# Delta v of a trans-Mars injection (TMI)

Why does it only take about 600 m/s more than Earth's escape velocity to have an encounter with Mars while it takes much more Delta v (about 3 km/s) from a solar orbit (same as Earth orbit) to have an encounter with Mars?

What explains this difference?

Is it the Oberth effect or something else that explains this?

• Presumably to reduce speed when approaching Sun. Commented Sep 3, 2022 at 10:06

Yes, it seems related to the Oberth effect. Starting from the solar orbit, we need $$\Delta v$$ to enter an orbit to Mars. By energy conservation the velocity of the spaceship at launch from Earth has to be $$v_0 = \sqrt{v_{\text{esc}}^2+\Delta v^2} \ ,$$ where $$v_{\text{esc}}$$ is the escape velocity. For $$v_{\text{esc}} \gg \Delta v$$, we find that the $$\Delta v^{\prime}$$ on top of the escape velocity is $$\Delta v^{\prime} \simeq \frac{1}{2}\frac{\Delta v^2}{v_{\text{esc}}} \ .$$ This formula relates your Delta-v's in your question. We observe that $$\Delta v^{\prime}$$ becomes smaller the bigger the escape velocity from Earth. Bigger escape velocity means that you're deeper in the gravitational well. By the Oberth effect, a small Delta-v deep in the gravitational well has a bigger effect outside of the well.
In fact, irrespectively of the above approximation, we always have $$\Delta v^{\prime} = \sqrt{v_{\text{esc}}^2+\Delta v^2}-v_{\text{esc}} \leqslant \Delta v \ .$$