# What do the axis labels mean in this plot (apparently from Sutton 8th edition)?

In the Analytical Graphics Inc. video Low-Thrust Space Trajectory Design and Optimization the speaker Pradipto Ghosh of AGI shows a slide at around 07:00 shown below.

On it there is a plot with the vertical axis labeled $$\Delta u / v_c$$ and the horizontal axis labeled $$v/v_c$$.

What do the variables used in these axis labels represent?

Any other explanation of how the plot should be interpreted or used is appreciated!

• @Paul Thanks! I wonder if that's almost a complete answer, and worth posting as such? These are normalized with respect to $v_c$, if you can find out what $v_c$ is then you're probably good to go! – uhoh Apr 2 at 4:33

## 1 Answer

The caption in the book (Sutton) reads:

Normalized vehicle velocity increment as a function of normalized exhaust velocity for various payload fractions with negligible inert mass of propellant tanks. The optima of each curve are connected by a line that represents Eq. 17–9.

The text reads:

For a given mission, theoretically there is an optimum range of specific impulse that maximizes $$\Delta u / v_c$$ and thus a most favorable propulsion system design.

...This characteristic speed ($$v_c$$) does not represent a physical quantity but rather a grouping of parameters that has units of speed; it can be thought of as the speed a power plant’s inert mass $$m_{pp}$$ would attain if its full power output were converted into kinetic energy.