# How much delta-v have I used here? What's the “official” equation for delta-v from parametric thrust?

I took a break from Stack Exchange, jumped in my spacecar and flew the following squiggle:

$$a_x = \cos(10 \ t)$$ $$a_y = \sin(5 \ t)$$ $$a_z = \cos(2 \ t)$$

starting at xyz = [-0.01, 0, -0.05] and v_xyz = [0, -0.2, 0] with a total flight time of $$2 \pi$$.

When I got home I was told "Oh that was a lovely lissajous squiggle, but how much delta-v did you put on the car?"

I said "Oh, not much" and made a beeline to my computer to get back on Stack Exchange.

Question: How much delta-v DID I use?

1. If I have an acceleration vector (same as thrust vector; lets assume mass doesn't change) as a function of time $$\mathbf{F}(t)$$ what is the general integral expression for total delta-v should I use?
2. If someone looked up my trip in Horizons and got my state vectors $$\mathbf{x}(t)$$ and $$\mathbf{v}(t)$$ and had a numerical integrator and interpolator, what is the general integral expression for total delta-v should they use?

3D plot of position (returns to origin) and plots of velocity components

import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
from scipy.integrate import odeint as ODEint

def deriv(X, t):
x, v = X.reshape(2, -1)
ax = np.cos(10*t)
ay = np.sin(5*t)
az = np.cos(2* t)
return np.hstack((v, [ax, ay, az]))

times = np.linspace(0, 2*np.pi, 1001)

X0 = np.hstack(([-0.01, 0, -0.05], [0, -0.2, 0]))

answer, info = ODEint(deriv, X0, times, full_output=True)

xyz, vxyz = answer.T.reshape(2, 3, -1)

fig = plt.figure()
ax  = fig.add_subplot(1, 1, 1, projection='3d', proj_type = 'ortho')
x, y, z = xyz
ax.plot(x, y, z)
ax.plot(x[:1], y[:1], z[:1], 'ok')
ax.plot(x[-1:], y[-1:], z[-1:], 'or')
plt.show()

for thing in vxyz:
plt.plot(thing)
plt.show()

• This feels more like code golf than a genuine space question... at best it's a math question about basic calculus. Either way, I don't think it's a good fit here. This is written like a homework assignment. This is Q&A, not Mechanical Turk. – J... Oct 19 '20 at 14:30
• @J... as far as I know the concept of "delta-v" is specific to spaceflight. If you can show otherwise I would be happy to find out. – uhoh Oct 19 '20 at 14:35
• That's not the point. – J... Oct 19 '20 at 14:40
• That's exactly the point. "is written like a homework assignment" just means that it is stylized. After writing over 2001 questions here you have to mix it up a bit to stay fresh :-) – uhoh Oct 19 '20 at 14:43
• @J... When a question is too big or complicatged to fit into one SE question post, we break it up into smaller answerable pieces. I'm currently still uncomfortable with this hand-waving answer using an unexplained and unsourced equation, so I first asked this question so that the equation could have a foundation. Next I made this plot In order to start thinking about extracting post-launch delta-v for deep space spacecraft using its state vectors. – uhoh Oct 19 '20 at 14:52

As $$\Delta v$$ is just change in velocity, we can just integrate the norm of the acceleration function over time:

$$\Delta v = \int|\mathbf{a}(t)| dt$$

You're out of luck getting a closed form of that integral though.

As far as analytical solutions goes, we can note that at $$t = \frac{\pi}{2}$$, all of $$a_x$$, $$a_y$$ and $$a_z$$ are maxed out, and hence $$\Delta v < 2\pi\sqrt{3}$$.

Similarly, the acceleration at all times is going to be greater than or equal to one of the components, and since they are trigonometric functions, their integrals are trivial.

$$4 < \Delta v < 2\pi\sqrt{3}$$

I can't see that there's much more to it from here than just putting the acceleration function into a numerical integrator. It's a smooth curve, so they are good at this.

Integral(sqrt(cos(10*x)^2 + sin(5*x)^2 + cos(2*x)^2),0,2*pi)
-> 7.5279


Or, by the definition of acceleration, if what you have is velocity data:

$$\Delta v = \int\left|\frac{d\mathbf{v}}{dt}\right| dt$$

Which if you have tabular data and don't bother with interpolation, is simply:

$$\Delta v =\sum |d\mathbf{v}|$$

Which is just summing up all the velocity differences between the discrete data points.

• Great answer. The analytic solution of the line integral seems to be an elliptic integral, is that right? – 0xDBFB7 Oct 24 '20 at 14:37