Archive | April 2014

Hitching a ride on a black hole, part IV: Going on an 10gnt

You might be wondering why I picked 10g as the star drive’s acceleration and whether it is gives you much vs a leisurely 1g or less. Let us deal with this question.

First, assuming perfect mass-to-energy conversion, how long can we keep accelerating before all the fuel is used up? (We will not be considering ramjet-style refueling by scooping interstellar medium.) An honest calculation would require some calculus, but we can skip that by noting that when rate of change as a fraction of magnitude is a constant, the magnitude decreases exponentially. A well-known example is radioactive decay.

So, suppose we start with a rocket ship with mass M. If every T second we eject m kg of fuel with velocity v, then the thrust pushing the rocket forward is \frac{mv}{T} and its acceleration is a=\frac{mv}{MT}. After T seconds the rocket’s mass is M-m, so to keep the acceleration constant we need to reduce thrust correspondingly, and continue to do so as the rocket gets lighter and lighter. As a result, the rocket’s mass decreases slower and slower with time, getting to about 37% of its original weight after t=\frac va seconds (and to 37% of that after another t seconds). For a light-speed exhaust this works out to be only 35 days at 10g acceleration.

This should give you pause. It sure made me check this simple calculation, just in case. With our star drive we would have to basically annihilate 60% of the star in just one month. For scale, the Sun converts not even 10% of its mass into energy by burning Hydrogen then Helium for over 10 billion years! And we plan to burn that much in one week! Or at least one week by the ship’s clock, as we would be close to the speed of light after only a few weeks, when time dilation firmly sets in. Still, this is very much comparable with supernova explosions, only going on for weeks non-stop.

Now would be a good time for environmental considerations. If the exhaust consists mostly of electromagnetic radiation, then whatever is in the path of the exhaust would not fare well, not within a few light years, anyway. Says Wikipedia:

Gamma rays induce a chemical reaction in the upper atmosphere, converting molecular nitrogen into nitrogen oxides, depleting the ozone layer enough to expose the surface to harmful solar and cosmic radiation

And that’s a few dozen light years away. Using near-light speed massive particles, like Hydrogen plasma jets is just as bad. CEPA, the Cosmic Environmental Protection Agency, might be a bit put out with us for destroying all life on a planet or ten.

Fortunately, there is a way out. Kind of. If the exhaust mostly consisted of neutrinos, the impact on the surrounding Cosmos would be minimal. Neutrinos go through everything almost completely unaffected, so even at a few million km the radiation exposure from them is quite small.

Interestingly enough, most of the energy released in a core-collapse supernova, the kind where a large star runs out of gas and collapses on itself until everything in it turns into neutrons, is in the form on neutrinos. However, this is just way too weak by our standards, as most of the energy is not released, but remains as its neutron core or the resulting black hole. Yes, that’s right, one of the most dreadful star explosions we observe, visible a few galaxies over, just doesn’t produce enough energy to power our star drive.

There are good reasons that it is very hard to turn normal matter into neutrinos completely, most of them are related to conservation laws. Specifically, we currently do not know of any way to turn “quarks”, the stuff of which atomic nuclei are made into “leptons”, which is what neutrinos are. It is entirely possible that as-yet-unknown laws of physics come into play at very high energies, but, if so, this energy would have to be higher than anything a mere supernova can produce, since we don’t see supernovae disappear in a puff of neutrinos.

One bit of good news is that this is not an issue for black/white holes, as quark number is not conserved (or at least not visible) after the black hole is formed. Only mass, angular momentum and electric charge are preserved by the collapse. So there is a hope that if we give up on “real” stars, like white dwarfs and neutron stars, and build our star drive out of the black/white hole combo instead, then there is no restriction on the type of material emitted.

Anyway, back to our calculations. Suppose we have run at full power for some time, and burned the fraction X of our star drive. How fast are we going now? Well, almost at the speed of light if X is large enough. What is more interesting is not the exact speed, but the time dilation/space contraction factor. Because this factor is what tells you how fast you get to where you are going. Like, if the factor is 10, then you get across 10 light years distance in only in one year local time.

So the space contraction factor for a 100%-efficient rocket with light-speed exhaust is \gamma=\frac{X^2+1}{2X}. While the expression itself is simple, I have not found an easy way to derive it, without cranking through the relativistic rocket calculations, so please forgive me for just dropping it here without a proper motivation. And if you know of a way, please comment.

Let’s plug in some numbers! As we have seen earlier, after one month of travel at 10g (or after almost a full year at just 1g) we are left with 40% of the fuel we had at the start. Plugging in X=0.4 in the time-dilation formula above, we get \gamma=1.45, not a very impressive number. It only reduces our subjective travel time by barely 30%. If we give it another month, we are down to X=0.16 and \gamma=3.2. Only when we burn away 95% of the star we get 10 times for our time dilation effect. How long will it take at 10g? About 3 months ship time.

So, we accelerate until our time dilation factor is 10, then what? Turn off the star drive and cruise? But then we are left weightless for most of the flight, until we start decelerating, which kind of defeats the purpose of our ingenious setup to provide a steady 1g for the crew. Also, how long would we have to cruise weightless? With the time dilation factor of 10 it would take full 3 years to get to, say, Vega, if you count acceleration and deceleration. Including 2.5 years with the drive off. And after decelerating we are left with 5% of 5% or just one quarter of one percent of the original star. Somewhat wasteful, is it not? We used up a whole star in a most efficient way imaginable to get not across the Universe, not across the Galaxy, but across less than one one-thousandth of the Milky Way. And it took us three years to do it.

To summarize the disappointing answer to our original question, 10g acceleration gets us the time dilation/space contraction factor of 10 if we accelerate for 3 months and burn away 95% of the fuel. But wait! Not all is lost quite yet. This result is for a conventional rocket, not a black-hole joy ride, where the effects of General Relativity are also important. We will see if it makes any difference next time.

Hitching a ride on a black hole, part III: Edge of your seat

Last time we designed our literal-star-drive-based space ship:

star surfing

and figured out that to provide 10g of acceleration in the frame of the star while surfing its gravitational wake in free-fall, we would have to trail the star drive by about a million kilometers (or miles, the calculations were not very precise). We also noticed that the tidal gravity from the star limits the size of the crew module to tens of kilometers, or the size of a really small moon, like Phobos. Anything larger is going to be stretched beyond breaking point and ripped into pieces. This size is certainly nothing to sneeze at, but it has certain limitations. For example, if you want to travel comfortably without leaving your home planet and instead use it at the crew module, it is probably too large to remain in one piece. So even the largest possible crew module does not provide enough natural gravity to feel comfortable.

Of course, gravity does not have to be natural. One can always produce enough centrifugal forces by spinning the module fast enough. Many sci-fi stories rely on that. A ring of 1 km in radius only needs to make a full turn once a minute to give the illusion of 1g gravity on the inside of its surface. “Only”? Seems a bit dizzying, to see the world around you complete a full rotation in barely one minute. Just over 3 min for a 10 km ring. Still not a lot of fun. Imagine all those distant stars zipping around you every minute. So, just for fun, let’s consider other alternatives. After all we’ve been playing with gravity for some time in this series, why stop now?

But first… Let us return to the issue of stability. If you ever surfed, skied or even walked (and I hope you have done at least that last activity, otherwise you are probably just a brain in a jar), you realize that it requires some effort to remain on the sweet spot of the wave. If you goof, you are either too far down or too far up and probably off your board. And that’s assuming you don’t fall over sideways. This is because the equilibrium you achieve is unstable. It’s more like balancing a long stick vertically on your finger than holding it hanging down. You can do it, but you have to constantly compensate for the damn thing trying to fall. It is the same with our star-drive setup. If the crew module lags a bit behind the sweet spot, it would tend to keep lagging farther and farther, until it is left behind in the darkness, forever lost. If the crew module slides a bit forward, the star’s gravity will pull it harder and it will plunge toward it and disintegrate in the short order, which is also a suboptimal outcome. So we have to maintain constant vigilance and compensate for random deviations from the optimal surfing location. How would we do that? One obvious solution is to have small rocket engines — maneuvering thrusters — to provide course corrections. This means having to carry extra fuel, maintain the engines, etc. While inescapable in other circumstances, using on-board engines for course correction seems a bit silly when we are literally blowing up a star for fuel just ahead of us.

Note that in the diagram above I carefully drew the exhaust skirting the crew module. Mainly because I did not want the crew members to be incinerated by a stream of radiation strong enough to accelerate a star-massed object at freaking 10 gravities! We will get to the question of how terrifyingly awesome this exhaust must be later in this post series. For now let us note that if the star drive goes out of alignment only a little bit and the propellant hits the crew module, the crew does not stand a chance in hell, because hell is what it would feel like for the scant few seconds until they get evaporated. However, in homeopathic doses this same exhaust might be helpful, rather than harmful. (Who said homeopathy doesn’t work?)

If only a small amount of exhaust hits our crew module, it nudges us backward. This can be good if we stray too close to the star. A useful side effect is that it provides some acceleration, which to the people inside would feel like gravity. Isn’t it nice to kill two birds with one stone? Umm, this metaphor might be a bit cruel and dated. Throwing stones at birds might have been considered good clean fun a couple of centuries ago, but not by the enlightened standards of the present day. Anyway, I digress. We want to use a small amount of starlight to stabilize the crew module and to simulate gravity. Again, this idea is not new, using solar sail for interplanetary travel has been discussed for some 100 years. A German-Russian-Soviet scientist and engineer Friedrich Zander was apparently one of the first to do a serious calculation of the solar sail propulsion, during his downtime between designing one of the first liquid-fuel rockets (well, second, after Goddard in the US, but before von Braun in Germany) and figuring out the details of a gravitational slingshot, a now-standard technique for interplanetary travel.

There is at least a couple of ways we can design our solar sail: we can put it in front of the crew module, like a rocket:

Star-surfing with reflector

or behind it, like a parachute:

Star-surfing with parachute

There are advantages and disadvantages in both cases, as evident by the fact that both approaches are in use. A parachute-style sail provides stability for free, while a crew module equipped with a reflector would have to be actively stabilized, the way planes and rockets are. On the other hand, a heavy module precariously suspended on something like ropes… What if one of them breaks? And how will they fair while being bombarded by the exhaust rays? In either case, as you can see, the direction toward the star becomes “down”. The preferred location of the crew module would be somewhat ahead of the free-fall point. How far ahead? Not very much. If the star drive provides 10g acceleration, we only need to get to the 11g zone to get 1g apparent gravity. And since the gravitational force is proportional to the inverse square of the distance, we only need to be 5% closer to the star to get that extra 1g, or around 50,000 km closer.

Interestingly, the solar sail provides us something like a neutral equilibrium: if we get closer to the star, its gravity gets stronger, but so does the exhaust density, which also falls off as inverse square of the distance, or close to it. So we still need to do some course correction, but not nearly as much. There are also some other small effects which affect the crew module, like a slight blueshift of the exhaust during the time it took for it to reach the sails, the gravitational effects of the exhaust left behind, and a few others. However, we cannot properly address them without doing a fully general-relativistic analysis, and they are too small to matter, anyway.

So, as promised, we have improved our star ride to make it more stable, easier to control and more comfortable by diverting a tiny fraction of the ejecta toward the solar sail. We have not yet calculated how big or strong the sail would have to be. Hopefully next time.  We will also look into how far 10g acceleration gets us and how fast. In the meantime, please feel free to comment or ask questions!

 

Hitching a ride on a black hole, part II: The basics of star-surfing

Last time we started by trying to reduce the effects of g-forces on the crew and ended up considering a star corpse as our star ship. Here is a schematic drawing of how it looks:

star surfing

Yes, this is a 5-min drawing in Google docs. If you are inspired to do better, let me know and I will gratefully replace this “drawing” with yours, and give you credit, of course.

Here is what is happening on this picture: A star is used as a propulsion source controlled by as-yet-unspecified technology and emits its content at or near light speed in one direction. As a result, it is pushed in the opposite direction, like a rocket. A crew module is positioned at a respectable distance (which we will shortly calculate) and is in a free fall toward the star. While the star in question keeps getting away. If you position the crew ship just right, it will be stationary relative to the star, even if the acceleration of the star itself is many times Earth’s gravity.

Let’s do a few simple calculations to see how far from the star we should place our ship. We assume that the star used as a star-drive is roughly one solar mass. This is not an unreasonable assumption, as all white dwarfs and neutron stars and even newly formed stellar-remnant black holes are in that range. Instead of dealing with the large and inconvenient numbers, like the gravitational constant, or the Sun’s mass, I will use only a few basic small ones I remember:

  • Free-fall acceleration at earth’s surface (1g).
  • Earth’s speed in its orbit around the Sun (v = 30 km/s).
  • Distance from Earth to Sun (r = 1 astronomical unit = 150 million kilometers).
  • Newton’s law of gravitation: the attraction force falls as the square of the distance between two bodies.

For now, we will do only the basic Newtonian physics, no relativistic corrections whatsoever. Because none are needed just yet: we are dealing with relatively slow speed and weak gravity. First, let’s calculate the centripetal acceleration due to the Earth-Sun attraction force: it is \frac{v^2}{r} = \frac{g}{1600}. So, to get a 1g acceleration we need to get 40 times closer to the Sun,down  to only 4 million kilometers from its center, or barely 3 million km from its surface, 10 times closer than Mercury. Clearly this is not a healthy place to be in. That’s one reason that a white dwarf would work better. Hopefully. If we want to have our star drive to work at something like 10g, we’d need to be 3 times closer still, down to just over 1 million km from the center. This close even relatively dim objects like white dwarfs and neutron stars would probably make your life uncomfortable, no matter how good your radiation shielding is. So, we are likely down to our last star drive candidate, the black hole. Unless we use a huge crew module… like an asteroid-sized one? Made from an actual asteroid, maybe? But will a relatively large object like that withstand the tidal forces exerted on it by the star’s gravity? Let’s do another simple calculation!

Given gravitational acceleration a and distance r from the massive object, what is the tidal force per unit length? Well, dimensional analysis to the rescue! There is only one way to get acceleration per unit length from these two numbers, and it is to divide them: \frac{a}{r}! (An honest calculation which involves a small amount of calculus gives us an extra factor of 2.) For an asteroid 100 km in diameter the resulting tidal gravity between its “bow” and “stern” is \frac{2 \cdot 100 km \cdot 10 g}{1,000,000 km} = \frac{g}{5000}. This is a rather inconvenient number. On one hand, it is too small to provide a comfortable level of gravity to the crew, on the other hand, it is large enough to tear our gravitationally-bound asteroid apart, or at least make it shed most of its mass, until it is small enough for the cohesion forces to dominate self-gravity, a few dozen kilometers at best. How did I calculate that last number? I didn’t. I cheated. I looked up shapes of some moons and asteroids, like Phobos, and picked the size where they stop being more-or-less round.

Let’s review what we have figured out so far. If we get a white dwarf-powered star drive provide a 10 g acceleration, we can make a Phobos-sized crew module surf its gravitational wake some 1 million km behind while nearly weightless. This answers our first question posed in the previous post: “How far behind the star-ship is the sweet spot for the crew?”. We shall discuss the next question, how stable the ride is, in the next post. We will also talk about ways to provide Earth-like gravity for the crew without expending a lot of on-board fuel.

 

Hitching a ride on a black hole, part I

We have been talking about white holes and black holes and how one cannot exist without the other (or at least white without black). But let us return to the slow and familiar world of Sir Isaac Newton for the moment.

Suppose you have a massive star ship. Maybe one carved from an asteroid or even a planetoid. To protect the crew from the dangers of the outer space. And it has plenty of material for propulsion, assuming some day the rocks and metals can be used as fuel and/or propellant.

Now, once in flight at a 1 g acceleration the crew inside would feel the normal Earth’s gravity. However, someone on the surface of the ship would feel either heavier or lighter, depending on where exactly they are standing. On the bow of the ship the asteroid’s gravity would add to the ship’s acceleration, while on the stern the apparent gravity would be less than the ship’s acceleration by the same amount. Just take care to not get too close to the engines and exhaust. Imagine how powerful and probably hot they must be to accelerate something like a Death Star at 1g.

Now, the surface gravity of an asteroid is pretty small. Even on the largest known one in the solar system, Ceres, it is not even 1/30 of that on Earth, so it would not significantly affect the apparent gravity on the ship. But if the asteroid was made of a denser material (and be correspondingly smaller for the same mass), its gravity would be stronger. Same mass in half as much radius means four times as much gravity, etc.

This would present an opportunity. Suppose we want to accelerate to the cruising velocity faster than just at 1g. Instead of subjecting the crew to strong g-forces for an extended period of time, we could relocate them toward the stern of the ship, where the ship’s own gravity would counteract the g-forces. Unfortunately, no known material is dense enough to provide 1g worth of surface gravity in a relatively small object. So, unless we want to just take a whole Earth-sized planet, our hope of counteracting the effects of acceleration are in vain. Plus, any self-respecting planet keeps itself in shape thanks to its own gravity. It would quickly rearrange its shape or even fall apart if we were to do something as violent to it as accelerating beyond the gentlest nudge.

But wait, not all is lost! Why think small? We know of a few of celestial objects which are dense enough and durable enough to withstand a bit of rough handling. Alas, none lend themselves easily to carving a star ship out of. But let’s see where this leads us. One such dense object is a white dwarf, a remnant of a sun-like star. Its surface gravity is some millions of g, so an extra g required to accelerate it would hardly make a dent. Literally. Of course, there is the small matter of making the white dwarf into a rocket engine, but let’s suppose we solved this minor problem and the cooling star corpse is made to emit its guts in the right direction at something close to the light speed to provide acceleration.

Now, our crew cannot, of course, make their quarters inside or on the surface of the dying star, it is still way too hot, Sun’s surface hot. It is also as massive as the Sun. So our initial plan to ride an asteroid has been inflated somewhat. Still, size-wise a white dwarf it is relatively small compared to a “real” star, it is only maybe Earth-sized. So we need to keep our distance, or radiation and gravity will do us in.

Before we do some calculations regarding the practicality of star-riding, let’s look at other options. One object even denser than a white dwarf is a neutron star. They are only slightly heavier than white dwarfs, but much denser and smaller (a city-size, rather than a planet-size) and so emit a lot less radiation in total, even though they are hotter. Also, turning one into a rocket could be somewhat problematic, given how strongly it is gravitationally bound. On the other hand, pulsars manage to fling a lot of energetic matter and radiation into space, so maybe the task of rocketizing a neutron star is not as impossible as it looks at the first glance.

The last option for high-acceleration star riding is, of course, the object dear to my heart, the amazing black and white hole combo. Black holes are roughly as massive as the other two super-dense objects, only smaller, maybe a dozen city blocks in size. And black holes usually do not emit anything, so that is both convenient and annoying. On one hand, you don’t need to block the potentially harmful radiation from the star, on the other hand you cannot harness this radiation as a source of energy. On the third hand, if our project involves milking stars for fuel, the scale of the energy sources required to do that is probably rather larger than what solar batteries can provide.

So, we are back to black holes, bye-bye Newton, hello Einstein. So, how the heck would we extract energy from a black hole, let alone shape it as radiation emitted in a specific direction as exhaust? Well, the actual mechanism is a bit fuzzy, just like it is for white dwarfs and neutron stars, but the god news is that, just like Tsiolkovsky was the first one (well, not really first, but hey, it’s Stigler’s world out there) who figured out the rocket equation for the non-relativistic propulsion, one William Kinnersley did that for relativistic one, at least when the propellant is massless. This is known as the Kinnersley photon rocket. And it just so happens that it describes the last case we discussed, at least to some extent: light, or something like light is emitted from a black/white hole preferentially in one direction, accelerating the hole in the opposite direction.

Now it is almost time for some basic calculations. But first let’s review what we have figured out so far. We wanted to use starship’s natural gravity to reduce the effects of g-forces on the crew and quickly realized that the ship would have to be very dense for this to work. And, as far as we know, dense means heavy, Sun-mass heavy. So we’d have to turn a star into a star-ship, literally. And this means we have to keep our crew a ways out, behind the star-ship, surfing its wake. Maybe it should be called star-surfing? Things we still need to figure out are manifold, but here are some of them, in no particular order:

  • How far behind the star-ship is the sweet spot for the crew?
  • How large and stable is that sweet spot?
  • How fast can we accelerate and how far can we travel until the star-ship is all used up?
  • The twin paradox and all, how much will the crew age during a round trip, compared to those left behind?
  • Is it even ethical to kill stars for fuel, even if they are already [almost] dead?

To be continued…