A gram is a fortune

The mass defect

A bound system weighs less than its parts, because the energy that holds it together left when it formed. The difference is the mass defect, which accounts for the energy released by nuclear reactions.

Take four hydrogen atoms and make one helium-4 atom. The masses measured best are atomic masses, electrons included, from the AME2020 evaluation:

• one hydrogen-1 atom: 1.00782503 u, so four of them: 4.03130013 u
• one helium-4 atom: 4.00260325 u

The helium weighs 0.0287 u less. One atomic mass unit is 931.494 MeV of energy, so the energy released is

$$Q = (4 m_\mathrm{H} - m_\mathrm{He})\,c^2$$

which comes to 26.73 MeV. The fraction of the hydrogen's mass that left is

$$\frac{4 m_\mathrm{H} - m_\mathrm{He}}{4 m_\mathrm{H}}$$

which is 0.712 %. The ladder rounds that to 0.71 %.

That number includes both the nuclear reaction and subsequent annihilation. Inside the star, four protons become a helium nucleus, two positrons and two neutrinos. Four hydrogen atoms bring four electrons and the helium atom keeps two; the other two meet the two positrons and annihilate. So the difference of atomic masses counts everything at once: 24.69 MeV from the nuclear reaction, the energy the neutrinos carry included, plus 2.04 MeV from the annihilation. The neutrinos pass straight through the star and out into space, and no radiator will ever catch them, so the heat a star can use is a little less than the full 0.712 %.

Why about one percent and not ten?

A hydrogen atom has a rest energy of 938.8 MeV. The helium nucleus, held together by the strong force, has a total binding energy of 28.30 MeV, which is 7.07 MeV for each of its four particles. Divide the second by four times the first and you get 0.754 %. The nuclear force is strong, but its energy is still tiny next to the rest energy of the particles it binds.

The figure is a little smaller than that because two of the four protons must become neutrons on the way. Each neutron outweighs a hydrogen atom by 0.782 MeV, and the cost comes off the top: 28.30 MeV minus two of those is the 26.73 MeV above.

Fusion can be pushed further, but not by much. Iron-56 has an atomic mass of 55.9349355 u. Building it from 56 hydrogen atoms frees 0.892 % of their mass. No nucleus weighs less for each particle it holds, so that is the most any nuclear process can free from hydrogen: a maximum of 1.25 times what hydrogen to helium frees.

Falling to the last stable orbit

Gas near a black hole cannot fall straight in. It has angular momentum, so it circles. Friction in the disc robs it of that momentum slowly, and it drifts inward through a series of nearly circular orbits, giving up energy as heat and light at each step. That is the furnace.

How much energy does a unit mass have on each orbit? In units where G = c = 1 and the hole's mass is 1, a particle in the field of a non-spinning hole obeys

$$\left(\frac{dr}{d\tau}\right)^2 = e^2 - V(r), \qquad V(r) = \left(1 - \frac{2}{r}\right)\left(1 + \frac{\ell^2}{r^2}\right)$$

where e is its energy per unit rest mass and ℓ its angular momentum per unit rest mass. A circular orbit needs V = e² and V′ = 0, which gives

$$\ell^2 = \frac{r^2}{r - 3}, \qquad e = \frac{1 - 2/r}{\sqrt{1 - 3/r}}$$

The orbit is stable only if V″ > 0, and that works out to r > 6. Inside r = 6 the gas has no stable circular orbit to settle on and plunges into the hole with its energy intact. So the gas radiates as it slides down to r = 6 and not after. Put r = 6 into the formula for e:

$$e = \frac{1 - 1/3}{\sqrt{1 - 1/2}} = \sqrt{\frac{8}{9}}$$

which is 0.94281. The gas started far away and at rest, with e = 1, so what it has given up is the difference:

$$\eta = 1 - \sqrt{\tfrac{8}{9}}$$

That is 5.72 %. As a check, Newton's formula for the binding energy of a circular orbit is 1/(2r), which at r = 6 gives 8.3 %. General relativity gives a smaller number of the same order.

With spin

For a spinning hole of spin a, in the same units, the energy of a circular prograde orbit and the radius of the last stable one are

$$e(r) = \frac{r^{3/2} - 2 r^{1/2} + a}{r^{3/4}\sqrt{r^{3/2} - 3 r^{1/2} + 2a}}$$

$$r_\mathrm{isco} = 3 + z_2 - \sqrt{(3 - z_1)(3 + z_1 + 2 z_2)}$$

with $z_1 = 1 + (1 - a^2)^{1/3}\left[(1 + a)^{1/3} + (1 - a)^{1/3}\right]$ and $z_2 = \sqrt{3a^2 + z_1^2}$. At a = 0 this gives r = 6. At Vesper's a = 0.7 it gives 3.39, and at a = 0.998 it gives 1.24. The efficiency η = 1 − e(r_isco) is then the 10.4 % and 32.1 % of the ladder. Spin lets the gas orbit closer before it plunges, and the deeper it gets, the more energy it gives up.

Efficiency against spin

The formulas above are from James Bardeen, William Press and Saul Teukolsky (1972), who worked out the orbits around a spinning hole. For gas that orbits against the spin, the same formulas hold with a negative and the minus sign in front of the square root in r_isco turned to a plus. Their result gives the whole curve, and its ends are worth knowing:

Spin	Efficiency
Against the spin of a hole turning as fast as it can (a = −1, last stable orbit at r = 9)	3.8 %
No spin	5.7 %
0.7 (Vesper)	10.4 %
0.998	32.1 %
With the spin of a hole turning as fast as it can (a = 1, last stable orbit at r = 1)	42.3 %

A disc that turns against the hole's spin gets the worst deal there is, and one that turns with it gets the best. Between the two the efficiency changes by a factor of 11.

Why 0.998 and not 1?

The gas a disc feeds in carries angular momentum the same way the hole turns, so feeding alone would spin a hole all the way up to a = 1. Kip Thorne showed in 1974 that the disc's own light stops it short. The disc shines in every direction, and some of that light falls into the hole. Light moving against the hole's spin is swallowed more readily than light moving with it, so what the hole catches carries angular momentum that opposes its spin. The spin climbs until that drag balances the spin-up from the gas, at a = 0.998. The ladder's rung of 32 % sits at that limit.

The limit assumes a thin disc and leaves out magnetic fields. Charles Gammie, Stuart Shapiro and Jonathan McKinney ran relativistic magnetohydrodynamic simulations in 2004 and found that, for one sequence of flows, the spin stopped rising well short of Thorne's limit; they noted that this need not hold for every flow, thin discs especially. Where real holes settle is still argued. Vesper's 0.7 is a figure of this world and does not depend on the argument.

Radiated against total

The efficiency η in the table is the energy the gas has given up by the time it reaches the last stable orbit. It is an upper limit on what shines, and in a thin, cool disc nearly all of it does. Not every flow is like that. When the gas is tenuous and hot, it cannot radiate quickly enough, and the heat is carried into the hole with the gas instead. Ramesh Narayan and Insu Yi worked out these advection-dominated flows in 1994: most of the energy is stored in the gas as heat rather than radiated. A hole fed too gently shines far less than the ladder promises, which is why the feed rate is part of the design and not an afterthought; the next page, Feeding a black hole, takes this up.

Jets, and more than 100 percent

Some of the energy leaves as neither light nor heat, but as a magnetised outflow. In 1977 Roger Blandford and Roman Znajek showed that magnetic fields threading a spinning hole can pull energy out of its rotation. Efficiency in this sense is the energy leaving divided by the rest-mass energy of the gas fed in. In 2011 Alexander Tchekhovskoy, Ramesh Narayan and Jonathan McKinney simulated a hole spinning close to the limit, wrapped in strongly magnetised gas, and found jets and winds carrying away more energy than the gas brought in. The difference came from the hole's spin, which slowed. The extra energy is drawn from the hole's stored rotation, as Mining a spinning hole explains. This result comes from a simulation; it has not been measured for a particular hole.

The top rung

All the rungs above the nuclear ones are for gas falling into a hole. The top rung needs the mass of a hole to be turned into light altogether, which happens only through Hawking radiation over times far longer than anything the Vigil plans, or by meeting antimatter, which has to be made first. Neither is a power source in 2540.

Further reading

1. Einstein (1905), Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig? (Does the inertia of a body depend upon its energy content?), Annalen der PhysikA body that gives off energy loses mass: the first statement of mass-energy.
2. Eddington (1920), The internal constitution of the stars, NatureSuggests that stars shine by turning hydrogen into helium, since a helium atom weighs less than four hydrogen atoms.
3. Bethe (1939), Energy production in stars, Physical ReviewWorks out the reaction chains that make helium from hydrogen in stars.
4. Huang, Wang, Kondev, Audi and Naimi (2021), The AME 2020 atomic mass evaluation (I), Chinese Physics CHow the atomic masses behind the mass defects are evaluated.
5. Wang, Huang, Kondev, Audi and Naimi (2021), The AME 2020 atomic mass evaluation (II), Chinese Physics CThe table of atomic masses used on this page.
6. Bardeen, Press and Teukolsky (1972), Rotating black holes: locally nonrotating frames, energy extraction, and scalar synchrotron radiation, The Astrophysical JournalThe orbits around a spinning hole, including the innermost stable orbit and its energy.
7. Thorne (1974), Disk-accretion onto a black hole. II. Evolution of the hole, The Astrophysical JournalThe disc light a hole swallows stops its spin just short of the maximum.
8. Gammie, Shapiro and McKinney (2004), Black hole spin evolution, The Astrophysical JournalMagnetised accretion simulations in which the spin settles below Thorne's limit.
9. Narayan and Yi (1994), Advection-dominated accretion: a self-similar solution, The Astrophysical Journal LettersWhen gas cannot radiate fast enough, most of the energy is stored and carried into the hole.
10. Blandford and Znajek (1977), Electromagnetic extraction of energy from Kerr black holes, Monthly Notices of the Royal Astronomical SocietyMagnetic fields can draw energy out of a spinning hole.
11. Tchekhovskoy, Narayan and McKinney (2011), Efficient generation of jets from magnetically arrested accretion on a rapidly spinning black hole, Monthly Notices of the Royal Astronomical Society: LettersSimulated jets that carry out more energy than the gas brings in.