earth is held a bit less strongly. What happens then?

I’ll spare you the math, but it has to do with angular momentum and it being a constant. What happens in the end is that the Earth’s orbital radius increases as the Sun loses mass, and it does so linearly with the mass loss. In other words, if the Sun loses 1 percent of its mass, the Earth’s orbit increases in size by 1 percent.

The Sun is losing about 6 x 1012 grams per second, and has a mass of 2 x 1033 grams. So the fraction of its mass it loses every year is about 10-13. The Earth’s orbit is 150 million kilometers, and if you multiply that by 10-13 you get about 1.5 centimeters. That’s how much bigger the Earth’s orbit gets every year! Less than an inch. It would take 65,000 years for the Earth to move away one kilometer (I can walk that far in a few minutes). Assuming the mass loss is constant, the Earth has only moved out from the Sun 70,000 km in a billion years! That’s only a few times Earth’s own diameter. Even over its lifetime, the Earth’s orbit hasn’t changed much due to the Sun’s mass loss. To be fair, the Sun’s mass loss may have been higher in the past, but even then the Earth hasn’t moved much from this process.

solar storm
A watched Sun never boils. Wait. No. A watched Sun always boils.

Photo by Discovery Communications/Big Wave Productions

A Lifetime of Mass Loss

But wait! How much mass has the Sun lost over its lifetime? It loses about 5 or 6 million tons of material every second, and that sounds like a lot.

The Sun is about 4.5 billion years old, and a year is about 31 million seconds long. Multiplying all that out, the Sun has lost a total of about 1024 tons of material. That’s more than 100 times the mass of the Earth!

But like I said, the Sun is big. That’s still a tiny fraction of its total mass: Over its lifetime, it’s only lost about 0.05 percent of its mass. That’s a pretty poor weight loss plan.

But it means that the Sun is good to go for a long, long time yet. It could merrily fuse matter and blow off a solar wind for trillions of years at this rate.

But it won’t. Long before then, conditions in the Sun’s core will change. It will run out of hydrogen to use for nuclear fuel, swell up into a red giant, consume Mercury and Venus (gaining some mass back), fry the Earth, then blow off a far more intense solar wind. It’ll shed matter at a rate that will completely outstrip a lifetime of dieting, losing half its mass in just a few million years.

After that all that will be left is a white dwarf; a hot, dense, Earth-sized ball of material that will slowly cool over billions of years and fade away.

If there’s a life lesson about living for today, diets, and thinking in very long timescales, you’re free to find it for yourself. As for me, I think I’ll go out on my bike now and enjoy some sunshine. There’s only a few billion years of it left.

I’ve thought about writing this article for a long, long time, but a tweet by mxyzplx made me think about it again. I made a note for myself, and I’m glad I finally wrote this up. The Sun lost more than 100 trillion tons of mass in the meantime. I need to schedule myself better.

Also, if you like thinking about stuff like this, I give a lot more detail about the Sun’s aging in my book, Death From the Skies!

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