fusion – Cosmic Reflections

Star Stuff

The elements that make up the stars also exist here on Earth. In fact, our Earth, and indeed all the planets, were created from the dust and gas produced by previous generations of stars that existed before our Sun and solar system formed. We truly are made of stardust!

Stars are made up almost entirely of hydrogen and helium. Here is a table of the most abundant elements in our Sun.

Element	% by atoms
Hydrogen	92.2%
Helium	7.7%
Oxygen	0.0473%
Carbon	0.0272%
Neon	0.0130%
Nitrogen	0.0065%
Magnesium	0.0033%
Silicon	0.0030%
Iron	0.0028%
Sulfur	0.0013%

Most abundant elements in the Sun

It is not a trivial matter to determine the abundance of elements in the Sun. For most elements, astronomers have to look at the strength of spectral absorption lines in the photosphere. Some elements, like fluorine, chlorine, and thallium, require looking for their spectral lines inside of sunspots, which are cooler-than-average regions of the photosphere. Other elements require that we look at spectral lines in the solar corona, or capture and analyze the solar wind. And some elements we are simply unable to detect.

The region of the photosphere that is amenable to spectral study represents only about 2% of the mass of the Sun. Since the Sun’s formation 4.6 Gyr ago, some gravitational settling of heavier elements and diffusion of hydrogen towards the surface means the Sun is not uniform in composition. Fortunately, the relative abundances of the elements heavier than helium are probably similar throughout the Sun.

Lithium, the third element in the periodic table after hydrogen and helium, is the odd element out. It has a relative abundance in the solar photosphere that is only 1/170th that found in meteorites. The Sun’s original supply of lithium has largely been destroyed by the high temperatures inside the pre-main-sequence Sun, and today at the hot bottom of the Sun’s convection zone.

Light pollution is a problem here on Earth, but on the Sun we have a problem with “line pollution”. There are so many spectral lines that the weak signatures from some elements become difficult or impossible to isolate and measure. There is much blending of overlapping lines, and some elements—most notably iron which is the ninth most abundant element in the Sun—are “superpolluters” with hundreds to thousands of spectral lines from both excited and ionized states.

Sometimes, the spectral lines of interest are in a region of the electromagnetic spectrum (ultraviolet, for example) that can only be observed from space, and that creates additional challenges.

Notably, the noble gases helium, neon, argon, krypton, and xenon have no photospheric absorption lines that can be observed, and we must look to coronal sources such as the solar wind, solar flares, or solar energetic particles for information about their abundances.

Helium—the second most abundant element in the Sun—requires an indirect approach combining a theoretical solar model and observational helioseismology data to tease out its abundance.

The following elements are undetectable in the Sun: arsenic, selenium, bromine, technetium, tellurium, iodine, cesium, promethium, tantalum, rhenium, mercury, bismuth, polonium, astatine, radon, francium, radium, actinium, protactinium, and all the synthetic elements above uranium on the period table.

Interestingly, helium was discovered in the Sun before it was discovered on Earth! That’s why this element is name after Helios, the Greek god of the Sun.

The energy source that allows stars to shine steadily, often for billions of years, is fusion. Fusion in a star can only occur where both the temperature and pressure are very high. Usually (but not always!), this occurs in the core of the star. When the element hydrogen fuses into helium, a huge amount of energy is released in the process. Lucky for us, fusing hydrogen into helium is difficult to do in a one-solar-mass star. On average, any particular hydrogen atom in our Sun has to “wait” about five billion years before having the “opportunity” to participate in a fusion reaction!

In order for sustained fusion to occur in the core of a star, the star must have sufficient mass so that the core temperature and pressure is high enough. Present thinking is that the lowest mass stars where sustained fusion can occur have about 75 times the mass of Jupiter, or about 7% the mass of the Sun.

References

Lodders, K. 2020 Solar Elemental Abundances, in The Oxford Research Encyclopedia of Planetary Science, Oxford University Press
arXiv:1912.00844 [astro-ph.SR]

Space Travel Under Constant 1g Acceleration

The basic principle behind every high-thrust interplanetary space probe is to accelerate briefly and then coast, following an elliptical, parabolic, or mildly hyperbolic solar trajectory to your destination, using gravity assists whenever possible. But this is very slow.

Imagine, for a moment, that we have a spacecraft that is capable of a constant 1g (“one gee” = 9.8 m/s²) acceleration. Your spacecraft accelerates for the first half of the journey, and then decelerates for the second half of the journey to allow an extended visit at your destination. A constant 1g acceleration would afford human occupants the comfort of an earthlike gravitational environment where you would not be weightless except during very brief periods during the mission. Granted such a rocket ship would require a tremendous source of power, far beyond what today’s chemical rockets can deliver, but the day will come—perhaps even in our lifetimes—when probes and people will routinely travel the solar system in just a few days. Journeys to the stars, however, will be much more difficult.

The key to tomorrow’s space propulsion systems will be fusion and, later, matter-antimatter annihilation. The fusion of hydrogen into helium provides energy E = 0.008 mc². This may not seem like much energy, but when today’s technological hurdles are overcome, fusion reactors will produce far more energy in a manner far safer than today’s fission reactors. Matter-antimatter annihilation, on the other hand, completely converts mass into energy in the amount given by Einstein’s famous equation E = mc². You cannot get any more energy than this out of any conceivable on-board power or propulsion system. Of course, no system is perfect, so there will be some losses that will reduce the efficiency of even the best fusion or matter-antimatter propulsion system by a few percent.

How long would it take to travel from Earth to the Moon or any of the planets in our solar system under constant 1g acceleration for the first half of the journey and constant 1g deceleration during the second half of the journey? Using the equations below, you can calculate this easily.

Keep in mind that under a constant 1g acceleration, your velocity quickly becomes so great that you can assume a straight-line trajectory from point a to point b anywhere in our solar system.

Maximum velocity is reached at the halfway point (when you stop accelerating and begin decelerating) and is given by

The energy per unit mass needed for the trip (one way) is then given by

How much fuel will you need for the journey?

hydrogen fusion into helium gives: E_fusion = 0.008 m_fuel c²

matter-antimatter annihilation gives: E_anti = m_fuel c²

This assumes 100% of the fuel goes into propelling the spacecraft, but of course there will be energy losses and operational energy requirements which will require a greater amount of fuel than this. Moreover, we are here calculating the amount of fuel you’ll need for each kg of payload. We would need to use calculus to determine how much additional energy will be needed to accelerate the ever changing amount of fuel as well. The journey may well be analogous to the traveler not being able to carry enough water to survive crossing the desert on foot.

Now, let’s use the equations above for a journey to the nearest stars. There are currently 58 known stars within 15 light years. The nearest is the triple star system Alpha Centauri A & B and Proxima Centauri (4.3 ly), and the farthest is LHS 292 (14.9 ly).

I predict that interstellar travel will remain impractical until we figure out a way to harness the vacuum energy of spacetime itself. If we could extract energy from the medium through which we travel, we wouldn’t need to carry fuel onboard the spacecraft.

We already do something analogous to this when we perform a gravity assist maneuver. As the illustration below shows, the spacecraft “borrows” energy by infinitesimally slowing down the much more massive Jupiter in its orbit around the Sun and transferring that energy to the tiny spacecraft so that it speeds up and changes direction. When the spacecraft leaves the gravitational sphere of influence of Jupiter, it is traveling just as fast as it did when it entered it, but now the spacecraft is farther from the Sun and moving faster than it would have otherwise.

Reference: https://www.daviddarling.info/encyclopedia/G/gravityassist.html

Of course, our spacecraft will be “in the middle of nowhere” traveling through interstellar space, but what if space itself has energy we can borrow?

Two Paths to Low Mass

A brown dwarf (also known as an infrared dwarf) is, in a way, a failed star. Early in their lives, these ultra-low-mass stars (13+ M_J) fuse deuterium into helium-3, and in the highest mass brown dwarfs (65-80 M_J) lithium is depleted into helium-4, as shown below.

But the mass is too low for fusion to be sustained (the temperature and pressure in the core aren’t high enough), and soon the fusion reactions peter out. Then, only the slow process of thermal contraction provides a source of heat for the wanna-be star.

There is another, very different, path to a brown dwarf star. A cataclysmic variable usually consists of a white dwarf and a normal star in a close binary system. As material is pulled off the “donor star” (as the normal star is called) onto the white dwarf, the donor star can eventually lose so much mass that it can no longer sustain fusion in its core, and it becomes a brown dwarf star.

When we see a white dwarf / brown dwarf binary system, how do we know that the brown dwarf wasn’t always a brown dwarf? Strong X-ray and ultraviolet emission provides evidence of an accretion disk around the white dwarf, and astronomers can calculate the rate of mass transfer between the two stars. Often, this is billions of tons per second! Using other techniques to estimate the age of the binary system, we sometimes find that the donor star must have started out as a normal star with much more mass than we see today.