Warp
Time taken to warp
It is possible to work out how long it should take for a ship to complete warp (once it enters warp) based on formulae released by CCP.
Warp consists of 3 stages:
- Acceleration
- Cruising
- Deceleration
Calculating the time taken to warp is done by calculating the time spent in each of these phases and adding them together. This requires calculating acceleration and deceleration time first, followed by cruise time.
Total time in warp is given by:
[math]\pagecolor{Black}\color{White}t_{total} = t_{accel} + t_{decel} + t_{cruise}[/math]
Acceleration
The formulae
CCP provided formulae for both distance traveled and velocity reached after t seconds of acceleration. If d is distance in meters, v is speed in meters per second, k is a (sort of) constant defined as the warp speed (in AU/s) and a = 149,597,870,700 meters (1 AU).
[math]\pagecolor{Black}\color{White} \begin{align} d & = e^{kt} \\ v & = k*e^{kt}\\ v_{max} & = k * a\\ \end{align} [/math]
Distance
To calculate distance traveled while accelerating
[math]\pagecolor{Black}\color{White} \begin{align} d & = e^{kt} \\ v & = k*e^{kt}\\ & = k*d\\ \therefore d & = \frac{v}{k} \end{align} [/math]
The distance covered while accelerating to vwarp is
[math]\pagecolor{Black}\color{White} \begin{align} d_{accel} & = \frac{v_{warp}}{k} & = \frac{k*a}{k} & = a \end{align} [/math]
This means that every ship covers exactly 1 AU while accelerating to its maximum warp speed.
Time
To calculate the time spent accelerating to warp speed, the equation for v should be rearranged to be in terms of t, and then solved for the case of v being equal to the warp speed (in m/s)
[math]\pagecolor{Black}\color{White} \begin{align} v & = k*e^{kt}\\ \frac{v}{k} & = e^{kt}\\ kt & = \ln{(\frac{v}{k})}\\ t & =\frac{\ln{(\frac{v}{k})}}{k}\\ \end{align} [/math]
We want to find the time taken to maximum warp:
[math]\pagecolor{Black}\color{White} \begin{align} v_{warp} & = k * a\\ t_{accel} & = \frac{\ln{(\frac{v_{warp}}{k})}}{k}\\ \end{align} [/math]
This formula can be simplified further to [math]\pagecolor{Black}\color{White}\frac{\ln{a}}{k}[/math].
Deceleration
Deceleration is calculated slightly differently. Instead of using k to calculate distance and velocity, it uses j, which is defined as [math]\pagecolor{Black}\color{White}\min(\frac{k}{3},2)[/math]. A different rate of deceleration is used to prevent ships suddenly transitioning from "many, many AU away" to "on grid and out of warp" more rapidly than other pilots (or the server / client) can keep up with.
There is a complication with deceleration calculations. Ships do not drop out of warp at 0 m/s. Instead, they drop out of warp at s m/s, after which normal sub-warp calculations take over.
[math]\pagecolor{Black}\color{White}s = \min(100, v_{subwarp}/2)[/math]
Where vsubwarp is the maximum subwarp velocity of the ship; this varies greatly depending on the ship hull and pilot skills.
Distance
This changes the formulae used slightly. Remember that distance travelled is the integral of velocity.
[math]\pagecolor{Black}\color{White} \begin{align} v & = k * e^{jt}\\ d_{decel} & = \int_{0}^{\infty}k*e^{jt}\,dt = \frac{k*e^{jt}}{j}\\ & = \frac{v}{j} \end{align} [/math]
The distance covered while decelerating from vwarp is
[math]\pagecolor{Black}\color{White} \begin{align} d_{decel} & = \frac{v_{warp}}{j} = \frac{k*a}{j} \end{align} [/math]
Note that for ships that travel at up to 6 AU/s, k / j = k / (k/3) = 3, so these ships cover 3 AU while decelerating. The complication of not stopping warp at 0 can be safely ignored for distance calculations, because the distance that would be covered while decelerating from 100 m/s is insignificant compared to the ~450 billion meters it takes to decelerate from warp speed to warp drop speed.
Time
As with acceleration, time to decelerate from maximum warp velocity is worked out by rearranging the velocity equation.
[math]\pagecolor{Black}\color{White} \begin{align} v &= k*e^{jt}\\ \frac{v}{k} & = e ^ {jt}\\ t & = \frac{\ln{(\frac{v}{k})}}{j} \end{align} [/math]
While the deceleration from s to 0 was insignificant in terms of distance, it is significant in terms of time. This means that the time to decelerate is calculated as follows:
[math]\pagecolor{Black}\color{White} \begin{align} t_{decel} & = t_{decel\_warp} - t_{decel\_s}\\ & = \frac{\ln{(\frac{v_{warp}}{k})}}{j} - \frac{\ln{(\frac{s}{k})}}{j}\\ & = \frac{\ln{(\frac{v_{warp}}{k})} - \ln{(\frac{s}{k})}}{j}\\ & = \frac{\ln{v_{warp}} - \ln{k} - \ln{s} + \ln{k}}{j}\\ & = \frac{\ln{v_{warp}} - \ln{s}}{j}\\ & = \frac{\ln{(\frac{v_{warp}}{s})}}{j} \end{align} [/math]
Cruising
Distance
The distance covered while cruising is the total warp distance minus any distance covered while accelerating or decelerating.
[math]\pagecolor{Black}\color{White}d_{cruise} = d_{total} - d_{accel} - d_{decel}[/math]
For all but the fastest ships, this will be dtotal - 4 AU.
Time
This one is easy. Time spent cruising is simply
[math]\pagecolor{Black}\color{White}t_{cruise} = \frac{d_{cruise}}{v_{warp}}[/math]
Short Warps
The above calculations work as long as some time is spent at maximum warp speed. If the warp is short enough that the ship never reaches top speed, a different set of calculations are needed.
[math]\pagecolor{Black}\color{White} \begin{align} d_{accel} & = \frac{v_{max}}{k}, d_{decel} = \frac{v_{max}}{j}\\ d_{total} & = d_{accel} + d_{decel} = v_{max}(\frac{1}{k} + \frac{1}{j})\\ v_{max} & = \frac{d_{total}*k*j}{k + j} \end{align} [/math]
This enables the calculation of new acceleration and deceleration times using the formulae described in the previous sections, but substituting in the new vmax
[math]\pagecolor{Black}\color{White} \begin{align} t_{accel} & = \frac{\ln{(\frac{v_{max}}{k})}}{k}\\ t_{decel} & = \frac{\ln{(\frac{v_{max}}{s})}}{j}\\ t_{total} & = t_{accel} + t_{decel} \end{align} [/math]
Implementation
The following python code is one possible implementation of the above. It attempts to generate the same data as presented by CCP in the forums. It matches with their numbers, except for 50 AU titan warps, which are one second out. Note that the sub warp speed of the ship is fixed at 200 m/s. This is because the CCP-produced tables assume that every ship drops out of warp at 100 m/s. If trying to run calculations for actual ships, this value will need to be replaced with a more appropriate one.
import math
AU_IN_M=149597870700
def get_distance(dist):
if dist > 1e9:
return (dist / AU_IN_M, "AU")
else:
return (dist/1000, "KM")
# Warp speed in AU/s, subwarp speed in m/s, distance in m
def calculate_time_in_warp(max_warp_speed, max_subwarp_speed, warp_dist):
k_accel = max_warp_speed
k_decel = min(max_warp_speed / 3, 2)
warp_dropout_speed = min(max_subwarp_speed / 2, 100)
max_ms_warp_speed = max_warp_speed * AU_IN_M
accel_dist = max_ms_warp_speed / k_accel
decel_dist = max_ms_warp_speed / k_decel
minimum_dist = accel_dist + decel_dist
cruise_time = 0
if minimum_dist > warp_dist:
max_ms_warp_speed = warp_dist * k_accel * k_decel / (k_accel + k_decel)
else:
cruise_time = (warp_dist - minimum_dist) / max_ms_warp_speed
accel_time = math.log(max_ms_warp_speed / k_accel) / k_accel
decel_time = math.log(max_ms_warp_speed / warp_dropout_speed) / k_decel
total_time = cruise_time + accel_time + decel_time
distance = get_distance(warp_dist)
return total_time
distances = [150e3, 1e9, AU_IN_M * 1, AU_IN_M * 2, AU_IN_M * 5, AU_IN_M * 10, AU_IN_M * 20, AU_IN_M * 50, AU_IN_M * 100, AU_IN_M * 200]
speeds = [1.36, 1.5, 2, 2.2, 2.5, 2.75, 3, 3.3, 4.5, 5, 5.5, 6, 8]
result = {}
for speed in speeds:
for dist in distances:
result[(dist, speed)] = calculate_time_in_warp(speed, 200, dist)
print("{:9s}".format(""), end="")
for speed in speeds:
print("{:9.2f}".format(speed), end="")
last_dist = 1e999
for x,y in sorted(result.keys()):
dist = get_distance(x)
if (y < last_dist):
print("\n{:7.5n} {:s}".format(dist[0],dist[1]), end="")
last_dist = y
print("{:9.0f}".format(math.ceil(result[x,y])), end="")
print()
