Decelerating: laser vs gun vs rocket

In our pa­per on ex­plor­ing the uni­verse, some of our probes re­quired huge quan­tities of re­ac­tion mass in or­der to de­cel­er­ate on ar­rival.

This is due to the tyranny of the rocket equa­tion: to de­cel­er­ate our fi­nal mass by , we need an ini­tial mass so that:

  • , where is the ex­haust ve­loc­ity.

Given this equa­tion, the ini­tial mass needed grows as ex­po­nen­tial in the re­quired de­cel­er­a­tion .

The rel­a­tivis­tic rocket equa­tion is similar but even worse, with an ex­tra term in it.

So the most effec­tive way of al­low­ing high speed ex­plo­ra­tion of the uni­verse is to some­how get around the rocket equa­tion. I was vaguely think­ing about ways of us­ing the tar­get galaxy or so­lar sys­tem to do that—maybe the probe could scoop up in­ter­stel­lar dust or use grav­ity sling­shots. But I re­al­ised that we can get around the rocket equa­tion more di­rectly.

De­cel­er­at­ing with guns

Sup­pose you are shoot­ing through space, and you want to de­cel­er­ate by point­ing a gun in the di­rec­tion of travel. Be­cause of rel­a­tivity, we can con­sider that you are at rest, and that you are ac­cel­er­at­ing by shoot­ing a gun in an op­po­site di­rec­tion.

You have two bul­lets, and you can shoot the bul­lets se­quen­tially or si­mul­ta­neously; imag­ine that you had two guns strapped to­gether. If you shoot the bul­lets one af­ter an­other, the first will start mov­ing at ve­loc­ity of while you re­coil at some . Then when you shoot the sec­ond bul­let, it will start mov­ing at (if we stay in the clas­si­cal model for the mo­ment). So the to­tal back­wards mo­men­tum is , where is the mass of a bul­let.

If you fire the bul­lets si­mul­ta­neously, the to­tal back­wards mo­men­tum is , how­ever, and . By con­ser­va­tion of mo­men­tum, you will there­fore be re­coiling faster than if you shot the bul­lets se­quen­tially.

What’s hap­pened? When you shot the bul­lets se­quen­tially, part of re­coil of the first bul­let went into mov­ing the sec­ond bul­let at the same speed as you, which you ac­tu­ally didn’t want. When you shot both bul­lets to­gether, the re­coil of both went purely to mov­ing you. There­fore si­mul­ta­neous fire is more effec­tive at ac­cel­er­at­ing/​de­cel­er­at­ing. The real tyranny of the rocket equa­tion comes from the fact that the early fuel needs to move the later fuel that needs to move the even later fuel. And most of that mo­men­tum gain is com­pletely wasted: we don’t ac­tu­ally care that ex­haust fuel has gained mo­men­tum through the pro­cess. We’d like that ex­tra mo­men­tum to be ap­plied to the pay­load or probe, not to the fuel.

In prac­tice: laser and so­lar sail

So there’s a the­o­ret­i­cal way around the rocket equa­tion; can we do this in prac­tice? Ex­pend­ing all fuel si­mul­ta­neously would help (the equiv­a­lent of shoot­ing all your bul­lets at once), but that ex­treme discharge might tear the probe and the rocket to pieces.

In space, there’s no differ­ence be­tween the gun and the bul­let—they’re both just pieces of mass that fly off in op­po­site di­rec­tions due to an ex­plo­sion. So now imag­ine that there are ten thou­sand guns, float­ing in­de­pen­dently in space, point­ing at you. Every­thing is at rest with each other, and all the guns will fire in some se­quence, and you will catch all the bul­lets (com­pletely in­elas­tic col­li­sion). As­sume each gun, of mass , will re­coil with ve­loc­ity . Then the guns will have a to­tal mo­men­tum of , and, by con­ser­va­tion of mo­men­tum, you and the bul­lets will have the same mo­men­tum in the op­po­site di­rec­tion. If the mass of the bul­lets (and you) is small com­pared to , this will be an effec­tive way of ac­cel­er­at­ing you. And note that your to­tal fi­nal mo­men­tum de­pends on your mass, the mass of the bul­lets, the num­ber of guns, , and . So it does not de­pend on the guns be­ing fired at the same time, or any de­tails of when they were fired. As long as you can catch ev­ery bul­let, your fi­nal ac­cel­er­a­tion/​de­cel­er­a­tion will be the same. So you don’t need to burn all your en­ergy at once.

Catch­ing bul­lets is hard, and we want to min­imise their mass. So it’s even bet­ter if we do this with lasers! Un­furl a so­lar sail around your­self, and have ten thou­sand free-float­ing lasers shoot at you in some se­quence. This will gain you all the mo­men­tum of the lasers, in­de­pen­dently of the se­quence of firing.

The only real prac­ti­cal con­sid­er­a­tion is that you can cool down fast enough that each laser can fire be­fore your sail moves out of their fo­cus range; but a big­ger sail can make both cool­ing and long dis­tance firing eas­ier.

Ex­tra, the­o­ret­i­cal, efficiency

What if your sail doesn’t perfectly ab­sorb all the laser light, but re­flects some of it back? That’s even bet­ter! In terms of bul­lets, that’s the equiv­a­lent of elas­tic col­li­sions, and you’ll ac­cel­er­ate/​de­cel­er­ate even faster, los­ing less en­ergy. Think in terms of con­ser­va­tion of mo­men­tum again: some light is now mov­ing back­wards, away from you. This can only hap­pen if you’ve your­self gained some for­ward mo­men­tum.

In fact, the perfectly effi­cient way of de­cel­er­at­ing would be for you to de­ploy a gi­ant mir­ror, and for a sin­gle gi­ant laser to do the same, then for the laser to blast you. The laser beam would bounce back be­tween your mir­ror and the laser’s mir­ror, grad­u­ally get­ting red­shifted as you and the laser move faster and faster apart. This setup pre­serves both mo­men­tum and en­ergy, and is the most perfectly effi­cient way of de­cel­er­at­ing—and it doesn’t de­pend on how fast the laser fires, a slow burn reaches the same con­clu­sion as a swift burst. Why? Be­cause con­serv­ing en­ergy and mo­men­tum dic­tates the speeds at which you and the laser will end up.

Of course, in prac­tice, the mir­rors would not be perfectly re­flec­tive, the beam would lose fo­cus, there would be some cos­mic dust, and so on. Still, it’s in­ter­est­ing to note that, in the­ory, we can com­pletely do away with the rocket equa­tion and ac­cel­er­ate/​de­cel­er­ate in the most effi­cient way pos­si­ble, while us­ing up en­ergy ar­bi­trar­ily slowly to do so. This hints that there may be prac­ti­cal meth­ods that could get very effi­cient as well.