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 October 25 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 October 25 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 October 25 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 October 26 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 October 28 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 October 29 by Gian

[Halc]

7 hours ago, Gian said:

if I just flew the Millennium Falcon 2 in a big circle by firing the stabiliser jets to keep us on the circular heading

Acceleration in physics is a vector change in velocity over time (and is not a scalar increase in speed, the dictionary definition). That change can only be in one direction, so if you're using lateral jets and still are maintaining 1G of proper acceleration then you're just finding an inefficient way of accelerating at some angle.

Anyway, if you're getting up to .99c in the solar system frame, your circle at 1G would be at minimum far larger in diameter than the distance to AC, and you'd be using your main engines pointed sideways to maintain the turn radius, with no increase in speed.  This would take at least 7 years to get up to this speed linearly and far more if you maintained a circle centered on something like Earth. It would take decades to do one 'orbit', one lap of this huge circle.  Once you 'cruise' and go straight, you'd be heading away from your destination since at no point in that circle are you heading toward it.

Fastest way to get there at fixed acceleration is straight out and back, but getting up to a speed higher than your ability to stop in time is just wasted effort.

 

7 hours ago, Gian said:

Either that or my gf and me would have to bear switching the acceleration to 2G

It would take at least 3g to get up to .99c and still stop 4 light years away

 

Edited October 30 by Halc

[Gian]

15 hours ago, Halc said:

ok so 3½y to Centauri, well I guess me and the gf will just have to find stuff to do on the way😋💘
Cheerz
GIAN🙂

 

 

[Gian]

16 hours ago, Halc said:

 

 

Hi Mr Halc, one last query

the Falcon2 has a stationary mass of 200,000kg 

To get the mass of the Falcon when she gets to 99%c looks like I have to use tau: √1-v²/c²  which seems to work out about 0.14

Am I right that I have to divide the stationary mass by this number to get the mass of the Falcon at 99%c? which gives the Falcon's mass as about 1,430,000kg
 

If so am I right that using E=mc² gives the amount of energy needed to shift her at this point, which my calculator gives as 1.28571390E+17 joules

which I think means the Falcon's fusion engines need to make about 1.29¹⁷joules for the last second before 99%c ??

Cheerz

GIAN🙂XXX

[Halc]

1 hour ago, Gian said:

Am I right that I have to divide the stationary mass by this number to get the mass of the Falcon at 99%c? which gives the Falcon's mass as about 1,430,000kg

The Lorentz factor is almost exactly 7. Some call the 1.4 million kg the relativistic mass but technically the mass of the ship is the frame invariant proper mass of 200,000 kg.

That's just a terminology thing. The ship relativistic momentum goes up to γvm, so you can compute its kinetic energy by multiplying that by v.

Also note that once you've given your ship a specific mass, one has to wonder how it accelerates without losing any of it, which is why most hypothetical scenarios avoid any mention of masses since it introduces so many problems that are not illustrative of the point of the exercise. Not sure how your fusion engine produces thrust. Most fusion reactors only produce heat, and maybe electricity from that, neither of which immediately translates to thrust.

So anyway, unless you are firing this object out of an insanely long rail gun, any calculation of energy usage needs to factor in where the energy is produced and where it is going, and what is left at each point.

1 hour ago, Gian said:

energy needed to shift her at this point,

Don't know what that phrase means. You talk about energy, but then follow it up with a specification of power (so many joules per sec).  If the ship is accelerating at 1g, then it's power consumption is presumably proportional to its current mass, so integration is needed. One also has to factor in energy imparted to the reaction mass, and all that scaled by some kind of efficiency rating. The most efficient engine might be something like an ion drive, but those don't put out enough force for 1g of acceleration.

 

200 tons is seemingly not enough to provide life support for Joanne for all those years, let alone any left over for trivialities like propulsion. Perhaps it is really good at recycling.

Edited October 30 by Halc

[Gian]

1 hour ago, Halc said:

...You talk about energy, but then follow it up with a specification of power (so many joules per sec).  If the ship is accelerating at 1g, then it's power consumption is presumably proportional to its current mass, so integration is needed. One also has to factor in energy imparted to the reaction mass, and all that scaled by some kind of efficiency rating. The most efficient engine might be something like an ion drive, but those don't put out enough force for 1g of acceleration.

 

200 tons is seemingly not enough to provide life support for Joanne for all those years, let alone any left over for trivialities like propulsion. Perhaps it is really good at recycling.

Thanks Mr Halc, the Millennium Falcon 2 has super-advanced fusion engines given to me by the Time Lords.

I just got the impression that mass increases the faster an object travels, so more energy is needed to continue the acceleration.

I read it in Nicholls; The Science In Science Fiction (1982) pp68-9 attached   GIAN🙂XXX

Edited October 30 by Gian

[swansont]

1 hour ago, Gian said:

I just got the impression that mass increases the faster an object travels, so more energy is needed to continue the acceleration.

I read it in Nicholls; The Science In Science Fiction (1982) pp68-9 attached

The problem with relying on pop-sci writing.

The m in E=mc^2 is at rest (it’s a condition in Einstein’s derivation). The full equation is E^2 = p^2c^2 + m^2c^4   The energy of translational motion is include in the p^2c^2 term

Mass can increase from motion, but not translational motion of the center of mass. Vibrational or rotational energy will increase the mass of the system. An atom or nucleus in an excited state is more massive than in the ground state. 

[Gian]

7 minutes ago, swansont said:

The problem with relying on pop-sci writing.

The m in E=mc^2 is at rest (it’s a condition in Einstein’s derivation). The full equation is E^2 = p^2c^2 + m^2c^4   The energy of translational motion is include in the p^2c^2 term

Mass can increase from motion, but not translational motion of the center of mass. Vibrational or rotational energy will increase the mass of the system. An atom or nucleus in an excited state is more massive than in the ground state. 

🤣So I have it totally wrong lol

What's the p stand for?

And can you refer to me to something easy-to-read but accurate about it?

GIAN🙂XXX

[Mordred]

45 minutes ago, Gian said:

🤣So I have it totally wrong lol

What's the p stand for?

And can you refer to me to something easy-to-read but accurate about it?

GIAN🙂XXX

P is momentum the equation Swansont posted is called the energy momentum relation. 

https://en.m.wikipedia.org/wiki/Energy–momentum_relation

 

[Halc]

4 hours ago, Gian said:

engines given to me by the Time Lords.

Time lords live in a universe with significantly different laws of physics. Same for Star Wars/Trek.

The problem is that there are two kinds of mass: proper mass (frame invariant) and coordinate mass (frame dependent). Use of the term 'mass' was sort of ambiguous until around 1950 when the term was formally assigned to mean proper mass. But using it to mean coordinate mass (as Einstein did) persists in pop sources to this day, and chatbots will likely still use it since those pop sources are the larger percentage of its training data.

 

The article is nice, but makes a lot of errors, some of which are just in clarity, but some actually wrong. The wonderland example for instance treats a bicycle as a brick instead of a system with moving parts.

The cycle wheels contract if they rotate, so the spokes would become slack and the bicycle would fail (collapse) if a plodding walking pace was approached.

The article treats speed as absolute, which violates special relativity. You're moving at 0.99c right now relative to some muon.  No fancy acceleration or Time Lord engine required. Do you mass more because of that?  No. Hence their choice of what 'mass' means.

It starts with: "nothing can move faster than [the speed of] light", which should read that it's only true relative to an inertial frame.  For instance, if I shine a light to a reflector on Mars and measure the round trip, it will go faster than c. Not much, but it will, and it is due to spacetime not being flat between here and there, so inertial frames are not applicable.

 

Reading more, I see SR, 1905 "begins with the astonishing experimental fact that c never changes". This is wrong. It is a premise, not a fact, and to date it has never been experimentally verified.

It says that an object's mass increases with speed, but that's a coordinate effect, not a physical one, so only the coordinate mass thus increases, and they don't correctly say that.

It later says 'ship time is flowing at less than 7/8 the normal rate' which is confusing. Ship time is at 100% per the first postulate of SR. The wording makes it sound like time dilation is a physical effect instead of a coordinate one. Again, does your clock run slow because it is moving at .99c relative to some muon?  No, it doesn't.

Discussion of a more general version of E=mc2 is given by other posters.

 

The bottom left of the 2nd page says something crazy about a journey to 283 light years away taking just over a year from Earth's point of view. That's totally wrong. It might take that much proper time on the ship, but that's not an Earth observer.

 

Edited October 31 by Halc

[swansont]

24 minutes ago, Halc said:

Reading more, I see SR, 1905 "begins with the astonishing experimental fact that c never changes". This is wrong. It is a premise, not a fact, and to date it has never been experimentally verified.

Not directly, but relativity is based on it and it’s also found in electrodynamics so there’s a whole lot of physics that wouldn’t work without it.

[Gian]

15 hours ago, Mordred said:

P is momentum the equation Swansont posted is called the energy momentum relation. 

https://en.m.wikipedia.org/wiki/Energy–momentum_relation

 

Thanks Mr Mordred! guess I've got to get started studying GCSE physics!!🤔
GIAN 🙂XXX

13 hours ago, Halc said:

The bottom left of the 2nd page says something crazy about a journey to 283 light years away taking just over a year from Earth's point of view. That's totally wrong. It might take that much proper time on the ship, but that's not an Earth observer.
 

Thanks Mr Halc, the bit I think you're quoting says;
 

"Stories of ships accelerating to 237 light-years in 48 hours, as in EE Smith's Skylark In Space, or half a light-year every minute, as in AE van Vogt's The Storm, simply will not do. Light takes exactly 1 year to travel 1 light-year, and without infinite energy we can never go quite that fast."

Cheerz GIAN🙂XXX

PS I'm still digesting the rest of what you said and will get back to you🙂

Edited October 31 by Gian