Dot Physics

If you have been on the internet, you surely have seen this website showing the sizes of different starships. Way cool. Here is a small sample – but it doesn’t do the site justice. You really need to go browse around.

i-0b9109e45aa1aad7b5ef680d59e3cda6-2010-04-16_sm.jpg

Of course, I can’t let something like this just go. One of the things I always think is interesting is to consider objects of different size. Perhaps the general idea is that you can just scale stuff up or down as you like. But, it doesn’t work this way. Let me start with my own spaceship. It is a sphere with a thruster on the back. It just holds one person.

i-eead2c58e4a170d16032fdf074c7d441-2010-04-16_untitled_9.jpg

Now, what happens if I want to make a bigger version. Let me go ahead and get some related points out of the way.

What does the thruster do?

Great question. Really, what do they do? In real spacecraft, these thrusters are used to change the momentum of the spacecraft. You could think of these as exerting a force on the craft which would make it accelerate. What if you kept the thruster on for a long time? You would keep accelerating. Do you see the problem? Most of these ships in the science fiction shows fly with their thrusters on at constant speed. This is what would happen if you have some resistive force (like air resistance).

The point is: I am going to assume that the thruster accelerates the ship – which may not agree with the movie. Moving on.

Other Assumptions

I know someone is going to point out some of these issues (even though I am stating now that they are assumptions). Some of these might not be completely valid – but it will be close enough to show my point.

  • Force from the thruster is proportional to the area of the thruster. I know – you could come up with all sorts of reasons why this would not be true, but I am still going with this assumption.
  • The density of big and small starcrafts are about the same. Yes, maybe the walls of a ship are the same thickness, and this would make bigger ships have a lower density – but then they have more internal walls also.
  • Small starcraft and large starcraft have thrusters to produce the same (or similar) acceleration.
  • The following all mean the same thing: space ship, starship, starcraft, space craft, etc…

The Physics

Let me look at my small craft above. If the only force on it is from the thruster, then:

i-7938e04786353e42062afd24e7e9cccf-2010-04-16_la_te_xi_t_1_13.jpg

If Fthrust is the only force on the ship, then the acceleration and the thrust are always in the same direction. This means I can just look at the magnitude of thrust and acceleration. In my above ship, it is a sphere with a radius R. The thruster is circular with a diameter L. If thrust is proportional to the area of the thruster, I can say:

i-70fb838ce8cac43a090104cde3c17d83-2010-04-16_la_te_xi_t_1_14.jpg

Where K is some constant. The mass is related to the density and the volume so that:

i-d7abddd650dfa1890b67d77bd3efa3be-2010-04-16_la_te_xi_t_1_15.jpg

Here, C is another constant (not to be confused with the konstant K). C includes info about the shape and the density – both of which are not too important. This means:

i-c340b09c385aa6b15182388d97bbc954-2010-04-16_la_te_xi_t_1_17.jpg

One more thing – let me get L as a function of R.

i-45fed17665915c602d41bf1c9b368393-2010-04-16_la_te_xi_t_1_18.jpg

Notice – I made a new constant (alpha). Just because.

A bigger spherical ship

I am now going to build an even bigger spherical ship. It is going to be 10 times the radius of the first one. It is going to have about the same density and be capable of the same acceleration. How big would it’s thruster be? If the constant alpha is the same, and I increase R by a factor of ten then:

i-e05b919055f26909adfdc5b89631a380-2010-04-16_la_te_xi_t_1_19.jpg

Increasing the radius by a factor of 10 means I would have to increase the diameter of the thruster by a factor of 31.6ish. So, it wouldn’t “look” the same. This is what it would look like.

i-965de0fced5977aaf4bfa433873e2348-2010-04-16_untitled_10.jpg

Bigger ships need way bigger thrusters.

An example in Star Wars

The Star Wars universe has two ships that are a great “case study”. These are the Star Destroyer and the Super Star Destroyer. What makes these two ships great for a comparison? Well, they are in the same universe. They have the same shape. I can safely assume that they have similar densities. Finally, if they are going to be in the same fleet, it seems reasonable that they would have similar accelerations. Here is a side by side partial comparison (also from Jeff Russell’s starship dimensions page)

i-eb691550a5409af355a1253c29a123dc-2010-04-16_untitled_11.jpg

Apparently, there are some others debating the size of the Super Star Destroyer. I will go with the dimensions from Jeff Russell’s diagram. This gives the Star Destroyer with a length of 1.6 km and the Super Star Destroyer with a length of 19.0 km.

What about the thrusters? According to this page at theforce.net, there are 13 thrusters on the back of the Super Star Destroyer. And there are 3 on the back of the plain old Star Destroyer.

i-889cf8155849f933ae3109a8db978b9d-2010-04-16_sdfs.jpg

I will first calculate the constant alpha for the Star Destroyer. I know it is not a sphere, but here R is like the length. So how big are the thrusters? By examining Jeff’s diagram, I get the length of the thruster is L = 0.126 km. This gives:

i-9d188490a98cd59114deb57d06b1c8e6-2010-04-16_la_te_xi_t_1_21.jpg

Note the square root of 3. That is because the three thrusters increase the thrust by three. If I increased the length of a thruster by a factor of three, the area would increase by a factor of 9. Now. if I assume 13 identically sized thrusters on the back of the Super Star Destroyer, how big should they be? Using the same alpha for the Star Destroyer and R = 19.0 km:

i-0c3c188d1186aae9d731ac8f2f4534b7-2010-04-16_la_te_xi_t_1_22.jpg

How big are the Super Star Destroyer (SSD) thrusters according to Jeff’s diagram? From my measurements, they are 0.42 km wide. Let me fix the diagram so that it has a thruster that is 2.48 km wide.

i-81e0c8b9a69be86db05a19f2ae99db09-2010-04-16_untitled_14.jpg

Go ahead. I am sure you could think of some valid reasons why this is wrong. I agree with you. I drew they thrusters as being 20 times wider. Maybe they are only 15 times wider. Still big.

Pre-emptive comments:

I am not saying don’t comment. But here is my prediction for comments.

  • Leave George Lucas alone!
  • Dude, get a life. Why don’t you go outside or get a date or something
  • Are you serious? Don’t you know this is a movie? It is not even science fiction, it is science fantasy
  • I have just placed a large bounty on your head
  • Bounty hunters? We don’t need this scum
  • Well, actually according to the Star Wars Universe, the Super Star Destroyer has super thrusters. This is why it is called super. They stole this technology from the Ewoks
  • Speaking of Ewoks, the battle on Endor was staged. There is no way rocks could penetrate storm trooper armor. The Emperor set it up.
  • I have checked your figures and you are way off. I get an alpha of 0.13342 instead of your value of 0.108

Comments

  1. #1 Chris Granade
    April 16, 2010

    I love these kinds of posts. I especially love how it demonstrates that simply naming an unknown quantity is often sufficient to allow one to remove it from a problem. That’s something I often struggle to demonstrate to people who aren’t accustomed to thinking algebraically.

    Anyway, I would like to pick slightly at one of the assumptions: that big spaceships and small spaceships have the same acceleration. It seems to be a quite common trope that big ships have lower acceleration, but higher top speed, which would help somewhat, though still not enough to get from 20× wider to something more reasonable, as you don’t want big ships having 1/400th the thrust. I know, it’s a simplifying assumption made for a Fermi problem, but what can I say… I tend to nitpick.

  2. #2 D. Vader (esq.)
    April 16, 2010

    Don’t underestimate the power of the Dark Side.

  3. #3 Alan B
    April 16, 2010

    Think about it like this: the ship with the “correct” thruster size is actually the larger one, and the smaller ship has thrusters that are much LARGER than they need to be. This affords the smaller ship a much higher maximum acceleration, which is used only when necessary; at other times, all ships’ accelerations match.

    This explanation violates your acceleration assumption, but in a way that sort of makes sense for ships in a fleet.

    Now, write a post about the effect of ship size on performance of FTL drives…

  4. #4 Thomas
    April 16, 2010

    Your assumption that acceleration is proportional to the area of the thrusters, but this may not be the case. Maybe using the technology of Star Wars the limiting factor is the mass of the internal engine and amount of fuel they can carry while the size of the thrusters have a minor impact. You can add more thrust/area with a minor loss of efficiency and some more expensive design. Then it makes sense that you add as many thrusters as the geometry of the ships allows.

  5. #5 NoAstronomer
    April 16, 2010

    I find your lack of faith disturbing.

  6. #6 Eric Lund
    April 16, 2010

    Force from the thruster is proportional to the area of the thruster.

    This is technically true, but cross-sectional area is not the only variable in play. These thrusters work by spitting mass in the direction opposite to that in which the spacecraft is accelerated. The larger the spacecraft, the more length you have available to accelerate particles. That buys you a factor of L^1/2 in the nonrelativistic limit, and even more if, as is likely, you are shooting rocket exhaust at relativistic speeds so that your starship will reach the next star system within the lifetime of the crew. The result is that bigger really is better.

  7. #7 Rhett Allain
    April 16, 2010

    @Chris,

    I think in general – you could be correct about the acceleration of big ships. The first problem is that these thrusters seem to be used for constant velocity. Second, the example I picked – those two ships should have similar characteristics if they are in the same “fleet”.

  8. #8 Rhett Allain
    April 16, 2010

    @D.

    I am tired of your sorcerer ways.

  9. #9 Rhett Allain
    April 16, 2010

    @Eric,

    That is a very good point. I didn’t think about that.

  10. #10 L. Skywalker
    April 16, 2010

    Noooooooo!!

  11. #11 IanW
    April 16, 2010

    You keep using that word “Force”. I do not think it means what you think it means! (How about that for mixing genres?)

    You appear to be assuming that these thrusters have the same mechanics on the inside whether they be on the larger or the smaller ship. Worse than that, you’re assuming Earth technology! There was a reason that Lucas started that first movie with “…a galaxy far, far, away..”. That doesn’t mean they have different physical laws from us (as far as we know!), but it does mean they’re probably not using exactly the same technology we are!

  12. #12 Dale Ferguson
    April 16, 2010

    This is the same discussion as the reason why very large animals (super-elephants, for instance) can’t exist, because their mass and weight scale as the cube of the dimension, while the strength of bones scales only as the square of the dimension. Your arguments are persuasive. Thanks for clarifying things.

  13. #13 Warren
    April 16, 2010

    One big problem: There are many, many different star/space-ship/craft discussed here from many different genres and shows and movies and series and TV networks, and their dates of manufacture span a good 20-30 years of technological improvement, and I am sure that during that time better thrust technologies have developed throughout the various planetary and galaxiary (?) areas. I mean, 30 years is a lifetime in technological terms, as the technologies develop and advance … oh … I just saw the problem with my argument. Nevermind. I’ll go home now.

  14. #14 HP
    April 16, 2010

    I’m a complete dilletante at this stuff, so I’m likely to say something stupid, but here goes.

    What if you think about it in terms of velocity rather than acceleration? Let’s suppose these are pulsed thrusters. The small ship, starting at velocity X, fires a single pulse and accelerates to velocity Y (can you tell I’m not a physicist yet?).

    If the 10r ship had the same size thruster as the smaller ship, starting at velocity X, how many pulses would it take to reach velocity Y? If it’s linear (and I’d be surprised if it were), the difference between one pulse and ten pulses might not be that significant in terms of the vessel’s performance. After all, an aircraft carrier takes much longer to reach a given velocity than a frigate, but the Navy seems to take it in stride.

    (This isn’t some kind of maths challenge. I’m honestly curious, and have no idea how to solve the problem.)

  15. #15 HP
    April 16, 2010

    Or, to put it another way, we’ve got two ships, R and 10R. They have identically sized thrusters. These are pulsed thrusters. R fires its thruster 10 times a second. 10R fires its thruster 1000 times a second. If they both leave Aldeberan for Gliese at the same time, who will get there first?

  16. #16 Laurence Topliffe
    April 17, 2010

    Possible and perhaps only solution it that new technology allows to alter the mass of the ship, possibly using knowledge about dark energy to do a phase shift of the mass itself so it becomes weightless or almost. Surround the ship with a field of coherent energy that separates it from the forces that govern the relative field of existence. Of course the crew could all be yogis. What does that have to do with it? Do a search for “yogic flying.”

  17. #17 XavierL
    April 17, 2010

    Your article has created a disturbance in the force.

  18. #18 Ian Kemmish
    April 18, 2010

    Kudos to comment #2! However, I’ll content myself with pointing out that your third assumption, of similar acceleration, may be unjustified.

    Even we earthlings need a lot more ocean to turn an aircraft carrier around than a frigate. Otherwise, a battle group wold consists of ten ships of the line instead of one big ‘un and lots of little ‘uns. Horses for courses – not all ships are called upon to perform the same functions.

    Or again, early Spitfires used one Merlin engine, and Lancaster bombers (almost relevant, since George is said to have based his battle scenes on the movie “The Battle of Britain”) had four of them. The Spitfire was either the most or the second most nimble aircraft on Earth; a fully loaded Lancaster could barely get off the ground (although the noise is enough to put the fear of god into anyone along its route).

  19. #19 IanW
    April 18, 2010

    “….a fully loaded Lancaster could barely get off the ground…”

    Especially when it was carrying a Gland Slam bomb:
    http://en.wikipedia.org/wiki/Lancaster_bomber

  20. #20 ilan
    April 18, 2010

    my first blush at Prof. Rhett
    I loved the “reductive” thinking, and enjoyed the post

  21. #21 DrO
    April 18, 2010

    ok, ok! its got big pipes.
    How much bigger will the fuel tanks be?

    also How bout ram jets?

    Nice Stuff
    thanks

  22. #22 Nova lys
    April 19, 2010

    We might wonder whether the above calculations do not violate the laws of the BLUE FLOWER

  23. #23 Kyle
    April 19, 2010

    I have 8 problems with your post!

    1. Leave George Lucas alone!
    2. Dude, get a life. Why don’t you go outside or get a date or something
    3. Are you serious? Don’t you know this is a movie? It is not even science fiction, it is science fantasy
    4. I have just placed a large bounty on your head
    5. Bounty hunters? We don’t need this scum
    6. Well, actually according to the Star Wars Universe, the Super Star Destroyer has super thrusters. This is why it is called super. They stole this technology from the Ewoks
    7. Speaking of Ewoks, the battle on Endor was staged. There is no way rocks could penetrate storm trooper armor. The Emperor set it up.
    8. I have checked your figures and you are way off. I get an alpha of 0.13342 instead of your value of 0.108

  24. #24 John
    April 19, 2010

    First I’ll say that I have no problem suspending my understanding of physics to enjoy a great sci-fi/fantasy movie. But, since we’re on the nit-picking Ewoks, here’s my big gripe/question about what’s missed by the movie and the above rationalization. All of the apparent or at least the Wookie’s share of the thrust goes to pushing the ships in one direction, with very little emphasis placed on either thrust vectoring/side thrusters for turning, and most importantly for slowing down and stopping. Wouldn’t a ship require similarly sized thrust capability in the reverse direction to slow down (in a timely fashion)?

    Another thought to ponder with respect to density. I suspect (postulate) that density for ships would vary greatly within a fleet, but more as a result of their mission or speed they wanted to go, rather than their size. Akin to naval warships vs. recreational/cruise ships, whose hulls I think, are quite different in thickness, numbers of bulkheads, thickness of inner walls etc. With a starship that you are looking to accelerate to absurd velocities, wouldn’t you want this same level of protection (energy/power shielding aside) as a last resort option, so that proverbial grain of sand or even small meteoroid traveling just as fast in the opposite direction whacks into your ship. I can’t begin to imagine how thick of a hull this would require. Granted some sort of ‘self-healing’ advanced ceramics active armor system with a multi-ply layering of various metals might be an early options (cooled to a few thousandths of a degree above Kelvin of course).

    Thus, the engineering question that needs to be answered first and foremost is how much dust, sand and very small rocks would be present and likely collided with during a trip through stellar and interstellar space. This combined with the answer of the speed at which you would like to travel and the required hull thickness/overall density of the craft, and whether or not you want to be able to slow down will all have a direct if not proportional effect on the number and or size of thrusters.

  25. #25 John
    April 19, 2010

    What if in the place of thrusters quantum state generators were installed? These might put mass of the whole ship into and a quantum superposition and then ‘tipped’ to one position or the other, a lot of times, really fast. When slowing down was desired, just cycle the superposition and the negative direction. This would in effect make the ship appear to be in two places as once, but always pushing it into one of the other states, thus moving the ship forward. Increasing the frequency of position change should have a direct and immediate correlation with achieved velocity.

  26. #26 Prof. Abrams
    April 20, 2010

    I have checked your figures and you are way off. I get an alpha of 0.13342 instead of your value of 0.108

  27. #27 grant
    April 20, 2010

    Do these calculations hold true for similar scale differences between Earth-motorboats and Earth-cruise ships?

    If not, why not?

  28. #28 MSM
    April 20, 2010

    Apart from scaling the mechanics of zipping around in these big ships, I’ve always thought the scaling of the thermodynamics seems to be a glaring oversight (pun intended).

    How are these things dumping their waste heat? The waste heat from internal systems alone ought to make for some large glowing radiators. Any active technology manipulating the thruster mass and energy transfers has got to have some waste, and the scale of thrusting energies implies lots and lots of waste heat to dump. (Chemical rockets dump heat in their plumes, but the rocket equation says one must take along a lot of mass, giving an upper limit. Ion propulsion handles lots of power, as I would expect nucleon force analogs to do as well…)

    Can you really make something much larger than an order of magnitude or so larger than the 2001 Space Odyssey ship?

  29. #29 J.J.B.
    April 20, 2010

    Meesa tink you-sa tinkin ’bout thrusters good!

  30. #30 Rhett Allain
    April 20, 2010

    @MSN,

    I like the thermo idea. This also would be a problem since cooling via radiation is proportional to the surface area and I would assume thermal energy is proportional to the volume.

    Same thing happens with planets and muffins. The smaller they are, the greater the surface area to volume ratio and so they cool off faster. That is why mini-muffins are so awesome.

  31. #31 Rhett Allain
    April 20, 2010

    @grant,

    I would assume there would be something similar. The big difference is a boat uses the thrust to move at a constant speed because of the resistance from the water – but still, larger boats have a much larger weight.

  32. #32 Max Udargo
    April 20, 2010

    For crying out loud, have you ever even watched Star Wars?

    It takes place in a distant galaxy that is very different from our own. For one thing, there’s air in space. We know this because we can hear sounds and see fire in space. Also, small spaceships move like airplanes and Han Solo only needs a breathing mask when he steps outside of his spaceship to kill space-bats.

    So the atmosphere in space isn’t breathable, but it’s dense enough that you don’t need a pressure suit and it must contain some oxygen because you can light things on fire.

    Obviously, this fact changes everything. It’s why you need to maintain constant thrust to keep moving forward against the resistance of the space-air.

    And the Super Star Destroyer doesn’t need bigger engines because it has all that lift!

    I don’t see that variable represented in any of your fancy equations, college boy.

  33. #33 Gustavo Carreno
    April 20, 2010

    If we are talking imaginary universes with the same physics let’s have a look at this one:
    I once read a Disney cartoon where Goofy was Leonardo da Vinci. After loads of going around he pushed a skinny person and a fat person from the Tower of Pisa and they both crashed the floor at the same time.
    Every body knows that, in a vacuum, a feather and a piece of lead will fall at the same time. So, without resistance, and applying the same force(gravity) it doesn’t really matter what the mass is, you will get the same acceleration.

    So why the feck would you need larger thrusters on larger ships… Meeza sain’ makes no sense that!!

  34. #34 John
    April 20, 2010

    @Gustavo

    Sure, the posts are going along all civil like and wouldn’t you know it, someone would have to quote one of George’s worst creations/cretins, old Binx-ee boy.

  35. #35 rob
    April 20, 2010

    i felt a great disturbance in the scienceblog, as if 35 commenters suddenly cried out in terror and were suddenly silenced. i fear something terrible has happened.

  36. #36 JTDC
    April 20, 2010

    What you really established was that their sub-light acceleration is substantially less than an IDS’s, which few would argue. Furthermore, it is entirely possible that a larger-sized ship will use a different propulsion system, one that might require more internal space to achieve efficiency required for practical use. If you’ll recall, larger ships used larger cannons because smaller ships literally could NOT use larger cannons because they could not handle the recoil. So there are alternate explanations to this conundrum.

    @Max: Sound is added by internal cockpits to give another means of determining location and direction of opposing starships. As for the interior of the asteroid, it’s not hard to imagine that they had basic sensors that were able to determine relative pressure on the exterior before leaving. It wasn’t the interior of an asteroid, but the interior of a creature. Also, the only reason Han stepped outside was that there were creatures on the exterior of his hull–which meant that he already had reason to believe that there was plenty of pressure there.

    If you’re going to nit-pick, please be good at it.

  37. #37 John
    April 20, 2010

    My apologies to Gustavo, JJB beat you to the Binx quotable.

    @JTDC
    ha ha…TOUCHE!

  38. #38 Jon
    April 20, 2010

    This is all assuming the thrusters are burning at a constant relative power between the various craft, correct? Has anyone hypothesized that the thrusters, while powered as evidenced by a constant glow, might be merely idling instead of creating a constant thrust? Or simply providing enough thrust to counteract the resistance created by particulates (interplanetary dust, microscopic particles and radiation, etc…) but not the primary acceleration of the craft?

    I shouldn’t read posts such as this after drinking.

  39. #39 Ryland
    April 20, 2010

    Sorry, but while your math makes sense, Star Wars is a terrible choice to illustrate your point. Space craft in Star Wars are powered by magic, not physics.

    Star Destroyers, and in fact almost all Star Wars ships, move like submarines, not space ships; they accelerate and decelerate facing the same direction, they go from zero to top speed in a very short time; they come to a full stop, again still facing forward. They have to have some kind of gravity/inertia manipulation to keep the ship from collapsing like an accordion during acceleration/deceleration.

    A real space ship (as anyone who’s read a Heinlein novel knows) travels by aiming precisely in the direction you want to go, and then firing the engine for a certain length of time at a certain power level to accelerate, and then flipping around in the exact opposite direction and firing the engine for the exact same length of time and power as your acceleration burn to decelerate.

  40. #40 Max Udargo
    April 21, 2010

    @JTDC

    Nitpick? Since when is space-air nitpicking? Space-air changes everything!

    And, I’m sorry, but most of the sounds we hear (not to mention the flames we see, which you failed to address) are heard from the outside, not inside a cockpit. Space-air is the only explanation.

    I know, you’re saying, “Okay, if there’s all that air in space, how does a big, aerodynamically challenged object like the Death Star move at high speeds from system to system?”

    One word: buoyancy.

    And people say physics is hard.

  41. #41 TinFoilEnthusiast
    October 11, 2010

    Obviously you haven’t been abducted by aliens yet.

  42. #42 Robbie
    January 2, 2011

    Yeah, I have pretty much stopped watching Star Wars. The slow as a bullet laser pulses and the spacecraft behaving like airplanes, and the warships trading broadsides like 18th century sailing ships, really makes me cringe.

    I had a nerdgasm when I watched the opening sequence of Killzone 2 on the PS3 for the first time. The good guy’s spacecraft flip over backwards to point their engines in the other direction, and then fire missiles over a distance of thousands of kilometers at the enemy spacecraft. A tear rolled down my cheek, finally some one got it right!

  43. #43 Robbie
    January 2, 2011

    I forgot to add that it took 2 weeks for the good guys to arrive at the enemy planet, the good guys being located in the same star system as the bad guys.

  44. #44 Robbie
    January 2, 2011

    But I think we’re forgetting an important fact here, the Star Wars civilization is over 25,000 years old, meaning their engines are tremendously advanced over our pathetically weak thrusters that can barely lift a few tons into low orbit. We have a long way to go my friends, before we can say whether or not the things depicted in Star Wars is truly impossible.