Saturday, January 20, 2007

Attack Day

More lazy blogging: another excerpt from my (unpublished) SF novel.

2048-06-30: 1625 GMT. Location: the bridge of the USS Nimitz.

Rear Adm. Samuel J. Scott and his battle group, after three and a quarter hours at 3g, are now a more comfortable 2 million kilometres from the incoming asteroid, at their post-operation coordinates.

While Scott struggled to breathe in his acceleration couch, his similarly-impaired technical staff had been reviewing the telemetry to prepare the briefing the Admiral had ordered, and would surely need.

Now proceeding at 1g in a parking trajectory, Scott gets the picture from his strategy officer.

“They hit us, sir. Hit us with five kinetic energy rounds. We only have it on the fastest imagery, but they launched five antimissiles 114 milliseconds before projected impact. At that time our stealthed warheads were about 115 kilometres out from the asteroid and closing at 1,000 kilometres per second. In the next 14 milliseconds the five hostile antimissiles each covered 100 kilometres and each one scored a direct hit on one of our missiles, totally destroying it.”

Admiral Scott looks nonplussed. Leaving aside considerations of navigation, like how did they know precisely where our missiles were, the timing seems wrong.

“Let me get this straight. They launched kinetic energy kill weapons, which covered 100 kilometres from a standing start in 14 milliseconds?”

“Yes, sir.”

“And how fast were these antimissiles going when they hit our warheads? And what was their acceleration?”

“Sir, we didn’t believe it either, but it comes straight out of the sensor data. Speed of the antimissiles at impact was 4.7 percent of the speed of light. Their acceleration was one hundred million gee.”


The plot to this point involves an asteroid (from the asteroid belt) being de-orbited on a powered trajectory to impact the earth - clearly a hostile act! Our protagonist has a subtle, psychological attack against the enemy, but the military just want to smash it.

They launch five 12 Gigaton thermonuclear missiles designed to hit the asteroid from orthogonal directions simultaneously at 1,000 km/sec. I did the maths. This would disrupt the asteroid beyond gravitational reassembly.

I wanted the alien attackers to have the best possible defences against such an attack, leaving it to the last possible moment. Here are the antimissile specs.

Antimissile mass = 10 kg
Acceleration: 100 million G (neutronium at 100 billion G)
Time to 100 km = 14 milliseconds
Velocity at 100 km = 14,000 km/sec (4.7% of the speed of light)
KE at 100 km (as tons of TNT) = 0.33 Megatons (+ any remaining antimatter)
Antimatter required to accelerate missile to this velocity >= 12 gm (this is a lower-bound, as the propellant-ejecta will be travelling much faster, but will not mass so much)

According to Joseph Lazio, an astronomer who posts to the web, the largest possible acceleration at the surface of a neutron star before it collapses into a black hole is not known precisely, but it’s somewhere between 250 billion and 1 trillion gee. I’m comfortably inside that limit. I think the kinetic energy kill missile would stay normal matter, not even collapsing to neutronium!

Joseph Lazio’s note

Re: What is the maximum possible gravity of a neutron star?
Date: Sun Mar 23 07:25:09 2003
Posted By: Joseph Lazio, Radio Astronomer
Area of science: Astronomy
ID: 1047696071.As

I'm going to start by spelling out quite carefully how one arrives at the answer. Neutron stars are sufficiently massive and compact that general relativity has to be taken into account in describing it, which means that answers can differ from the ones to which we are accustomed in our low-gravity experience.

A more natural way of describing the gravitational acceleration of a body is the gravitational red-shift that it produces. Imagine standing on the surface of an object (or in the case of a black hole being able to hover just above its event horizon). Shine a light of a particular wavelength upward. As the light leaves your beacon, it must lose energy in leaving the gravitational field of the object.

Of course light always travels as the same velocity, about 300,000 km/s, so as it loses energy it cannot slow down. Rather, its wavelength increases. The difference between the wavelength emitted at the surface and the wavelength received by a distant observer divided by the emitted wavelength is the gravitational red-shift or

z = (wave-length-emitted - wave-length-received)/wave-length-emitted.

For reference, here are the gravitational red-shifts from the surfaces of various objects.

Sun 0.000002
white dwarf ~ 0.0002
neutron star ~ 0.35
black hole ~ to infinity

So what does this mean in terms of the gravitational acceleration at its surface? Provided that the red-shift z is "small," i.e., much less than 1, we can use the standard Newtonian formula for the gravitational acceleration.

The neutron star gravitational red-shift is large enough, though, that we have to use the general relativistic formula, which Pons et al. (2002, Astrophysical Journal, vol. 564, p. 981) give as

g = GM / R^2 sqrt (1 - 2GM/ [Rc^2])

Using the canonical values of M = 1.4 solar masses and R = 10 km (and making sure that one does the calculation in consistent units!), I find g = 2.4 x 1012 m/s^2, or expressed in units of the gravitational acceleration at the Earth's surface (which is 9.8 m/s^2),

I find 250 billion g's. If anything, your science fiction story used too low a value!

If a neutron star acquires too much mass, it is not stable against collapse into a black hole. The current best estimates are that this happens around 3 solar masses. The formula for the gravitational acceleration is not linear in the mass, because the mass appears in the square root in the denominator. As a result, as the mass of the neutron star approaches its maximum value, the gravitational red-shift will increase rapidly.

The "canonical" values for the neutron star mass and radius seem reasonable, based on the limited observational evidence that we have, e.g., Thorsett and Chakrabarty (1999, Astrophysical Journal, vol. 512, p. 288). However, if one wants to be extremely cautious and take into account all of the possible theoretical uncertainty, the maximum mass of a neutron star could be as large as about 6 solar masses before it collapsed into a black hole. As one can see from above, that means the gravitational acceleration could be more than four times larger than what I cite as the "canonical" value. A TRILLION GEES.

For more, though quite technical, reading, see Shapiro and Teukolsky, Black Holes, White Dwarfs, and Neutron Stars (1983, John Wiley and Sons, Co.).