Hohmann Δv calculation — both maneuvers

WHAT is being calculated?

We have three orbits:

Orbit 1: circular, radius $r_1$ (starting).
Transfer: ellipse with periapsis $r_1$ , apoapsis $r_2$ .
Orbit 2: circular, radius $r_2$ (target), with $r_2 > r_1$ .

HOW: Derive every speed from first principles

Step 1 — Circular orbit speed (from Newton)

Why? We need the speeds on the two circles to compare against the transfer ellipse.

Set gravity = centripetal force for a circle of radius $r$ : $\frac{GMm}{r^2} = \frac{mv^2}{r}$

Why this step? On a circle the net inward force is gravity, and the required inward force is $mv^2/r$ . Cancel $m$ , multiply by $r$ : $v_{\text{circ}}^2 = \frac{GM}{r} = \frac{\mu}{r} \quad\Rightarrow\quad \boxed{v_{\text{circ}} = \sqrt{\tfrac{\mu}{r}}}$ where $\mu = GM$ is the standard gravitational parameter.

Step 2 — Speed on the transfer ellipse (vis-viva)

Why? The burns compare a circular speed to an elliptical speed at the same radius, so we need speed on an ellipse.

Start from energy conservation on the ellipse. Specific orbital energy: $\varepsilon = \frac{v^2}{2} - \frac{\mu}{r} = -\frac{\mu}{2a}$

Why is $\varepsilon=-\mu/2a$ ? Total energy of any Keplerian orbit depends only on the semi-major axis $a$ (a standard Kepler result). Rearranging for $v$ gives the vis-viva equation: $\boxed{v^2 = \mu\left(\frac{2}{r} - \frac{1}{a}\right)}$

Why this step? This one formula gives speed at any point $r$ of an orbit of size $a$ . Circles ( $a=r$ ) fall out as a special case: $v^2=\mu(2/r-1/r)=\mu/r$ . ✔ (consistency check with Step 1).

Step 3 — Semi-major axis of the transfer ellipse

Why? vis-viva needs $a$ . The ellipse touches $r_1$ at periapsis and $r_2$ at apoapsis. The major axis spans periapsis-to-apoapsis: $2a_t = r_1 + r_2 \quad\Rightarrow\quad \boxed{a_t = \frac{r_1 + r_2}{2}}$ Why this step? Major axis = periapsis distance + apoapsis distance (both measured from the focus at the planet's center), summed across the ellipse.

Step 4 — The four key speeds

Location	Formula	Meaning
Circle 1	$v_{c1}=\sqrt{\mu/r_1}$	speed before burn 1
Transfer periapsis	$v_{p}=\sqrt{\mu\left(\tfrac{2}{r_1}-\tfrac{1}{a_t}\right)}$	speed after burn 1
Transfer apoapsis	$v_{a}=\sqrt{\mu\left(\tfrac{2}{r_2}-\tfrac{1}{a_t}\right)}$	speed before burn 2
Circle 2	$v_{c2}=\sqrt{\mu/r_2}$	speed after burn 2

Step 5 — The two Δv's

Burn 1 (at $r_1$ ): speed up from circle to transfer periapsis. Since $a_t > r_1$ , the ellipse is faster here: $\boxed{\Delta v_1 = v_p - v_{c1} = \sqrt{\frac{\mu}{r_1}}\left(\sqrt{\frac{2r_2}{r_1+r_2}}-1\right)}$

Burn 2 (at $r_2$ ): speed up from transfer apoapsis to the (faster) outer circle: $\boxed{\Delta v_2 = v_{c2} - v_a = \sqrt{\frac{\mu}{r_2}}\left(1-\sqrt{\frac{2r_1}{r_1+r_2}}\right)}$

Total: $\Delta v_{\text{tot}} = \Delta v_1 + \Delta v_2$

Figure — Hohmann Δv calculation — both maneuvers

Worked Example 1 — LEO to GEO (Earth)

Given $\mu_\oplus = 3.986\times10^{5}\ \text{km}^3/\text{s}^2$ , $r_1=6\,678$ km (300 km LEO), $r_2=42\,164$ km (GEO).

$a_t=\frac{6678+42164}{2}=24\,421$ km. Why? midpoint of periapsis+apoapsis.
$v_{c1}=\sqrt{398600/6678}=7.726$ km/s. Why? circular speed at $r_1$ .
$v_p=\sqrt{398600(2/6678-1/24421)}=10.15$ km/s. Why? vis-viva at periapsis.
$\Delta v_1 = 10.15-7.726 = \mathbf{2.42}$ km/s.
$v_{c2}=\sqrt{398600/42164}=3.075$ km/s.
$v_a=\sqrt{398600(2/42164-1/24421)}=1.61$ km/s.
$\Delta v_2 = 3.075-1.61 = \mathbf{1.47}$ km/s.
Total $\Delta v \approx 3.89$ km/s. Why sum? two independent impulsive burns.

Worked Example 2 — Which burn is bigger?

For LEO→GEO, $\Delta v_1 > \Delta v_2$ . Why does this feel surprising? Because the outer orbit is slower, students expect the outer burn to be small... and it is, but note the first burn does most of the energy lifting: it must inject enough energy to reach far-away apoapsis, near where speeds are largest and gravity strongest. Lesson: the burn deep in the gravity well is the expensive one (this is also why the Oberth effect makes low burns efficient for energy gain).

Common mistake Steel-manned errors

Mistake A: "Just subtract the two circular speeds, $v_{c1}-v_{c2}$ ." Why it feels right: You're moving between two circles, so surely the cost is the difference of circular speeds. Why it's wrong: You never actually ride both circles continuously — you ride an ellipse in between. The correct comparisons are circle↔ellipse at each radius. Using $v_{c1}-v_{c2}=4.65$ km/s here would be nonsense (wrong sign of reasoning and wrong number). Fix: always insert the transfer speeds $v_p,v_a$ via vis-viva.

Mistake B: Sign errors — writing $\Delta v_2 = v_a - v_{c2}$ . Why it feels right: You're "arriving" at apoapsis, so it seems natural to subtract the circle from the transfer. Why wrong: At apoapsis the ellipse is slower than the target circle, so you must add speed: $\Delta v_2 = v_{c2}-v_a>0$ . Fix: think physically — you always burn prograde in a Hohmann, both Δv's are positive speed increases (for raising the orbit).

Mistake C: Forgetting $\mu = GM$ , plugging $G$ alone. Fix: use $\mu$ of the central body directly; units km³/s² pair with km and km/s.

Recall Feynman: explain to a 12-year-old

Imagine you're on a merry-go-round (inner orbit) and want to hop onto a bigger, slower merry-go-round further out. You can't jump straight across. Instead you give one big push to fly outward along a curved path (an oval), coast up to the far edge, and then give a second push so you match the speed of the big ride and settle onto it. Two pushes, that's it. The first push (down where things spin fast) is the harder one.

Active Recall

What are the two orbits a classic Hohmann transfer connects?

Two coplanar, circular orbits.

Why exactly two burns?

One at periapsis to raise apoapsis onto the target radius, one at apoapsis to circularize.

State the vis-viva equation.

v^2=\mu\left(\dfrac{2}{r}-\dfrac{1}{a}\right)

Semi-major axis of the transfer ellipse?

a_t=(r_1+r_2)/2

Formula for the first Δv (raising orbit)?

\Delta v_1=\sqrt{\mu/r_1}\left(\sqrt{2r_2/(r_1+r_2)}-1\right)

Formula for the second Δv?

\Delta v_2=\sqrt{\mu/r_2}\left(1-\sqrt{2r_1/(r_1+r_2)}\right)

Why is

\Delta v_2 = v_{c2}-v_a

and not

v_a - v_{c2}

At apoapsis the ellipse is slower than the outer circle, so you must add speed (prograde burn), giving a positive Δv.

Which burn is larger for LEO→GEO and why?

The first (periapsis) burn, because it injects the energy to reach far apoapsis deep in the gravity well.

Circular orbit speed derived from?

Setting gravity = centripetal force:

GMm/r^2=mv^2/r \Rightarrow v=\sqrt{\mu/r}

r_2\to\infty

\Delta v_1

tends to?

v_{c1}(\sqrt2-1)

— the periapsis speed approaches escape/parabolic speed increment.

Connections

Vis-viva equation — the engine behind every speed here.
Circular orbital velocity — special case $a=r$ .
Specific orbital energy — source of $\varepsilon=-\mu/2a$ .
Oberth effect — why deep burns are efficient (explains Ex. 2).
Bi-elliptic transfer — beats Hohmann when $r_2/r_1 \gtrsim 11.94$ .
Standard gravitational parameter — the $\mu=GM$ used throughout.
Impulsive maneuver approximation — why we treat burns as instant Δv.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, Hohmann transfer ka idea bilkul simple hai: tumhe ek chhoti circular orbit se badi circular orbit par jana hai (jaise LEO se GEO). Seedha jump possible nahi, isliye beech mein ek elliptical transfer orbit use karte hain jo dono circles ko touch (tangent) karti hai. Isme sirf do burns lagte hain — pehla neeche (periapsis) par jahan tum speed badhake ellipse par chadhte ho, aur doosra upar (apoapsis) par jahan phir se speed badhake badi circle par settle ho jate ho.

Har speed hum first principles se nikaalte hain. Circular speed gravity = centripetal force se aati hai: $v=\sqrt{\mu/r}$ . Ellipse ki speed vis-viva se: $v^2=\mu(2/r - 1/a)$ , jahan transfer ellipse ka $a_t=(r_1+r_2)/2$ . Bas in chaar speeds ko compare karke do Δv nikal jaate hain: $\Delta v_1 = v_p - v_{c1}$ aur $\Delta v_2 = v_{c2} - v_a$ . Dono positive hote hain kyunki dono baar tum prograde (aage ki taraf) speed badhate ho.

Ek important baat yaad rakho — dono circular speeds ko seedha subtract mat karo, ye galat hai. Tum kabhi dono circles par continuously travel nahi karte, beech mein ellipse hoti hai. Aur LEO→GEO mein pehla burn (2.42 km/s) doosre (1.47 km/s) se bada hota hai, kyunki neeche gravity well mein energy dena mehnga padta hai. Yeh cheez rocket design ke liye critical hai — fuel budget isi Δv par depend karta hai.

Hohmann Δv calculation — both maneuvers

WHAT is being calculated?

HOW: Derive every speed from first principles

Step 1 — Circular orbit speed (from Newton)

Step 2 — Speed on the transfer ellipse (vis-viva)

Step 3 — Semi-major axis of the transfer ellipse

Step 4 — The four key speeds

Step 5 — The two Δv's

Worked Example 1 — LEO to GEO (Earth)

Worked Example 2 — Which burn is bigger?

Active Recall

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Orbital Mechanics & Astrodynamics

Connections