## Friday, November 7, 2014

### Relativistic Rocket Calculator

I have written a calculator for relativistic acceleration to make it easier to understand how relativity affects acceleration problems.

### Accelerating Rockets

Most people get exposed to equations like v = at and d½at² which let you compute velocity and distance for a rocket that accelerates at a fixed acceleration a for a time t. With a little algebra, you can play with those equations and compute other things, such as time given distance or velocity.

### Special Relativity

Most people also know that strange things happen when objects approach the speed of light: time slows down, lengths contract, and mass increases. The Lorentz equations that describe this are less-well known, but still quite accessible to anyone with an understanding of high-school algebra.

### Relativistic Acceleration

Things get complicated when you combine the two. Now instead of time, you have ship-time and earth-time, and as the ship goes faster, its time slows down, which reduces the acceleration, which affects the velocity . . .and you have this circular mess.

If you understand how to add relativistic velocities, then with a little work you can show that the velocity v = c tanh at/c, which fixes the circular definition, and you can then use calculus to compute the other numbers. Or you can use someone else's results.

But if you just want to explore different possibilities, you can use this calculator page.

### Using the Calculator

The relativistic acceleration calculator assumes that the ship accelerates at a fixed rate up to some peak cruising velocity and that it later decelerates at that same rate until it reaches the destination. That is, the ship is under power for the first and last parts of the trip and it cruises at constant velocity in the middle.

The calculator needs to know three things:
1. The acceleration. Default is 1g
2. The powered phase. Either how much time it takes (Earth or ship time) or how much distance it covers.
3. The cruise phase. Likewise, you need to specify either how long it lasts (default is zero) or how much distance is covers.
Answer any combination of these and the calculator will fill in all the others for you.

#### Example 1

A starship goes from Earth to Alpha Centauri, 4.366 light-years away. It accelerates at 1g until it's half-way there, then it decelerates a 1g until it arrives. How long does it take on Earth? How long does it take the crew? What's the max velocity?

To figure this out, open the calculator. Acceleration is already set to 1g, so leave that alone. Set "How far does it travel under power" to 4.366 light-years. Leave cruise time at zero.

The two "how long is the whole trip" numbers are 6 earth-years and 3.6 ship-years, and the max velocity is 95% the speed of light.

#### Example 2

A starship to Tau Ceti, 11.905 light-years away, spends most of the voyage coasting at 90% the speed of light. At 1 g, how long does it have to accelerate at the start and decelerate at the end? How long is the whole trip? How far is it from Tau Ceti when it starts to decelerate?

Again, leave acceleration at 1g. In the Powered Phase, Set "what velocity does it reach" to 0.9 c. In the Cruise Phase set "what is the total distance traveled" to 11.905 light-years.

Look at "how long does it accelerate?" in the Powered section. It's 3 years Earth-time and 1.8 years ship time.  Then look at "How long is the whole trip?" in the Cruise section. It's 14.4 earth-years and 7.4 ship-years. Finally, "how far does it travel under boost alone" tells you that it will be 1.25 light-years from Tau Ceti when it starts to decelerate.

1. This is great. Here's an example of what can be done with it.

On the evening of 2040/09/09 at 00:00:00 UTC (2040/09/08 at 8 PM local time), a spaceship takes off from Cape Canaveral accelerating at 1 g.

Encounter Time Distance Vis. EDT ETA UTC

Moon 0.1036 d 392.8 Mm Visible 09/09-02:29
Mercury 2.216 d 179.7 Gm Visible 09/11-05:11
Venus 2.429 d 216.0 Gm Visible 09/11-10:18
Mars 3.130 d 358.6 Gm Visible 09/12-03:07

Jupiter 5.119 d 959.3 Gm Visible 09/14-02:51
Saturn 6.571 d 1.581 Tm Visible 09/15-13:42
Uranus 8.858 d 2.872 Tm R 3:57 AM 09/17-20:36
Neptune 10.891 d 4.341 Tm R 9:56 PM 09/19-21:23

Rigil 2.317 a 4.37 ly V 4:52 PM 2043/01/03
Altair 3.486 a 16.73 ly Visible 2044/03/05
Vega 3.859 a 25.04 ly Visible 2044/07/19
Arcturus 4.218 a 36.7 ly Visible 2044/11/27

Spica 6.056 a 250 ly Visible 2046/09/29
Acrux 6.295 a 320 ly NV 2:39 PM 2046/12/26
Hadar 6.486 a 390 ly V 4:16 PM 2047/03/06
Antares 6.818 a 550 ly Visible 2047/07/05

M6 (Butterfly) 7.852 a 1.6 kly Visible 2048/07/16
M8 (Lagoon) 8.763 a 4.1 kly Visible 2049/06/14
M4 (Scorpio) 9.309 a 7.2 kly Visible 2049/12/31
M2 (Aquarius) 10.784 a 33 kly Visible 2051/06/22

M31 (Andromeda) 14.993 a 2.54 Mly Visible 2055/09/07
M81 (Bode's) 16.481 a 11.8 Mly Visible 2057/03/03
M101 (Pinwheel) 17.035 a 20.9 Mly Visible 2057/09/22
M51 (Whirlpool) 17.128 a 23 Mly Visible 2057/10/26

Virgo (M87) 17.946 a 53.5 Mly Visible 2058/08/20
Coma (NGC 4889) 19.642 a 308 Mly Visible 2060/05/01
Hercules (IC 1182) 20.128 a 509 Mly Visible 2060/10/25
Corona (PGC54846) 20.747 a 964 Mly Visible 2061/06/08

3C 273 21.648 a 2.0 Gly Visible 2062/05/03
3C 48 22.102 a 3.9 Gly R 8:26 PM 2062/10/16
3C 47 22.196 a 4.3 Gly R 8:58 PM 2062/11/20
3C 147 22.362 a 5.1 Gly R 11:17 PM 2063/01/19

3C 9 23.014 a 10.0 Gly Visible 2063/09/14

Edge of Universe 23.326 a 13.8 Gly

However, these calculations assumed that all the places visited are on a straight line from the place of launch. This is obviously not the case. How can I take into account a spaceship turning while moving?

2. Or if you want to go in another direction:

On the evening of 2040/09/09 at 00:00:00 UTC (2040/09/08 at 8 PM local time), a spaceship takes off from Cape Canaveral accelerating at 1 g.

Encounter Time Distance Vis. EDT ETA UTC

Moon 0.1036 d 392.8 Mm Visible 09/09-02:29
Mercury 2.216 d 179.7 Gm Visible 09/11-05:11
Venus 2.429 d 216.0 Gm Visible 09/11-10:18
Mars 3.130 d 358.6 Gm Visible 09/12-03:07

Jupiter 5.119 d 959.3 Gm Visible 09/14-02:51
Saturn 6.571 d 1.581 Tm Visible 09/15-13:42
Uranus 8.858 d 2.872 Tm R 3:57 AM 09/17-20:36
Neptune 10.891 d 4.341 Tm R 9:56 PM 09/19-21:23

Sirius 2.888 a 8.60 ly R 3:30 AM 2043/07/30
Procyon 3.143 a 11.46 ly R 3:35 AM 2043/10/31
Capella 4.366 a 42.9 ly R 11:06 PM 2045/01/20
Aldebaran 4.764 a 65.3 ly R 12:07 PM 2045/06/15

Achernar 5.490 a 139 ly R 1:27 AM 2046/03/07
Canopus 6.264 a 310 ly R 5:29 AM 2046/12/14
Pleiades 6.611 a 444 ly R 10:56 PM 2047/04/20
Betelgeuse 6.969 a 643 ly R 1:47 AM 2047/08/29

Rigel 7.251 a 860 ly R 1:40 AM 2047/12/10
M42 (Orion) 7.683 a 1.344 kly R 1:55 AM 2048/05/16
C106 (47 Tucanae) 10.12 a 16.7 kly Not visible 2050/10/24
C80 (Omega Centauri) 10.07 a 15.8 kly S 7:16 PM 2050/10/05

LMC 12.30 a 158 kly Not visible 2052/12/28
SMC 12.52 a 199 kly Not visible 2053/03/19
M83 (Southern Pinwheel) 16.73 a 15.21 Mly S 8:38 PM 2057/06/01
C77 (Centaurus A) 16.53 a 12.4 Mly S 7:38 PM 2057/03/21

Fornax (NGC 1399) 17.42 a 62 Mly R 1:15 AM 2058/02/08
Centaurus (NGC 4696) 19.07 a 170 Mly S 7:09 PM 2059/10/03
Shapely (NGC 5124) 20.37 a 650 Mly S 8:24 PM 2061/01/20
Horologium (IC 1933) 20.73 a 950 Mly R 2:31 AM 2061/06/03

3C 273 21.648 a 2.0 Gly Visible 2062/05/03
3C 48 22.102 a 3.9 Gly R 8:26 PM 2062/10/16
3C 47 22.196 4.3 Gly R 8:58 PM 2062/11/20
3C 147 22.362 5.1 Gly R 11:17 PM 2063/01/19

3C 9 23.014 10.0 Gly Visible 2063/09/14

Edge of Universe 23.326 a 13.8 Gly Visible 2064/01/06

3. I want a calculator that will tell me what relativistic acceleration and velocity a rocket will get if it ejects 1 mg. per second at .38 C and weighs 15 metric tons.