Taking my girlfriend to Alpha Centauri on the Millennium Falcon 2

[Gian]

I have a spaceship the Millennium Falcon 2. She can travel upto 99%c, and I want to take my girlfriend Joanne to Alpha Centauri which is about 4LY away.

Obviously we can't start off at 99%c, so we will be accelerating at 1G from 0 and when we get to 99%c we will switch off the engines and cruise at a constant speed until we need to turn the ship around and fire the nuclear fusion engines to slow down the ship at a comfortable deceleration rate of 1G.

What maths do I need to calculate our total journey time ship's time taking into account relativistic effects?

I think the factor tau has something to do with it but I'm not sure how exactly

Joanne wants to know how much luggage she needs to take.

And how do I work out how long it will take for us Earth time to get home for our parents back on Earth?

Any ideas?

Cheerz

GIAN😊XXX

[Genady]

While the parts with constant relative speed require just algebra, the acceleration and deceleration are more involved. E.g.,

[Janus]

Here are some equations that may help

Where:

t =Earth time

T= ship time

d= distance

v = velocity

a= acceleration

For brevity:

ch = hyperbolic cosine

sh = hyperbolic sine

th = hyperbolic tangent

 

t=(c/a) sh(aT/c) = sqrt[(d/c)² + 2d/a

d=(c²/a) [ch(aT/c)-1] = (c²/a) (sqrt[1+(at/c)²]-1)

v= c th(aT/c) = at/sqrt[1+(at/c)²]

T= (c/a) sh^-1(at/c) = (c/a) ch^-1[ad/c² +1]

[Halc]

I'm afraid that you cannot get to that speed (relative to Earth) in only half the distance to your destination (at least not at 1G). Perhaps a target further away like Tau Ceti, which is almost exactly how far you'd get if you got to .99c and immediately started slowing.

You'd be back home in 27 years 2.3 months.  To Alpha Centauri and back you'd be gone only 12 Earth years, but would only reach 0.95c

Tell Joanne to skip most of the luggage and have laundry facilities onboard.

[swansont]

7 hours ago, Halc said:

I'm afraid that you cannot get to that speed (relative to Earth) in only half the distance to your destination (at least not at 1G).

1g acceleration gets you to 0.99c in a little less than a year (earth frame)

https://rechneronline.de/g-acceleration/

[Halc]

1 hour ago, swansont said:

1g acceleration gets you to 0.99c in a little less than a year (earth frame)

That's coordinate acceleration, and 1G of that for almost a year would kill poor Joanne, and it could not continue at all for a full year.

I do realize the OP did not specify explicitly, but 'comfortable' was used, so my figure is based on a comfortable 1G proper acceleration, and that takes almost 2.7 years (ship time) to get to that speed and around 6.8 years Earth time.  Fixed proper acceleration can in principle be kept up indefinitely.

 

Oh, and your link is behind a paywall, or at least a subscribe wall. I could not view it.

Edited Friday at 12:59 PM by Halc

[joigus]

10 minutes ago, Halc said:

Oh, and your link is behind a paywall, or at least a subscribe wall. I could not view it.

It seems to be an either cookies, else subscribe wall.

They never guess right what adds I'm interested in, so I always take them with the default settings.

[Halc]

56 minutes ago, joigus said:

It seems to be an either cookies, else subscribe wall.

I won't accept such cookies normally, but I pasted the link in an incognito window and it let me in without challenge.

It seems to be a Newtonian calculator and yields 1.18c if I put in 10k hours at 1G

Really, there are very good calculators for relativistic space travel.  One of the best:

https://gregsspacecalculations.blogspot.com/p/blog-page.html

That one presumes fixed proper acceleration, not coordinate acceleration like a Newtonian calculator would use.

Edited Friday at 02:07 PM by Halc

[Gian]

16 hours ago, Halc said:

I'm afraid that you cannot get to that speed (relative to Earth) in only half the distance to your destination (at least not at 1G). Tell Joanne to skip most of the luggage and have laundry facilities onboard.

Thanks Mr Halc, ah I see never thought of that. Well I think Joanne's got her heart set on alpha centauri; so accelerating by 1G for 2LY and then turning around and decelerating by 1G for the remaining 2LY, how long do you think it would take ship's time to get to centauri?

THANKS TO EVERYONE ELSE I'M WORKING THROUGH YOUR REPLIES. JOANNE'S EXCITED TO GET STARTED

Cheerz

GIAN🙂XXX

 

 

Edited Friday at 09:24 PM by Gian

[Halc]

6 years 1 way Earth time, 3.58 years on the ship. The figures reflect more the actual distance and not just 4 light years exactly.

If it was 4 light years exactly, it would be 3.46 years on the ship.

Edited Saturday at 04:52 AM by Halc

[Gian]

On 10/26/2024 at 5:51 AM, Halc said:

... was 4 light years exactly, it would be 3.46 years on the ship.

Thanks Mr Halc! I'm going to use the math I've been sent to try and work it out for myself (honest!) 

But would there be anything to be gained in travel time ship's time if I first took the Millenium Falcon into orbit around our Sun, accelerating by 1G upto 99%c, then turned towards Alpha Centauri and cruised there in a straight line at a constant speed of 99%c, using the same method to slow down at the other end?

Cheerz

GIAN🙂XXX 

[swansont]

11 minutes ago, Gian said:

Thanks Mr Halc! I'm going to use the math I've been sent to try and work it out for myself (honest!) 

But would there be anything to be gained in travel time ship's time if I first took the Millenium Falcon into orbit around our Sun, accelerating by 1G upto 99%c, then turned towards Alpha Centauri and cruised there in a straight line at a constant speed of 99%c, using the same method to slow down at the other end?

Cheerz

GIAN🙂XXX 

It’s still going to take time to get up to speed, and you should look at what orbital radius you’ll have as you approach c. Hope you like it warm! (i.e. the orbital speed at the surface is a lot smaller than c)

[Halc]

34 minutes ago, Gian said:

But would there be anything to be gained in travel time ship's time if I first took the Millenium Falcon into orbit around our Sun, accelerating by 1G upto 99%c

You can't orbit anything much faster than its escape velocity at 1G, so no, dropping to the sun requires one to lose energy rather than gain it, which is what you want to get to AC.  OK, technically a ship always loses energy as its fuel drains, but I'm talking about the mechanical energy (potential and kinetic) of the payload.  So orbit of anything is likely just a waste of time since one cannot exceed some low speed while going in circles, at least not if acceleration is confined to 1G.

 

Suppose the sun could be condensed into a neutron star or black hole. Presuming Joanne doesn't mind a little (a lot) of tidal stress, one could orbit such a thing at super high speed near c but it would take a lot of time to drop into this orbit. And then it gains you nothing getting to AC since all that kinetic energy must be wasted over a long time just to get back to where Earth is orbiting, all the energy being expended not accelerating to AC, but rather just climbing back out of that gravitational well you were in.  When back at Earth, no net speed remains. The trip is no shorter.

Edited Monday at 02:52 PM by Halc

[Gian]

 

On 10/28/2024 at 2:48 PM, Halc said:

You can't orbit anything much faster than its escape velocity at 1G, so no, dropping to the sun requires one to lose energy rather than gain it, which is what you want to get to AC.  OK, technically a ship always loses energy as its fuel drains, but I'm talking about the mechanical energy (potential and kinetic) of the payload.  So orbit of anything is likely just a waste of time since one cannot exceed some low speed while going in circles, at least not if acceleration is confined to 1G...

Thanks again Mr Halc! I guess what I mean is if I just flew the Millennium Falcon 2 in a big circle by firing the stabiliser jets to keep us on the circular heading, as the engines accelerate the ship by 1G. Then when we're upto cruising speed of 99%c switch the jets off and head out in a straight line. But there would probably be no advantage in travel time. 

Either that or my gf and me would have to bear switching the acceleration to 2G which would be uncomfortable. Otherwise a 3½ year flight it is!
Cheerz

GIAN🙂XXX

Edited 6 hours ago by Gian