**Going Interstellar: Time, Distance, Velocity & Acceleration**

by Cam Wipper, CFHT remote observer

This is the first part of a multi-part series, “*Destination: Proxima Centauri*“. This series will explore the various issues surrounding interstellar travel and what would be found once we reach Proxima Centauri, the closest star to the Sun.

———————————————————————————————————————

“How long before you can make the jump to light speed?”

“It’ll take a few moments to get the coordinates from the navi-computer.”

*The ship begins to rock violently as lasers hit it.*

“Are you kidding!? At the rate their gaining!…”

“Traveling though hyperspace isn’t like dusting crops, boy! Without precise calculations we could fly right through a star or bounce too close to a supernova and that would end your trip real quick, wouldn’t it?”

*The ship is now constantly battered with laserfire as a red warning light begins to flash.*

“What’s that flashing!?”

“We’re losing the our deflector shield. Go strap yourself in, I’m going to make the jump to light speed.”

*The galaxy brightens and they move faster, almost as if crashing a barrier. Stars become streaks as the pirateship makes the jump to hyperspace. The Millennium Falcon zooms into infinity in less than a second.*

Some of you may recognize that passage (and recall the iconic elongating stars as the Millennium Falcon accelerates to the speed of light). For those that don’t, this is an excerpt from the script for the 1977 movie *Star Wars*; a scene where the protagonists—Han Solo and Luke Skywalker among them—escape from the Galactic Empire above the desert planet of Tatooine.

In *Star Wars*, faster-than-light travel is routine and allows characters to travel to star systems across their galaxy. *Star Wars is*, of course, science ﬁction, but even in this world of make-believe, faster-than-light travel is still a complicated technology; it isn’t as simple as dusting crops after all! The light speed travel in *Star Wars* requires a special computer—the ‘navi-computer’—to calculate the route through the galaxy so that collisions with stars, planets and black holes are avoided. Also, as seen in other ﬁlms in the *Star Wars* saga, the hyperdrive engines which propel the ships to faster-than-light speeds are rather unreliable and break down frequently.

In our real world, we are relatively stone age. The fastest current spacecraft is the venerable *Voyager 1*. It is currently traveling away from the Sun at a speed of 17 kilometers per second or 38,000 miles per hour. While this sounds fast, it is only 0.0057% of the speed of light. To put such a small percentage into perspective, it is roughly the same comparison as the top speeds of the world’s fastest production car, the Bugatti Veyron Super Sport, and that of a common garden snail. The Bugatti can reach 258 mph, while a garden snail races along at speeds between 0.003 and 0.010 mph (16-50 *feet* per hour)…and we are the garden snail. The following is another way to attempt to understand how fast the speed of light truly is: to travel at just 1% of the speed of light, 2,998 km/s (6.7 million mph), *Voyager 1* would need to accelerate to 175 times it’s current speed. To reach 20% of light speed—59,958 km/s (134 million mph)—*Voyager 1* would need to accelerate 3,500 times it’s current speed! We clearly have a long way to go, but we have reason to try to achieve these speeds. At 20% of the speed of light, it would be possible to reach other star systems in the span of one human lifetime.

The closest star to the Sun is a small red dwarf star named Proxima Centauri. It sits about 4.25 light-years away—roughly 25 trillion miles. At this distance it would take *Voyager 1*, traveling at 17 km/s about 75,000 years to reach Proxima Centauri. At 1% the speed of light it would still take almost 426 years to get there. If, however, we could get up to 20% of the speed of light, it would take only 21 years! A brave astronaut could leave Earth, visit Proxima Centauri and return in less than 50 years! True interstellar travel would be possible and the human race would seem to be destined to spread across the galaxy…or would we? These calculations have left out a very important consideration: acceleration.

It would only take 21 years if you left Earth at 20% of the speed of light and continued at that same speed until you reached Proxima Centauri. To put it another way, this presumes instantaneous acceleration. While convenient for our calculations, in the real world, such acceleration would be obviously fatal. The human body can withstand any constant velocity (You feel the same sitting in a chair—0 mph—cruising down a highway—55 mph—or at 35,000 feet in a commercial airliner—575 mph); it is during acceleration that the body struggles. This was succinctly described by ex-*Top Gear*, and current *Grand Tour* host Jeremy Clarkson when he stated, “Speed has never killed anyone. Suddenly becoming stationary, that’s what gets you.” The result of the sudden deceleration he is referring to is the same as our instantaneous acceleration: near-certain death. We should then factor acceleration into our travel time calculations.

Acceleration is often described in terms of gravitational force or g-force. Acceleration due to gravity on Earth is approximately 9.8 m/s^{2}. A person, such as yourself while reading this article, is experiencing gravity trying to accelerate you at this rate. You are (likely) not accelerating because of what is known in physics as the ‘normal force’. If you are sitting on a ﬂat surface, it is pushing back against gravity at the same rate as gravity is pushing down on you and your feel this as your weight. As a result of these forces, you are experiencing 1.0g—the g-force experienced by every stationary object on Earth and the force at which the human body is evolutionarily adapted to withstand. In space if you accelerated at this rate, 9.8 m/s^{2}, you would weigh the same and feel just as you do on Earth. Since this is what we know best, let’s use this rate for our acceleration calculations. At this rate of constant acceleration how long would it take to reach the speed of *Voyager 1*? 1% the speed of light? 20% the speed of light? The answer is surprising.

As we know *Voyager 1* is traveling at 17 km/s. From a resting position (i.e. velocity of 0, assumed for ease of computation. See equation list below article), if we accelerated at 9.8 m/s2, we would reach 17 km/s in just 29 minutes! That’s right: with the same acceleration as the force of gravity pushing down on you right now, a spacecraft could reach 38,000 mph in 29 minutes. It would be possible to reach 1% of the speed of light in 85 hours (3.5 days) and 20% of the speed of light in 71 days (just under 2.5 months)! That doesn’t seem so bad at all! But wait…there is a huge cost to this acceleration: energy. In order to accelerate an object this quickly, it takes a lot of energy. To escape the Earth’s gravity, the Space Shuttle needed to accelerate to at least 9.8 m/s^{2}. In doing so, it used almost 2 million pounds of fuel, which it exhausted in 8.5 *minutes*! With this incredible rate of fuel consumption, it would be impossible to carry enough fuel for the needed 29 minutes to reach *Voyager 1* speeds, never mind the 71 days required to reach 20% of the speed of light.

That’s not the end of the story for interstellar travel though. We don’t need to accelerate that fast. No where near that fast. Once we escape the gravitational pull of the Earth, we only need to accelerate at 0.2 m/s^{2} to reach Proxima Centauri! To put this in perspective, a car accelerating at this rate would take 2.2 minutes to go from 0 to 60 mph—most modern vehicles can do so in under 10 seconds.

On a journey to Proxima Centauri, a spacecraft would need to accelerate, cruise at the top speed and then decelerate again. Our spacecraft, accelerating at 0.2 m/s2 would take 10 years to reach ~20% of the speed of light. In doing so, it would cover 1.06 light-years, about a quarter of the distance to Proxima Centauri. It would then need 10 years to slow down at the same rate as it approached Proxima Centauri, again covering just over a light year. In the cruise phase, it would cover the remaining 2.13 light years while traveling at 20% of the speed of light (1.06 + 2.13 + 1.06 = 4.25 light years). This would take another 10 years. In total we could accelerate to, cruise at and decelerate from 20% of the speed of light and reach Proxima Centauri in roughly 30 years! A return journey would take 60 years, still within the realm of one human lifetime. It seems that interstellar travel may be possible after all!

Now, it is worth acknowledging that there are a number of issues that have been glossed over here. For one: we made no attempt to use the complex physics of orbital mechanics in our equations. While this would have certainly increased the accuracy of our results, it is beyond the scope of this article and in reality likely would not have made a huge difference over the timescales (decades) and distances (light years) we considered. Secondly, 0.2 m/s^{2} is actually not as slow as it seems. Our current ion engines, used by spacecraft to travel between planets today, produce so little thrust that they only accelerate at 0.0005 m/s^{2}. We still need signiﬁcant advances in propulsion technology to achieve these rates of acceleration continuously over many years. There may be solutions in the the future though. Some incredible technologies are under development. Additionally, we have issues with Relativity (that Einstein guy) and small grains of dust (yes, dust!). Ignoring these for now, what would we ﬁnd once we get to Proxima Centauri? Well, as was announced in late-August 2016, a planet, known as Proxima b, is orbiting this star.

It is in the habitable zone and may be like Earth…we might have neighbors just next door! More on Proxima b, propulsion systems and the other issues with interstellar travel coming up in future installments of the ‘Destination: Proxima Centauri’ series here on the CFHT Hoku blog!

———————————————————————————————————————

This article was rather heavy on the mathematics (Sorry!).

For those interested, here are the various equations and solutions written out.

Proper adherence to the rules of signiﬁcant ﬁgures and rounding were not always followed, these are simply “back of the envelope” calculations.

The approximate answer given in the article is listed in parentheses when different from the exact answer.

**Key Equations and Constants:**

a = Δv / Δt = (v_{f} – v_{i}) / (t_{f} – t_{i})

d = v_{i}t + (1/2)at^{2}

Speed of light = 299,792,458 m/s (299,792.5 km/s)

1% of the speed of light = 2,997,924.58 m/s (2,998 km/s)

20% of the speed of light = 59,958,491.6 m/s (59,958 km/s)

**1. Voyager 1 velocity as a percentage of the speed of light.**

17 km/s / 299,792.5 km/s = 0.00005671 x 100 (convert to percentage) = 0.0057%

**2. Garden snail velocity as a percentage of the top speed of a Bugatti Veyron.**

0.010 mph / 258 mph = 0.00003876 x 100 (convert to percentage) = 0.0039%*

*while the numbers are different, this is the same order of magnitude (within a factor of 10), as the *Voyager 1*/speed of light comparison. In astronomy, this is considered comparable.

**3. Voyager 1 velocity increase needed to reach 1% the speed of light (multiples of)**. 2,998 km/s / 17 km/s = 176.35 (175 times)

**
4. Voyager 1 velocity increase needed to reach 20% the speed of light (multiples of)**. 59,958 km/s / 17 km/s = 3,526.94 (3,500 times)

**
5. Travel time to Proxima Centauri at 17 km/s (Voyager 1 speed) with no acceleration:** 4.25 light years = 40,232,500,000,000 km (40 trillion km)

40,232,500,000,000 km / 17 km/s = 2,366,617,646,058 s = 75,045 years (75,000 years)

**
6. Travel time to Proxima Centauri at 1% the speed of light with no acceleration:** 40,232,500,000,000 km / 2,998 km/s = 13,419,779,853 s = 425.5 years (426 years)

**
7. Travel time to Proxima Centauri at 20% the speed of light with no acceleration:** 40,232,500,000,000 km / 59,958 km/s = 671,011,374 s = 21.3 years (21 years)

8. Acceleration to Voyager 1 speed from rest at 9.8 m/s^{2}:

**(assume v _{i} = 0, set t_{i} = 0)**

a = Δv / Δt = (v

_{f}– v

_{i}) / (t

_{f}– t

_{i})

9.8 m/s

^{2}= (17,000 m/s – 0 m/s) / (t

_{f}– 0 s)

t

_{f}= 17,000 m/s / 9.8 m/s

^{2}

t

_{f}= 1,734.7 s

t

_{f}= 28.9 mins (29 minutes)

**9. Acceleration to 1% speed of light from rest at 9.8 m/s ^{2}:**

**(assume v**

_{i}= 0, set t_{i}= 0)a = Δv / Δt = (v

_{f}– v

_{i}) / (t

_{f}– t

_{i})

9.8 m/s

^{2}= (2,998,000 m/s – 0 m/s) / (t

_{f}– 0 s)

t

_{f}= 2,998,000 m/s / 9.8 m/s

^{2}

t

_{f}= 305,918.4 s

t

_{f}= 84.97 hours (85 hours)

**10. Acceleration to 20% speed of light from rest at 9.8 m/s ^{2}:**

(assume v

_{i}= 0, set t

_{i}= 0)

a = Δv / Δt = (v

_{f}– v

_{i}) / (t

_{f}– t

_{i})

9.8 m/s

^{2}= (59,958,000 m/s – 0 m/s) / (t

_{f}– 0 s)

t

_{f}= 59,958,000 m/s / 9.8 m/s

^{2}

t

_{f}= 6,118,163.3 s

t

_{f}= 1699 hours

t

_{f}= 70.8 days (71 days)

**11. Rate of acceleration needed so that it takes 10 years to travel one quarter of the distance to Proxima Centauri assuming start at rest (v _{i} = 0).**

Distance to Proxima Centauri = 4.25 light years.

One quarter the distance = 4.25 / 4 = 1.0625 light years

1.0625 light years = 10,052,030,000,000,000 m (1.005203 x 10

^{16}m)

10 years = 315,360,000 s d = vit + (1/2)at

^{2}

equation if v

_{i}= 0 —> d = (0)t + (1/2)at

^{2}= (1/2)at

^{2}

d = (1/2)at

^{2}

a = 2d / t

^{2}

a = 2(1.005203 x 10

^{16}m) / (315,360,000 s)

^{2}

a = (2.010406 x 10

^{16}m) / (9.94519296 x 10

^{16}s

^{2})

a = 0.20215 m/s

^{2}(0.2 m/s

^{2})

**12. Final velocity after accelerating at 0.20215 m/s ^{2} for a quarter of the distance to Proxima Centauri.**

(assume v

_{i}= 0, set t

_{i}= 0)

a = Δv / Δt = (v

_{f}– v

_{i}) / (t

_{f}– t

_{i})

a = (v

_{f}– 0) / (t

_{f}– 0)

a = v

_{f}/ t

_{f}

v

_{f}= at

_{f}

v

_{f}= (0.20215 m/s

^{2})(315,360,000 s)

v

_{f}= 63,749,556.1 m/s

v

_{f}= 63,750 km/s

v

_{f}= 21.2% of the speed of light (conversion: 63,750 km/s / 299,758.5 km/s)

**13. Time to travel middle half of journey to Proxima Centauri, assuming ﬁrst and last quarter are spent accelerating and decelerating.**

Distance to Proxima Centauri = 4.25 light years.

One half the distance = 4.25 / 2 = 2.125 light years

2.125 light years = 20,104,100,000,000,000 m (2.01041 x 10^{13} km)

20,104,100,000,000 km / 63,749 km/s = 315,363.378 s = 10.0001 years (10 years)**

**It may be surprising that this results in such a round number, and one identical to the time spent accelerating, but see equation 11. In that equation, we needed to double the distance term (“2d”) while solving for the acceleration. This means that our rate of acceleration (and therefore our ﬁnal velocity) is directly related to double to the distance spent accelerating. Since we divided our distance into quarters, this means that our “cruise” distance and double the distance spent accelerating are the same length (2.125 light years). As a result, the time spent accelerating, cruising and decelerating are the same: 10 years each (The reason the answer isn’t exactly 10 years, is simply due to rounding error).