Note: I originally posted this on reddit, then thought it might be worth sharing here too. 
For many people who play Elite Dangerous, I guess a big motivating factor to venture out into the unknown is to discover brand new systems and be rewarded with a “First discovered by <insert your name here>” tag.
I am acutely aware however for the potential of the “early adopter crowd” or senior players – which I am not one – to poo-poo on my plans by laying their galactic beach towels on the best sunny spots in the Milky Way. Having come from an EVE Online background… well let’s just say once bitten, twice shy. The last thing I want to do is to venture through a hundred systems, over thousands of light years, over many laborious hours, and never find a patch of turf to call my own.
This may seem a crazy thing to worry about in a galaxy containing 400 billion systems, but that’s no good if the only spots left are at the opposite end of the Milky Way to my current location. No-one drives into a car park and picks the spot furthest from the shops, do they?
Of course, I am not that bothered. Honest. But I thought it might be a little fun to go away and crunch the numbers on this. I had three key questions I wanted to answer, as follows:
If you can't be bothered to read to the bottom, then here are the answers to the questions above:
(1) 150,000 YEARS TO EXPLORE THE GALAXY… REALLY?
To help answer this question, it helps to make a few simplifications and assumptions. Firstly, let’s assume that the transit time between systems is just 10 seconds (no stopping for a toilet break I’m afraid), and second that all the star systems can be reached without needing to re-trace steps and visit a system more than once. The maths is simple then,
Total time taken = 400 billion * 10 secs = 126,839 years
Well this number is similar – certainly the right order of magnitude – to the 150,000 years I’ve seen stated elsewhere. Interestingly though I can now attach this figure to a single pilot’s endeavour. But what about multiple pilots working together?
Consider the efforts of The First Great Expedition, which was hoping to enlist 1,000 commanders to map every system in the galaxy (or thereabouts). If we assume that each pilot takes a little more time to journey between the stars, say 60 seconds, and after a big recruitment drive they swell their numbers to 10,000… then,
Total time taken = 400 billion * 60 / 10000 = 76 years
Which, it just so happens, is also the orbital period of Halley’s Comet
Now this is still a bloody long time, but suddenly the Milky Way doesn’t seem so large anymore.
Let's assume now that the original estimate of 150,000 years is correct, even for multiple players engaged in exploration. How long then is each player spending in each system that they visit? If we assume that at any one time there are 1,000 players exploring, the average "residency" time in each system is,
t = 150,000 yrs / (1000 * 400 billion) = 3.3 hours per system
This seems quite a long time, but might be realistic if we assume that each pilot takes the time to also visit every planet/star in a system and scan it properly. Some of those "semi-major axis" distances can stretch to hundreds of thousands of light seconds, which can take 10-20 minutes to traverse.
(Note that I am not saying that it is the same pilots working 24/7, only that at any single point in time there will be a total of 1,000 pilots engaged in exploration).
I guess one question that does arise is whether or not it is possible to visit every system without re-tracing steps? I think the answer is a definite yes. There are two techniques that can be employed; 1) The onion method or 2) the lawn mower method.
The onion method requires starting at the centre of the galaxy and moving outwards in concentric spheres or shells. Each shell is fully explored, zig-zagging across the surface, before elevating to the next shell above, and repeating. The lawn mower method means zipping along in parallel lines, going back and forth across the length of the galaxy, moving onto different planes vertically once one plane has been completed.
Clearly there’s still a lot out there and almost certainly an unclaimed bit of rock for me to go looking for. The problem though is that I’m a bit lazy, and I want to something to look for in my own back yard.
(2) HOW FAR SHOULD I TRAVEL BEFORE I FIND MY FIRST UNDISCOVERED SYSTEM?
Imagine that all pilots start from a single system, and branch outward, equally spaced, along straight vectors. If they each travel the same distance, the locations of the pilots will describe vertices on the surface of a sphere. What would be the size, or radius, of this sphere if each pilot is to be guaranteed landing at a unique system to call their own? If all the star systems are equally spaced apart, then there will be an optimum distance to travel so that no two commanders have to share a system.
To answer this question, we must first set the average separation distance between star systems. From my in-game experience, I would judge this to be about 10ly. Imagine now that a patch of the night sky, on our sphere, contains one of those star systems. The exclusion zone around that star might be described as a circle, on the surface of that sphere, whose area is,
A[SUB]1[/SUB] = PI * R[SUB]s[/SUB]2
Where R[SUB]s[/SUB] is the radius of the circle, and equal to the average separation distance (i.e. 10 ly). Now the area of the surface of the sphere as a whole is
A[SUB]2[/SUB] = 4 * PI * R[SUB]d[/SUB]2
Where Rd is the radius, or distance, travelled by all pilots from the starting point. The number of stars then that rest on the surface of this sphere is
N = A[SUB]2[/SUB] / A[SUB]1[/SUB] = 4 * (R[SUB]d[/SUB] / R[SUB]s[/SUB])2
We can rearrange to find Rd,
R[SUB]d[/SUB] = 0.5 * R[SUB]s[/SUB] * N1/2
So if there are 1,000 commanders venturing out into space from a common starting point, they would need to travel…
R[SUB]d[/SUB] = 0.5 * 10 * 1,0001/2 = 158 light years ...in order to guarantee arrival at a unique system.
Since the beginning of this year, the player base is estimated to be about 300,000 strong. If we assume that just 10% of this population is actively engaged in exploration, then the optimum travel distance is,
R[SUB]d[/SUB] = 0.5 * 10 * 30,0001/2 = 866 light years
The current size of “civilised” space is about 300 to 500 light years in radius, outside of which is where undiscovered systems reside. This optimum “Rd” distance is perhaps 2 times as large as that, but this clearly isn’t insurmountable.
The reality is that far fewer pilots will likely pursue exploration this aggressively, and those that do are more than likely to overlap the well-worn paths of others (such as pilgrimages to certain famous nebulae). This leaves vast gaps in the night-sky to aim for.
So finally, what if I wanted to find a patch of sky with at least 100 local, adjacent, “undiscovered” systems? How far should I travel if there are 1,000 pilots doing the same thing?
R[SUB]d[/SUB] = 0.5 * 10 * (1,000 * 100)1/2 = 1,581 light years
So this is quite a bit further but again not an insurmountable goal. This is achievable with just a few hours game play. Most advice I have seen from others is to get at least 1,000 to 1,500 light years from civilised space for exploration, and this would seem to fit the estimate above. As for my own experience, I found my first truly unexplored system at around 480ly from Sol, but every-other star system I then jumped to seemed to be already “claimed”. I clearly need to go much further to find vast, empty, unclaimed territories.
The real problem here is time. As the clock ticks on, there are fewer nearby systems that remain “undiscovered”. As a late starter, how much harder is the task getting?
(3) CLAIMING “UNDISCOVERED” SYSTEMS NEAR CIVILISED SPACE – AM I TOO LATE TO THE PARTY?
As the in-game population expands, and existing pilots add further systems to their roster, how quickly is the Horizon of Knowledge (HoK), i.e. discovered systems around civilised space, growing?
Let’s create a simple expression for the number of new system discoveries as a function of time,
N(t) = k * p(t) * f * e * t + N[SUB]c[/SUB]
Here “k” is the rate of discovery (units of 1/t), “p” is the total number of players, “f” is a decimal fraction (i.e. what percentage of in-game pilots are active in exploration), “e” is an efficiency factor, “t” is time and finally “N[SUB]c[/SUB]” is the number of already discovered systems of civilised space.
I have no idea what the demographic for ED is like, and not a clue about what the “average” player might look like. However, I’ll make the big assumption (because I have to) that the average explorer spends 10 hours a week seeking out new systems, and that for every one of those hours he/she will visit 30 systems (i.e. about 2 minutes per system on average). If we assume each one of those systems was a new discovery, then the effective continuous discovery rate “k” is,
k = (30 * 10) / (7 * 24 * 60 * 60) = 0.000496 new system discoveries per second
(or the equivalent of 42.9 discoveries per day)
Now to estimate p(t).
We know from details published by Frontier (link) that there are forecasts for how the player-base (or units sold) is expected to grow over the next few years. Taking their “worst”, i.e. lowest, case estimate:
2015 - 250,000 players
2016 - 1,000,000 players
2017 - 2,000,000 players
It was recently announced that ED had gained 300,000 players, up from just 50,000 last year, so they are definitely on track to deliver their forecast.
By regression analysis (plotting a polynomial trend in Excel) we can find an expression for the population growth based on the forecast above. We find that “p”, as a function of time, is,
p(t) =1.5308 * t2 + 118.96 * t + 50000
(note that this expression assumes “t” is measured in days, not seconds)
Again we will assume that just 10% of the population is partaking in exploration, and so we set,
f = 0.1
For the efficiency factor “e”, we will try to account for the fact that pilots are very likely to stumble on systems already discovered by someone else as they go about a “random walk” of the galaxy. We might expect that only 1 in every 10 systems visited is genuinely “undiscovered”, and so,
e = 0.1
Finally, we have,
N(t) = 42.9 * (1.5308 * t2 + 118.96 * t + 50000) * 0.1 * 0.1 * t + Nc
This is great, but what I’m really interested in is the growth rate of HoK, or the increase in the radius of the sphere that contains all “discovered” systems.
An expression for the number of systems contained within a spherical volume of radius R[SUB]d[/SUB], where the average separation distance of systems is Rs, is
N[SUB]v[/SUB] = (R[SUB]d[/SUB] / R[SUB]s[/SUB])3
Re-arranging for Rd, we have,
R[SUB]d[/SUB] = R[SUB]s[/SUB] * N[SUB]v[/SUB]1/3
Into N[SUB]v[/SUB] we can substitute our earlier expression for N as a function of time, and so we have,
R[SUB]d[/SUB] = R[SUB]s[/SUB] * (0.429 * (1.5308 * t2 + 118.96 * t + 50000) * t + N[SUB]c[/SUB])1/3
Finally, we can estimate “N[SUB]c[/SUB]” using the expression for N[SUB]v[/SUB] above and a radius for civilised space of 500 ly,
N[SUB]c[/SUB] = (500 / 10)3 = 125,000
Therefore our final expression, with R[SUB]s[/SUB] = 10ly and with “t” in days, is,
R[SUB]d[/SUB] = 10 * ( 0.429 * (1.5308 * t2 + 118.96 * t + 50000) * t + 125,000 )1/3
The Horizon of Knowledge (HoK) expands as follows,
Year 0 - 500 ly
Year 1 - 3,601 ly
Year 2 - 6,683 ly
Year 3 - 9,820 ly
This, frankly, is a little worrying. In a very short space of time, a huge envelope of space around civilised space will be “claimed” and new pilots will have to travel extreme distances to find their first true “undiscovered” system. Of course, this is just a simple model based on some pretty woolly assumptions. It does however indicate that a concerted effort by a proportion of the player base could see a big chunk of our galactic back yard fully explored in just a few years.
Clearly, I’d better get a move on.
For many people who play Elite Dangerous, I guess a big motivating factor to venture out into the unknown is to discover brand new systems and be rewarded with a “First discovered by <insert your name here>” tag.
I am acutely aware however for the potential of the “early adopter crowd” or senior players – which I am not one – to poo-poo on my plans by laying their galactic beach towels on the best sunny spots in the Milky Way. Having come from an EVE Online background… well let’s just say once bitten, twice shy. The last thing I want to do is to venture through a hundred systems, over thousands of light years, over many laborious hours, and never find a patch of turf to call my own.
This may seem a crazy thing to worry about in a galaxy containing 400 billion systems, but that’s no good if the only spots left are at the opposite end of the Milky Way to my current location. No-one drives into a car park and picks the spot furthest from the shops, do they?
Of course, I am not that bothered. Honest. But I thought it might be a little fun to go away and crunch the numbers on this. I had three key questions I wanted to answer, as follows:
- Would it really take 150,000 years to explore all the systems in ED’s Milky Way, as suggested by the devs?
- How far would I need to travel from “civilised” space before my chances of discovering an uncharted system became reasonable?
- With a growing player base, and increasing competition in exploration, how rapidly would the Horizon of Knowledge (HoK), i.e. radius of discovered systems around civilised space, expand?
If you can't be bothered to read to the bottom, then here are the answers to the questions above:
- For one pilot taking 150,000 years to explore all the galaxy, each system stop would have to be only 10 seconds. For 1,000 pilots working simultaneously, they could afford a leisurely 3.3 hours at each system.
- Probably 1,500 ly is good bet.
- One year from now, expect the HoK to be about 3,600 ly. Three years from now, expect HoK to be 9,800 ly!! Better get a move on guys!
(1) 150,000 YEARS TO EXPLORE THE GALAXY… REALLY?
To help answer this question, it helps to make a few simplifications and assumptions. Firstly, let’s assume that the transit time between systems is just 10 seconds (no stopping for a toilet break I’m afraid), and second that all the star systems can be reached without needing to re-trace steps and visit a system more than once. The maths is simple then,
Total time taken = 400 billion * 10 secs = 126,839 years
Well this number is similar – certainly the right order of magnitude – to the 150,000 years I’ve seen stated elsewhere. Interestingly though I can now attach this figure to a single pilot’s endeavour. But what about multiple pilots working together?
Consider the efforts of The First Great Expedition, which was hoping to enlist 1,000 commanders to map every system in the galaxy (or thereabouts). If we assume that each pilot takes a little more time to journey between the stars, say 60 seconds, and after a big recruitment drive they swell their numbers to 10,000… then,
Total time taken = 400 billion * 60 / 10000 = 76 years
Which, it just so happens, is also the orbital period of Halley’s Comet
Now this is still a bloody long time, but suddenly the Milky Way doesn’t seem so large anymore.
Let's assume now that the original estimate of 150,000 years is correct, even for multiple players engaged in exploration. How long then is each player spending in each system that they visit? If we assume that at any one time there are 1,000 players exploring, the average "residency" time in each system is,
t = 150,000 yrs / (1000 * 400 billion) = 3.3 hours per system
This seems quite a long time, but might be realistic if we assume that each pilot takes the time to also visit every planet/star in a system and scan it properly. Some of those "semi-major axis" distances can stretch to hundreds of thousands of light seconds, which can take 10-20 minutes to traverse.
(Note that I am not saying that it is the same pilots working 24/7, only that at any single point in time there will be a total of 1,000 pilots engaged in exploration).
I guess one question that does arise is whether or not it is possible to visit every system without re-tracing steps? I think the answer is a definite yes. There are two techniques that can be employed; 1) The onion method or 2) the lawn mower method.
The onion method requires starting at the centre of the galaxy and moving outwards in concentric spheres or shells. Each shell is fully explored, zig-zagging across the surface, before elevating to the next shell above, and repeating. The lawn mower method means zipping along in parallel lines, going back and forth across the length of the galaxy, moving onto different planes vertically once one plane has been completed.
Clearly there’s still a lot out there and almost certainly an unclaimed bit of rock for me to go looking for. The problem though is that I’m a bit lazy, and I want to something to look for in my own back yard.
(2) HOW FAR SHOULD I TRAVEL BEFORE I FIND MY FIRST UNDISCOVERED SYSTEM?
Imagine that all pilots start from a single system, and branch outward, equally spaced, along straight vectors. If they each travel the same distance, the locations of the pilots will describe vertices on the surface of a sphere. What would be the size, or radius, of this sphere if each pilot is to be guaranteed landing at a unique system to call their own? If all the star systems are equally spaced apart, then there will be an optimum distance to travel so that no two commanders have to share a system.
To answer this question, we must first set the average separation distance between star systems. From my in-game experience, I would judge this to be about 10ly. Imagine now that a patch of the night sky, on our sphere, contains one of those star systems. The exclusion zone around that star might be described as a circle, on the surface of that sphere, whose area is,
A[SUB]1[/SUB] = PI * R[SUB]s[/SUB]2
Where R[SUB]s[/SUB] is the radius of the circle, and equal to the average separation distance (i.e. 10 ly). Now the area of the surface of the sphere as a whole is
A[SUB]2[/SUB] = 4 * PI * R[SUB]d[/SUB]2
Where Rd is the radius, or distance, travelled by all pilots from the starting point. The number of stars then that rest on the surface of this sphere is
N = A[SUB]2[/SUB] / A[SUB]1[/SUB] = 4 * (R[SUB]d[/SUB] / R[SUB]s[/SUB])2
We can rearrange to find Rd,
R[SUB]d[/SUB] = 0.5 * R[SUB]s[/SUB] * N1/2
So if there are 1,000 commanders venturing out into space from a common starting point, they would need to travel…
R[SUB]d[/SUB] = 0.5 * 10 * 1,0001/2 = 158 light years ...in order to guarantee arrival at a unique system.
Since the beginning of this year, the player base is estimated to be about 300,000 strong. If we assume that just 10% of this population is actively engaged in exploration, then the optimum travel distance is,
R[SUB]d[/SUB] = 0.5 * 10 * 30,0001/2 = 866 light years
The current size of “civilised” space is about 300 to 500 light years in radius, outside of which is where undiscovered systems reside. This optimum “Rd” distance is perhaps 2 times as large as that, but this clearly isn’t insurmountable.
The reality is that far fewer pilots will likely pursue exploration this aggressively, and those that do are more than likely to overlap the well-worn paths of others (such as pilgrimages to certain famous nebulae). This leaves vast gaps in the night-sky to aim for.
So finally, what if I wanted to find a patch of sky with at least 100 local, adjacent, “undiscovered” systems? How far should I travel if there are 1,000 pilots doing the same thing?
R[SUB]d[/SUB] = 0.5 * 10 * (1,000 * 100)1/2 = 1,581 light years
So this is quite a bit further but again not an insurmountable goal. This is achievable with just a few hours game play. Most advice I have seen from others is to get at least 1,000 to 1,500 light years from civilised space for exploration, and this would seem to fit the estimate above. As for my own experience, I found my first truly unexplored system at around 480ly from Sol, but every-other star system I then jumped to seemed to be already “claimed”. I clearly need to go much further to find vast, empty, unclaimed territories.
The real problem here is time. As the clock ticks on, there are fewer nearby systems that remain “undiscovered”. As a late starter, how much harder is the task getting?
(3) CLAIMING “UNDISCOVERED” SYSTEMS NEAR CIVILISED SPACE – AM I TOO LATE TO THE PARTY?
As the in-game population expands, and existing pilots add further systems to their roster, how quickly is the Horizon of Knowledge (HoK), i.e. discovered systems around civilised space, growing?
Let’s create a simple expression for the number of new system discoveries as a function of time,
N(t) = k * p(t) * f * e * t + N[SUB]c[/SUB]
Here “k” is the rate of discovery (units of 1/t), “p” is the total number of players, “f” is a decimal fraction (i.e. what percentage of in-game pilots are active in exploration), “e” is an efficiency factor, “t” is time and finally “N[SUB]c[/SUB]” is the number of already discovered systems of civilised space.
I have no idea what the demographic for ED is like, and not a clue about what the “average” player might look like. However, I’ll make the big assumption (because I have to) that the average explorer spends 10 hours a week seeking out new systems, and that for every one of those hours he/she will visit 30 systems (i.e. about 2 minutes per system on average). If we assume each one of those systems was a new discovery, then the effective continuous discovery rate “k” is,
k = (30 * 10) / (7 * 24 * 60 * 60) = 0.000496 new system discoveries per second
(or the equivalent of 42.9 discoveries per day)
Now to estimate p(t).
We know from details published by Frontier (link) that there are forecasts for how the player-base (or units sold) is expected to grow over the next few years. Taking their “worst”, i.e. lowest, case estimate:
2015 - 250,000 players
2016 - 1,000,000 players
2017 - 2,000,000 players
It was recently announced that ED had gained 300,000 players, up from just 50,000 last year, so they are definitely on track to deliver their forecast.
By regression analysis (plotting a polynomial trend in Excel) we can find an expression for the population growth based on the forecast above. We find that “p”, as a function of time, is,
p(t) =1.5308 * t2 + 118.96 * t + 50000
(note that this expression assumes “t” is measured in days, not seconds)
Again we will assume that just 10% of the population is partaking in exploration, and so we set,
f = 0.1
For the efficiency factor “e”, we will try to account for the fact that pilots are very likely to stumble on systems already discovered by someone else as they go about a “random walk” of the galaxy. We might expect that only 1 in every 10 systems visited is genuinely “undiscovered”, and so,
e = 0.1
Finally, we have,
N(t) = 42.9 * (1.5308 * t2 + 118.96 * t + 50000) * 0.1 * 0.1 * t + Nc
This is great, but what I’m really interested in is the growth rate of HoK, or the increase in the radius of the sphere that contains all “discovered” systems.
An expression for the number of systems contained within a spherical volume of radius R[SUB]d[/SUB], where the average separation distance of systems is Rs, is
N[SUB]v[/SUB] = (R[SUB]d[/SUB] / R[SUB]s[/SUB])3
Re-arranging for Rd, we have,
R[SUB]d[/SUB] = R[SUB]s[/SUB] * N[SUB]v[/SUB]1/3
Into N[SUB]v[/SUB] we can substitute our earlier expression for N as a function of time, and so we have,
R[SUB]d[/SUB] = R[SUB]s[/SUB] * (0.429 * (1.5308 * t2 + 118.96 * t + 50000) * t + N[SUB]c[/SUB])1/3
Finally, we can estimate “N[SUB]c[/SUB]” using the expression for N[SUB]v[/SUB] above and a radius for civilised space of 500 ly,
N[SUB]c[/SUB] = (500 / 10)3 = 125,000
Therefore our final expression, with R[SUB]s[/SUB] = 10ly and with “t” in days, is,
R[SUB]d[/SUB] = 10 * ( 0.429 * (1.5308 * t2 + 118.96 * t + 50000) * t + 125,000 )1/3
The Horizon of Knowledge (HoK) expands as follows,
Year 0 - 500 ly
Year 1 - 3,601 ly
Year 2 - 6,683 ly
Year 3 - 9,820 ly
This, frankly, is a little worrying. In a very short space of time, a huge envelope of space around civilised space will be “claimed” and new pilots will have to travel extreme distances to find their first true “undiscovered” system. Of course, this is just a simple model based on some pretty woolly assumptions. It does however indicate that a concerted effort by a proportion of the player base could see a big chunk of our galactic back yard fully explored in just a few years.
Clearly, I’d better get a move on.
