The Warping of SpaceTime

One moment you are there, hanging out calmly on the fringes of a remote star system, minding your own business in the near-interstellar vacuum, and in the next there is a vast disturbance of space-time all around you. There is no matter here more dense than a handful of atoms of hydrogen per square meter of space, yet space itself folds at a quantum level.

Rather, space and time alike are unfolding around you, flattening back into normality. You aren’t aware of it having been folded, because you were folded along with it, but there is a definite sign that something dramatic has changed in the local volume.

A starship has arrived.

The starship has traveled a long way, twelve-hundred light-years from its home, yet it has done so over the relatively short time of only three years. That sounds suspiciously like faster-than-light travel, yet it can’t be, because that’s impossible, right? Einstein’s theory of special relativity tells us so.

Yet, Einstein’s theory of general relativity tells us something else. We still cannot travel faster than light in flat space, but space (and time) can be curved.

flat and curved planes
image credit: Dale Gray, PhD Physics, University of North Texas

Indeed, space-time is always curved, regularly distorted by the mass of any objects within it, and we all experience that curvature regularly in our daily life. We experience it as gravity.

Two-sided_spacetime_curvatures
image credit: user:Tokamac / wikimedia.org under CC BY-SA 4.0

Per special relativity, an object with mass cannot move at the speed of light without requiring infinite energy, and at the same time becoming zero mass.

In general relativity, however, in a curved space-time, the equations allow for a warping or folding of space in such a way that a bubble is formed, and as the bubble moves, space in front of it is highly compressed while space behind it is highly expanded.

Alcubierre
image credit: user:AllenMcC / wikimedia.org under CC-BY-SA 3.0

The bubble appears to an outside observer as if it has moved faster than light, but really it has just warped space around itself. No matter how much warping occurs, anything inside the bubble experiences no acceleration, and what’s more, no funny time dilation effects (more on this in a later post, I think).

This is not to say that there are no inherent problems with warping space to this dramatic extent. For one thing, in the original equations demonstrating this as a possibility, developed by Miguel Alcubierre in 1994, it appeared that a massive negative mass-energy state is required to generate the bubble. Negative energy implies exotic matter, but more troubling is that the amount of mass-energy required to transport a very small ship across the galaxy would be greater than the mass of the observable universe.

That’s a problem. We can’t have starships destroying the entire universe every time they fire up their warp drive.

In 1999, Chris Van den Broeck modified the equations to require the equivalent of just three solar masses. A great improvement! Now firing up the engine just destroys three star systems. With about 250 billion stars in our galaxy, who’s going to notice a few of them disappearing now and then?

Well, presumably the inhabitants of any planets around those stars will notice, and if we assume that the starship consumes the nearest three stars whenever it starts up the drive, then we could extrapolate that our own Sun (and us along with it) would be the first to go. Perhaps ok if we’re fleeing a Sun going nova about five billion years from now, but in the meantime we’d like to keep our home, thank you very much.

There are other problems with the Van den Broeck solution, and also with Serguei Krasnikov’s modification, despite the latter’s reduction of the mass requirement to mere milligrams. Basically, they require keeping the surface area of the warp bubble microscopically small while expanding the interior volume, which seems contradictory at first glance, and even so, they are only able to transport a few atoms of matter in this way. These are interesting modifications of the equation, but not truly useful.

In 2012, all that changed. Harold White showed (on paper) that modifying the shape of the bubble into a torus, or a rounded doughnut, dramatically reduces the mass-energy requirement into the hundreds of kilograms range, and with this the drive could propel a starship of useful (macroscopic) size… more than just a few atoms. No longer do we need to dismantle a gas giant planet — or a sun — for every voyage. White’s equations excited serious physicists enough that NASA is funding lab experiments to prove the concept.

This is where I would love to insert the beautiful image of IXS Enterprise developed by Mark Rademaker and then used in Harold White’s presentation materials as a concept design for how such an Alcubierre-White starship might look. However, all such images are copyrighted with all rights reserved, and at this time I cannot justify the expense of purchasing rights to display them here. Nevertheless, I encourage you to go view the originals.

So, we’ve solved the energy requirement (ignoring for the moment that we’re talking about negative energy and exotic matter), but issues still remain. Stefano Finazzi, Stefano Liberati, and Carlos Barcelo argued in 2009 that a classic Alcubierre warp bubble might be just fine if it is eternally moving at a stable superluminal speed — which makes it rather difficult for travelers to “hop on the bus,” so to speak — but the switching on of quantum effects to spin up such a bubble from flat space (a more realistic and usable scenario) would create enough Hawking radiation inside itself to completely fry any occupants.

Hawking radiation is strange stuff, and so far remains theoretical in nature. Then again, so does pretty much everything we’re discussing here.

Engineers love a good challenge, however, and we shall assume for the moment that the problem of internal radiation is solvable. After all, you’ve just observed an Alcubierre-White starship arriving in your immediate vicinity.

Luckily for you, you were not in the starship’s direct path when it unfolded space-time around itself and allowed the warp bubble to dissolve. Why?

During the ship’s three-year voyage to arrive here, it has folded up and compressed a great deal of space ahead of itself. While the ship’s navigator carefully plotted a course to avoid any significant obstacles along the way, the vacuum of interstellar space is not completely void of matter. Free hydrogen atoms exist between the stars, and while estimates vary, they average anywhere from one-quarter (0.25) to one-thousand per cubic meter of space (though more likely on the lower end of that range). As our warp bubble transited the Sagittarius Arm, it collected all those atoms on its leading edge, folding them into the highly energetic bubble wall itself.

If the torus has a cross-section of about two-hundred meters, then on the low end of the estimate our starship has pushed about fifty atoms for every meter it traveled. Over three years, the starship traveled twelve-hundred light-years; how far is that in meters?

The speed of light is a hair under 300,000 kilometers per second (you may be more familiar with the number 186,000 miles/second). A light-year is the distance it travels in one year, which is about 31,500,000 seconds, and thus roughly 9.46 trillion kilometers (that’s 9,460,000,000,000 in case you are counting the zeroes). Multiply that by 1200 light-years, and our starship traveled 11.35 quadrillion kilometers.

A long distance indeed.

We gathered 50 hydrogen atoms per meter, or 50,000 per kilometer. I think you see where I’m going with this.

Upon arrival at its destination and deceleration to subluminal (please, autocorrect, stop making that subliminal) velocities, the warp bubble has compressed about 568 quintillion hydrogen atoms.

That’s a lot of atoms.

Of course, atoms are very small. It takes 602,000,000,000,000,000,000,000 (602 sextillion) hydrogen atoms to make up one gram of matter. So, all those quintillions of atoms gathered over the course of three years still only amount to about a milligram of matter.

Surely that’s not enough matter to, well, matter, right? Let’s find out.

It’s only a milligram, but it is moving very, very fast. We know the basic equation:

E=mc^2

Where:
E = energy
m = mass
c = speed of light

One milligram moving right at the speed of light should be pretty simple to solve for. That is 0.001 g x 300,000 kps squared, or 90 million grams… er, hold on, something funny is happening here. Before we go through all that math, we’ll just approximate a few things. If the milligram of hydrogen atoms are moving at 99.9999% of the speed of light — very, very close, but not quite there — then anything they hit will experience the force equivalent to several kilotons worth of a nuclear bomb. It could definitely ruin your day if your spaceship happened to be right there, but it’s not going to pulverize a planet.

But our atoms aren’t moving at 99.9999% of c. They are moving at c. In our equation, when c = 1, we can simplify it to E = m. Put another way, energy and mass are equivalent, and at the speed of light, our mass becomes pure energy.

When the warp bubble suddenly decelerates, the atoms are ejected off the leading edge as extremely short-wavelength, high-frequency, and thus high-energy radiation, rather than as regular but fast-moving matter. In other words, our decelerating starship emits a gamma ray burst in its forward direction of travel, gamma rays which will easily penetrate almost any shielding and wreak havoc upon anything biological.

For the mathematically- and physics-inclined, I recommend perusing The Alcubierre Warp Drive: On the Matter of Matter by Brendan McMonigal, Geraint Lewis, and Philip O’Byrne.

The starship has a crack navigator, however, and she knows her theoretical physics. A little less theoretical in her case, since she is living it. So, she plots her course not only to avoid large obstacles along the way, but to decelerate far out on the edge of the destination star system. She doesn’t want to pulverize anyone or anything there. So, you survive the starship’s arrival.

There is another minor issue which the starship’s designers had to overcome before sending it on its long voyage. Michael John Pfenning demonstrated in 1998 that for a classic Alcubierre warp bubble there is an inverse relationship between bubble wall thickness and maximum superluminal velocity. At ten times c, the bubble wall can be no thicker than 10^{-32} meters. As the Planck length is 1.6x10^{-35} meters, our bubble wall is almost as thin as the thinnest possible measurement.

As a quick aside, the Planck length in quantum mechanics is the smallest size at which gravitational effects behave rationally. Below this scale, Euclidean geometry ceases to have any meaning, and spacetime becomes quantum foam. That’s a colorful term for saying it is no longer continuous, but has holes in it. So, in effect, the Planck length is the smallest size that anything can be and still effectively be part of our universe.

Our bubble wall is approaching the smallest possible size, and we’re moving at 10 c. That seems pretty fast, and indeed it is pretty fast, but for a voyage of 1200 light-years, and without any time dilation effects, that means… yep. 120 years to complete the voyage.

10 c may be sufficient for the nearest stars, but if our starship is to visit Kepler 62f, either our astronauts must be very patient (and long-lived), or be willing to risk cold sleep, or our ship is somehow going to have to go faster.

Pfenning made his observations during his doctoral thesis only two years after Miguel Alcubierre first published his equations. We’ve already explored how others have built upon the original equations in the intervening time, refining them in many ways so as to reduce the huge amounts of energy required, so it remains plausible that by employing the methods of Harold White and reshaping the bubble into a torus we may also find that we can effectively increase the speed of our starship. Can we do it enough so as to travel a light-year per day, as would be required to reach Kepler 62 in just three years?

I don’t know.

Can we effectively shield the interior of the bubble so that the astronauts are not fried by Hawking radiation or extremely blueshifted high-energy particles?

I don’t know.

So this, my friends, is what in hard science fiction terms we call a McGuffin.

I’ve rambled on far too long about the Alcubierre drive now, and doubtless put many of you to sleep, without ever getting on to the rest of the technology utilized by the crew of Aniara in The Silence of Ancient Light. I’ve also certainly made some significant technical or scientific errors in my attempts to explain (or simply understand) the complex physics and mathematics behind the idea of warping spacetime. For those, I apologize, and indeed, I invite discussion in the comments. Educate me!

If you enjoy these pseudo-scientific ramblings of mine, you may enjoy my previous similar posts of this nature:


header image credit: Les Bossinas / NASA under public domain

Are The Stars Beyond Our Reach?

via Stars Beyond Our Reach

We dream of reaching the stars. Indeed, it’s at the core of what I’ve been writing, and the same is true for many other science fiction authors. It’s also the subject of intensive research by some fairly serious scientists, even if they don’t quite get the billing and notoriety of NASA projects focused right here in our own Solar System.

But is it truly possible?

I like to think so, but I also understand that the challenges are incredibly daunting, more so than the majority of interstellar-themed science fiction stories would have us believe.

Bestselling author Kim Stanley Robinson tackles the challenges of so-called generation ships, in which people will be born, live, and die during the voyage, and only the grandchildren of the original astronauts will be alive at journey’s end in his 2015 novel Aurora. It’s a great read, and I highly encourage you to check it out. I won’t spoil it for you by talking about his conclusions in the novel.

But Robinson also wrote a blog post discussing his thoughts on the various challenges faced, Our Generation Ships Will Sink, and perhaps the title gives it away. He goes into some detail about the issues faced with biological, ecological, physical, sociological, psychological… lots of logicals there. Even upon arrival, the problems don’t cease.

Robinson’s article is a great read, but if you want a nicely wrapped up synopsis of it, I recommend Richard Rabil Jr’s Stars Beyond Our Reach, linked at the beginning of this post. Rabil is a technical writer, who writes both fiction and essays on subjects as diverse as technology and faith, and he tackles many interesting subjects on his blog (which I’ve only just discovered, but so far it’s very promising). He also does a great summary of the evolution of science fiction as a genre, another post I can strongly recommend.

If, like me, you are fascinated by realism in our quest to reach the stars, Rabil’s summary is a good place to start.

Stars Beyond Our Reach


header image credit: Reimund Bertrams (user:DasWortgewand) / pixabay.com under Pixabay License

In the Year of ’39

In the year of ’39 assembled here the volunteers,
In the days when lands were few;
Here the ship sailed out into the blue and sunny morn,
The sweetest sight ever seen.

Lately I’ve had this song running through my head, pretty much on constant repeat. It’s an old song, first released in 1975 on the album A Night at the Opera by Queen.

And the night followed day,
And the storytellers say
That the score brave souls inside
For many a lonely day sailed across the milky seas,
Ne’er looked back, never feared, never cried.

At first it seems to be telling a relatively ordinary story. Volunteers set sail in a ship for a dangerous journey. Is it 1939? Is this something to do with World War II? It’s not really clear yet.

Don’t you hear my call though you’re many years away,
Don’t you hear me calling you;
Write your letters in the sand
For the day I take your hand
In the land that our grandchildren knew.

Wait, what? The land that our grandchildren knew? Ok, there is something odd going on here. And what’s this about being many years away? The song seems to be playing around with time.

In the year of ’39 came a ship in from the blue,
The volunteers came home that day,
And they bring good news of a world so newly born,
Though their hearts so heavily weigh;

For the Earth is old and grey,
Little darling went away,
But my love this cannot be,
For so many years have gone though I’m older but a year,
Your mother’s eyes, from your eyes, cry to me.

Right, this is definitely not an ordinary ship sailing ordinary seas, and time is certainly being twisted. The volunteers bring news of a new world, while the Earth is old and grey? We’re talking about space travel, aren’t we? In fact, we’re talking about interstellar travel.

It’s definitely not 1939.

Many music lovers might have been confused by this song, but by now it should be obvious to readers of this blog what’s going on here. For astronauts to travel far enough to discover another world (“so newly born”), one capable of replacing the “old and grey” Earth as humanity’s home, and return back with the news “older but a year,” they must have traveled very fast indeed. Perhaps even approaching the speed of light?

At speeds this fast, the theory of special relativity tells us (and experimental research has shown) that odd things happen with time. Time appears to slow down for the traveler, at least relative to the stationary observer, so that by journey’s end the traveler will have aged far less than those who stayed home.

In the year of ’39 assembled here the volunteers

2139, perhaps? 2239? It’s not completely clear.

In the year of ’39 came a ship in from the blue,
The volunteers came home that day

Not the same ’39, but 100 years later. 2239? 2339?

For so many years have gone though I’m older but a year

As it happens, given the parameters of the song, we can calculate how fast the ship was traveling, and thus how far away they went, and perhaps even speculate what star they visited! This is because, despite being such a non-intuitive phenomenon, time dilation due to relativistic effects is well understood, and there is an equation to calculate it.

t'=t\sqrt{1-\frac{v^2}{c^2}}

Where:

t’ = dilated time
t = stationary time
v = velocity
c = speed of light

We want to know the ship’s velocity, so let’s parse this out (like traveling back in time to algebra class!):

t'^2=t^2\times({1-\frac{v^2}{c^2}})

\frac{t'^2}{t^2}=1-\frac{v^2}{c^2}

\frac{t'^2}{t^2}+\frac{v^2}{c^2}=1

\frac{v^2}{c^2}=1-\frac{t'^2}{t^2}

v^2=(1-\frac{t'^2}{t^2})\times c^2

v=\sqrt{(1-\frac{t'^2}{t^2})}\times c

Ok, let’s plug in some numbers! To keep things simple, we’ll express velocity as a percentage of the speed of light, and time in years, even though normally in physics equations velocities would be meters per second and time in seconds. But at this scale, those would be some big numbers, so we’re going to assume that c=1, and that v therefore is a percent of c.

v=\sqrt{(1-\frac{1^2}{100^2})}\times 1

v=\sqrt{1-\frac{1}{10000}}

v=\sqrt{1-0.0001}

v=\sqrt{0.9999}

v=0.99995

In order for 100 years to have passed on Earth while only 1 year passed for the astronauts, the ship had to be traveling approximately 99.995% of the speed of light. That is some extreme time dilation, and so that is some extreme speed. Quite the starship!

Time isn’t the only thing dilating here, as traveling at these speeds does some interesting things to the fabric of space as well. Distances ahead of the travelers will appear to shrink somewhat, though even at this high fraction of the speed of light, it’s a minimal effect. Add a few more 9s to the significant digits, however, and it gets very strange indeed.

Meanwhile, though, our travelers have spent a year journeying at very close to the speed of light. How far have they gone? One light-year?

Oh no. They’ve gone much farther than that. The distance traveled is at a speed relative to time for the stationary observers waiting patiently back on Earth, so our starship has traveled a hundred light-years, though it seems to the astronauts to take only one year to do so.

So, what star might they have visited to find a “world so newly born” to which humanity could relocate? First off, this was a round-trip, so with half the time spent journeying out and half spent returning, that would imply they went no more than fifty light-years away (“no more than,” I say, as if this is no big deal, but fifty light-years is a very big deal). Gliese 163 is 49 light-years away and has one potentially habitable world, but it’s not considered an absolutely prime candidate.

Let’s assume, for a moment, that our starship took a little bit of time to accelerate and then decelerate on its journey, so that instead of 50 light-years, perhaps it really only traveled about 40 light-years away.

They went to Trappist-1.

Artist’s impressions of the TRAPPIST-1 planetary system
image credit: ESO/M. Kornmesser (https://www.eso.org/public/images/eso1805a/) under CC-BY 4.0 (https://creativecommons.org/licenses/by/4.0/)

Trappist-1 is a cool red dwarf star 39.6 light-years away, and it has seven temperate and terrestrial planets, four of which are considered potentially habitable even by conservative estimates. Trappist-1 is obviously a prime candidate for finding life, or at least worlds on which humans could live, and being at about the right distance, is also a prime candidate for our volunteers on their desperate and lonely journey.

Not that this provided much consolation to our narrator, who returns to Earth to find his wife long dead, and only a memory of her in the eyes of his (presumably centenarian) daughter (or granddaughter?).

Your mother’s eyes, from your eyes, cry to me.

’39 was written by Brian May, lead guitarist for Queen, in 1975. May, as some of you may know, is also an accomplished astrophysicist, and while the planets of Trappist-1 had not yet been discovered in 1975, he certainly understood the effects and impacts of time dilation on travel at relativistic speeds. May studied physics and mathematics up through the time when his music career began to skyrocket to success, though due to focusing on music after that point, it took him 37 years to complete his doctoral thesis (A Survey of Radial Velocities in the Zodiacal Dust Cloud), finally earning his PhD in 2008. He was Chancellor of Liverpool John Moores University from then until 2013, and he was a science team collaborator for the NASA New Horizons mission to Pluto.

Not just a fantastic guitarist and songwriter, but a serious scientist!

Don’t you hear my call though you’re many years away,
Don’t you hear me calling you;
All your letters in the sand cannot heal me like your hand,
For my life
Still ahead
Pity me.

’39 (Youtube)

Final note: Although Freddie Mercury is far more well-known as lead vocalist, it was Brian May who sang the lyrics for the studio version of ’39 (though Mercury sang for most of their live performances).


’39 from the album A Night at the Opera by Queen, 1975
Songwriter Brian May, copyright EMI Music Publishing, Sony/ATV Music Publishing LLC

Special thanks to E=mc2 Explained for breaking down the physics of time dilation for us laypeople.

Header image credit: user:Les Chatfield / flickr.com under CC-BY 2.0

Astrodynamics

Three months ago I wrote about Orbital Mechanics, focusing on the ins and outs of how spaceships and satellites navigate their way around a planet’s orbital space. We got into the ins and outs of velocity vectors, inclination angles, and prograde vs retrograde thrust.

But what about when a spaceship leaves Earth — or whatever planet it happens to be in orbit around; Kepler 62f, perhaps? — and makes its way toward another planet in the Solar System. Mars, for instance. We’re all astute enough to realize that it’s not as simple as pointing the ship at Mars and firing the rockets. For one thing, of course, Mars is moving, as it is orbiting around the Sun just like Earth is, so even if a straight-line course were possible, we would need to “lead the target,” i.e. aim for the point in Mars’ orbit where Mars will be at the time we arrive.

Sphere of Influence

And, if that’s all it took, this would be a short post. But of course, it’s not as simple as that, as I’m sure you suspected by now. Once our spaceship leaves Earth’s sphere of influence, and until such time as we arrive in Mars’ sphere of influence, we aren’t just in some flat space with no gravitational pull on us. By sphere of influence, or SOI, I mean the region of space where the primary gravitational pull is from whatever celestial body “owns” that SOI. Close to Earth, or in orbit around Earth (even a very high orbit), the bulk of the gravity we feel is from Earth. And, if we don’t maintain sufficient velocity for the altitude of our orbit, then it is Earth into which we will fall. Likewise for Mars.

But in between Earth and Mars, we are still in an SOI — that of the Sun. So, once our spaceship leaves Earth’s SOI, we are primarily impacted by the Sun’s gravity, just as Earth and Mars are.

In fact, once we leave Earth’s SOI, we are still in orbit. It’s just that it’s no longer Earth that we’re orbiting, it’s the Sun.

Up, Down, and the Plane of the Ecliptic

If you’re like me, you probably grew up thinking of the Solar System as basically a flat plane (the plane of the ecliptic), with the Sun in the center and the planets in their various orbits sliding around that plane at a given distance from the Sun. Each of the planets has a north pole that points roughly in the same direction, “up” from the plane. Well, each of the planets except Uranus; Uranus is completely lying on its side, with an axial tilt of 98°, rolling through its orbit like a ball whereas the others are more like spinning tops (apologies to Fraser Cain of Universe Today for borrowing his imagery; I can’t think of a better way to describe it). Anyway, Uranus aside, this way of thinking about the Solar System and the plane of the ecliptic leads to the idea that the Solar System has something of an “up” that all the planets’ north poles somewhat point to, and a “down” that the various south poles point to.

This is not a useful way to think about space. Thinking like this is what leads to engineers constructing the Death Star, with such obvious design flaws that both the first and second versions suffered catastrophic failure with just a tiny nudge. Don’t be one of those engineers.

8631388149_2140d9df6b_o
image credit: Andy Langager / flickr.com via cc by-nc 2.0

Astronauts onboard the ISS don’t think of the general direction of Earth’s north pole as “up.” If anything, it’s sort of sideways. The effects of free fall notwithstanding, the ISS and everyone in it remain subject to Earth’s gravity at about 90% of what we feel on the ground — they just don’t notice it because they are falling all the time, just as I described in my previous post on this topic. If there’s a “down,” it’s toward the Earth’s surface, and “up” is away from Earth.

So likewise, when we are orbiting the Sun, and outside any planet’s SOI, it’s more useful to think of “down” as being toward the Sun, and “up” as away from the Sun. On a more technical level, the inner planets (Mercury and Venus) are in lower orbits, and the outer planets (Mars and beyond, though the belters of The Expanse might have a thing or two to say about referring to Mars as an “outer” planet) are in higher orbits.

Delta-V, Prograde and Retrograde

So, referring back to the earlier post, you’ll recall that it’s not possible (or at least not feasible without a very powerful torchship, but that’s beyond the current discussion) to travel from a lower orbit to a higher one by simply aiming the spaceship “up” and firing the rockets, and that likewise we cannot travel from higher orbit to lower orbit by aiming the spaceship “down” and firing rockets. Well, we could, but it wouldn’t give us the desired result.

Instead, we need to fire our rockets either prograde (to raise our orbit) or retrograde (to reduce or lower our orbit). Either way, what we want to achieve is the appropriate delta-v, or change in velocity (often expressed as Δv), required to match orbits with our target.

To get to Mars, once we have left Earth’s SOI (achieved an Earth orbit high enough that the Sun’s gravitational influence takes over), we don’t point our spaceship toward Mars at all. Instead, we point it forward along the direction of Earth’s orbit around the Sun, i.e. in a prograde direction, and fire the rockets. We are already orbiting the Sun at the same velocity as Earth, which just like the orbital velocity of a satellite around Earth is defined by the same equation, √(GM/r), that you surely recall from Orbital Mechanics. Without going into the equation’s details, the Earth moves along its orbit at a rate of approximately 108,000 kilometers per hour, which is fast.

So our spaceship already has this much velocity around the Sun. Mars, in contrast, orbits at an average of 86,760 km/hr (you’ll recall that objects in higher orbit move slower). What we need to do is accelerate prograde, expending delta-v to raise our orbit until it matches that of Mars.

Hohmann Transfer

In Orbital Mechanics I didn’t go quite far enough into the details of just how raising or lowering an orbit works, other than discussing about thrusting prograde or retrograde. However, the principle remains exactly the same, whether in orbit around the Earth or orbit around the Sun.

When we accelerate prograde, the orbit as a whole doesn’t just expand outward to the higher altitude. Instead, it takes our current mostly circular orbit around the Sun and reshapes it into an elliptical orbit instead, with the perihelion, or closest point to the Sun, being the distance (altitude) of our starting orbit, i.e. Earth’s orbit. As we apply thrust, either more powerfully or for a longer period of time, our orbit becomes more and more elliptical, with the aphelion raising higher and higher, or farther and farther out.

990px-Orbital_Hohmann_Transfer.svg
image credit: user:MenteMagica / wikimedia.org

The perihelion of our orbit doesn’t change, however. Eventually, after we have accelerated for a long enough time, our aphelion matches the orbit of Mars, but our perihelion remains at the orbit of Earth. We are now in what’s known as a Hohmann transfer orbit, and if we do nothing else, our spaceship will cycle endlessly back and forth between Mars’ orbit and Earth’s orbit. That, of course, could be quite useful if our intent is to set up some kind of shuttle or cargo transport back and forth, assuming we can match up the times of aphelion and perihelion to when Mars and Earth will be in the same space as our transport craft.

But we aren’t setting up a cargo shuttle. We want to get to Mars, and we want to stay there (for now). We could, upon arriving at aphelion (and thus the orbit of Mars), once again burn prograde. By burning at aphelion instead of perihelion, we won’t be further raising our aphelion; instead we’ll be raising our perihelion. This has the effect of (slowly) reducing the elliptical eccentricity of our orbit, i.e. of circularizing it. And indeed, this is precisely how we would set a satellite into, say, geosynchronous orbit around the Earth.

But, this would negate all the efficiencies of the Hohmann transfer, and require a lot of burning, a lot of propellant, and a lot of time (unless we use a continuous thrust engine, such as an ion drive; more about this later).

Instead, if we are clever and time our launch window so that our arrival at Mars is close to when it will be 180° around the Sun from where Earth was at launch time, then we arrive with least travel time (and least delta-v), and the last thing we want to do is now waste all that by raising our perihelion. Instead, by arriving at just the right time, we will insert our spacecraft into Mars’ SOI (sphere of influence, you’ll recall) and decelerate to slow the craft down enough that it is captured by Mars’ gravity.

MAVEN_orbital_path_rev-fi
image credit: NASA/JPL

Now we’re in Mars orbit! From this point, all the same rules apply about adjusting our orbit around Mars as did for adjusting it around Earth, or for that matter around the Sun. Or around Kepler 62f if that’s what we’re talking about.

Return to Earth

So now it’s time to go home. How is it different? It isn’t, really. In fact, it’s just the same as any orbital maneuver aimed at reducing altitude. We need to reduce our altitude from 228 million kilometers to 150 million kilometers. To do this, we burn retrograde, against the direction of Mars’ orbit. This has the reverse effect from before, in that it reduces the altitude (from the Sun) of our perihelion until it matches Earth’s orbit. We aren’t actually orbiting the Sun in the reverse direction now — we aren’t burning nearly hard enough for that to happen — we’re just pushing back against our orbit so that we start falling in toward the Sun.

Again, we want to time it with a launch window, one in which it will take us just about one-half of a revolution around the Sun to arrive at Earth’s orbital altitude just as Earth arrives to the same spot. And, upon reaching Earth’s SOI, we need to decelerate so that we are captured by Earth’s gravity, and just like that, we’re home! Well, in orbit around home, anyway, but we know what to do from here.

Interplanetary Transfer Summary

When rockets launch from the surface of Earth to reach orbit, or beyond, they usually do so in an easterly direction, and from as close to the equator as practicable, in order to take advantage of the velocity already imparted upon them by the Earth’s rotation. In other words, to reduce how much delta-v needs to be expended to obtain the velocity required for orbit.

When spacecraft leave Earth to go to Mars, they take advantage of the velocity already imparted upon them by Earth’s revolution around the Sun. Burn prograde, and you ascend to higher orbit, or to the outer planets. Burn retrograde, and you descend to lower orbit, or to the inner planets. It’s really no different, whether in Earth orbit, or interplanetary.

But what happens if our spacecraft leaves the Solar System entirely?

Interstellar Orbits

Leaving out for now questions of faster-than-light travel, Alcubierre warp engines, and other such exotic drives, interstellar maneuvers are no different from interplanetary maneuvers. It’s just a question of scale. Just as the Earth and the other planets in our Solar System revolve around the Sun, the Sun and the hundreds of billions of other stars that make up the Milky Way galaxy all revolve around the center of the galaxy. Just as each planet has an orbit around the Sun, each star has an orbit around the galactic center.

And just as there is a path, and a sequence of burns, that describe an efficient Hohmann transfer route between two planets, there is likewise a Hohmann transfer route between any two stars within the galaxy. There are even more or less ideal launch windows, though given the timescale of stellar orbits, we don’t have as much choice about them; we pretty much have to go when we’re ready to go and accept the less-than-ideal configuration.

Interplanetary and interstellar movement is really the same thing; it’s just a matter of scale. A rather large matter of scale, to be sure, but conceptually no different.

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image credit: NASA/Goddard

So when we’re ready to travel between the stars, if our means remains conventional — whether with a short-burn, high-thrust reaction drive, or a continuous-burn, low-thrust drive, or a continuous-burn, high-thrust torch drive — then the way in which we plot our course doesn’t change. We burn prograde or retrograde about the galactic center, and we utilize our home star’s orbital velocity to impart a delta-v advantage upon ourselves.

There are no straight lines. And our spacecraft is always, always in orbit around something.


header image credit: NASA/JPL