James Pailly's Question and my answer

On the Sci-Fi Ideas site

http://www.scifiideas.com/science-2/sciency-words-orbital-vocabulary/

James Pailly asked this question:

"I for one would love to hear more about how to get to Alpha Centauri. I have a somewhat abstract understanding of how transfer orbits work within the Solar System, but I'm kinda clueless about the mechanics of travel between stars."

It came about due to a discussion about Navigation in Space and how to calculate a course to another star.

To answer this I decided to use my Wiki as it allows me to include graphics and pictures. As well as links and animations.

This is going to be a long answer in several parts.

Calculating course for a hypothetical real space ship with current or near future technology (and completely grounded  in physics ) [1]
For this exercise let us calculate a trip to Proxima Centauri



For that we need some data.

First: Distance  Earth / Proxima
This is the distance from earth to your destination. Distance is approximate because the  positions change continuosly relative to the earth. observer time elapsed and the traveler's maximum velocity is calculated using this equation:[2]
 * 39734219300000000  meters

{cvt} over {c + sqrt(c^2-v^2)}

where

c = the speed of light,

v = maximum velocity,

t = time elapsed in observer timeframe.

Then we need Acceleration
This is the constant acceleration of the traveler's spacecraft. Half way through the journey, the spacecraft starts decelerating at the same rate.

it will be calculated using Newton's laws of motion

a = s over {t^2/4}

a = v^2 over s

a=2v over t

where

s = distance,

v = maximum velocity and

t = time elapsed in observer timeframe

This is increasingly inaccurate as you approach the speed of light, so for large distances,

If a spacecraft accelerates constantly at 1g --or 9.8m/s-- the travelers on board can experience earth-like gravity. Unfortunately interstellar travel at this acceleration will likely never be achieved because of the huge amount of energy required. It is not possible to travel to the nearest stars at this acceleration if the fuel must be carried onboard the spacecraft, no matter what kind of fuel is used.

Now comes Maximum velocity
This is the maximum velocity the spacecraft will reach, from the perspective of an observer on earth. This occurs when the spacecraft is half way to its destination. This is calculated using this equation:

{c} over {sqrt {1+{ {c^2} over {a^2{({T} over {2})^2}}}} }

where

c = speed of light,

a = acceleration and

t = time elapsed to end of journey in observer timeframe.

Observer time elapsed during journey
I know the theory of relativity is not entirely proven, but to keep this on the Hard Science ground. This is also a factor you need to consider:

This is the time elapsed for the whole journey from the observer on earth's time frame. This is calculated using this equation:

t = 2 sqrt{ {c^2 over a^2} left [ left (d over {2{c^2/a}}+1 right )^2 - 1 right ] }

where

c = speed of light,

d = distance of the journey and

a = acceleration.

Traveler time elapsed during journey
This is the time elapsed for the whole journey from the perspective of the spacecraft. This is calculated using this equation:

t = {{2c} over {a}} cosh^-1 left [ {d} over {2 {c^2/a} } + 1 right ]

where

c = speed of light,

d = distance of the journey and

a = acceleration.

Spacecraft mass at launch
This is the total mass of the spacecraft and its contents including fuel. For example  2,000,000kg is the mass of the space shuttle when it takes off, including all its fuel.

Also note that if the fuel mass is calculated to be more than the mass of your spacecraft, then your trip cannot be done unless you extract fuel from space. If your fuel mass is more than half the mass of your spacecraft, you're probably on a one way trip, so take enough food, books and episodes of Star Trek to last the rest of your life.

Energy
This is the amount of energy your spacecraft will need to constantly accelerate to half way to your destination and then decelerate at the same rate until you reach your destination. This is calculated using this equation:

e = 2mc^2 left [ 1 over sqrt(1-v^2/c^2) -1 right ]

where

c = speed of light,

v = maximum velocity and

m = spacecraft mass.

Fuel conversion rate
The fuel conversion rate is the the efficiency with which your spacecraft's fuel is converted into energy. At today's fuel conversion rates there is no prospect of sending a spacecraft to another star in a reasonable period of time. Advances in technologies such as nuclear fusion are needed first.

The default fuel conversion rate of 0.008 is for hydrogen into helium fusion. David Oesper explains that this rate assumes 100% of the fuel goes into propelling the spacecraft, but there will be energy losses which will require a greater amount of fuel than this. This is calculated using this equation:

r=e/mc^2

e = energy,

m = fuel mass and

c = speed of light.

And this is only the first part, now we go to the actual navigation bit
I never promised it would be an easy to understand explanation, that's why they call it Rocket science. Or more preciselly Astrophysics.

The Problems
One of the biggest challenges in space travel is the fact that, after people have traveled, they presumably want to come back again. Both Russia and the United States briefly entertained visions of winning the space race by dropping astronauts off on the Moon and having them bide their time there until their respective homelands could find a way to get them back. Similarly, some people have floated plans to colonize the Moon and Mars by shipping people one way. At least, though, the Moon and Mars are often in visual range of Earth. If people need to get back they can at least point the way. There's no such luck for eventual travelers in deep space. When the universe is spread out in three dimensions around us, and each part looks just about like every other part, how will future starships find their way?

The Milky Way galaxy is one hundred thousand light years across and a thousand light years thick. It has two hundred billion stars, some within its plane, and some around it. Outside the Milky Way are another, roughly, 8500 galaxies observable from Earth, but in a single image from the Hubble Ultra Deep Field , astronomers think that there are another 10,000. This adds up to hundreds of billions of galaxies in the Universe, meaning that it's hard to find consistent observable landmarks. In space, no one can remember where they parked.

Even if a space ship managed to leave a trail of breadcrumb galaxies on its way to wherever it was going, it's not necessarily going to get home again. Breadcrumbs don't move. Galaxies do. And so does home. Space agencies are used to moving targets, but as starships venture out farther, they have to navigate based on more and more moving points. One unexpected shift in gravity affecting a galaxy that was once a marker, and no one gets home again.

And while some people pin their hopes on wormholes, they'd have to be pretty specific. Put a spaceship down even slightly off course and the entire lay of the universe looks different. The parallax effect, the fact that objects in the foreground will shift compared to their background when looked at from a different position — similar to the way near objects jump back and forth when you look at them through first one eye and then the other — would make billions of stars shift in relations to billions of other stars.

The Deep Space Network
Since we already have a lot of stuff whizzing around space, there are existing navigational and control systems. Currently there is an International Deep Space Network, with three massive antennas placed on three different places on Earth, each roughly one hundred and twenty degrees from each other, checking position on various space craft. The antennas are in the Mojave desert, in the United States, just outside of Madrid in Spain , and outside of Canberra , Australia.

There are European, Indian, and Chinese Deep Space Networks as well, and they all take advantage of one of the few easy things about space: it's easy to make signals omnidirectional. Three stations on Earth are all that you need — get thirty thousand kilometers away from Earth, and you're always in view of an antenna. Place an antenna in space, and let it send out radio signals in all directions, and you've got a beacon that shines everywhere.

Of course, as explorers get farther and farther out they'd need a longer and longer chain of beacons sending out signals that can lead them home. And assuming that each of these beacons is dependent on signals from the last to keep from straying off course, then if there's even one break in the chain, the entire system could go down. If one antenna on Earth went down, we might lose one third of the starships out there.

Pulsars as Natural Signals
Pulsars are stars that have collapsed in on themselves in a specific way, which could turn out to be very handy.

The Earth's electromagnetic field is oriented from pole to pole. The Earth also spins around an axis that goes from pole to pole. Although the two are not exactly lined up, the rotation of the Earth doesn't involve a dramatic rotation of the Earth's electromagnetic field. It's like spinning a cylindrical bar magnet around its central axis. As it spun, any magnets around it wouldn't feel a huge change in the magnetic pull on them.

Pulsars, however, are stars whose magnetic poles and axes of rotation don't match up at all. They are more like a cylindrical bar magnet being twirled like a baton. Any other magnets around a pulsar would feel a strong variation in its pull as it was spun around. This causes pulsars to emit strong, regular beams of radiation.

Some pulsars measure their rotation in milliseconds, and with the accuracy of atomic clocks. If a pulsar's pulse sweeps past Earth at a specific time, and then sweeps past a space craft at another, it's possible to determine where the two are in relation to each other. This is a plan that has been proposed as a back-up system for space craft going to Mars, but there are a lot of pulsars out there, and their exact periods could help interstellar travelers figure out their position compared to other known natural objects. It's true that eventually even the best pulsars will wind down, but they're far more shock-proof than the average space craft carrying a human made beacon.

Now comes part III - How to navigate to Proxima
Just  make sure in case John Reither reads this far, no current technology exist to actually achieve the necessary data or variables to physically plot a course to Proxima.

The problem of interstellar navigation is tackled in the Project Ikarus study. Headed by the Tau Zero Foundation and British Interplanetary Society, a non-profit group of scientists dedicated to interstellar spaceflight, Icarus is working to develop a spacecraft that can travel to a nearby star.

Here is the scene: "Helm, come to heading of 250 Mark 3.5," says Captain James T. Kirk, and the USS Enterprise warps-off into the distance on another Star Trek adventure.

Now, what "250 Mark 3.5" actually means is far from certain, but if this fictional future is to be believed, navigating around the Galaxy is simple!

Tripple Triangulation
Sounds like a mouthful. But it postulates measuring the pulse frequency of 3 pulsars in refrence to the SOURCE (Earth in this case)

and also a "virtual clock" that is projected to the target. It is a mathematical projection.

Before the journey starts there are three clocks (adjusted for time diletation)  kept in the ships nav computer.

Clock 1) - is calibrated to the Pulsar frequence measured in regards to Earth

Clock 2)- is the Moving clock (ship)  measuring its spatial position in relation to the three  pulsars (triangulation) and it mathematically predicts where Earth is with a second triangulation calculation.

Clock 3 -Is mathematically projected and virtually at the target - and a third triangulation is made.

The results of all three triangulations provide the course plot (all three obects move but the positions of Source and Destination are known and can be predicted . So the third triangulation results in the spatial position in relation to 1 and 2. And thus the navigator aboard the ship knows where he is and where he needs  to go and correct its course.

This method requires no communication between the objects.

[1] Hard Science -- Hard SF

[2] Most Direct Derivation of Relativistic Constant Acceleration Distance Formula