Arrival: B-Plane Targeting and Orbit Insertion

Impact parameter, gravitational focusing, the B-plane, capture burns, optimal periapsis and aerobraking

Mars Reconnaissance Orbiter, 10 March 2006

After seven months of cruise, MRO approaches Mars on a hyperbola. To be captured it must fire its main engines for about 27 minutes, slowing down by roughly a kilometre per second — and part of the burn happens behind Mars, out of radio contact. The team on the ground watches the signal disappear on schedule and waits. The signal returns on schedule too: MRO is in a long, looping orbit of about 35 hours. Over the next five months it dips into the upper atmosphere hundreds of times, letting drag shrink the orbit instead of propellant. None of it works unless, months earlier and hundreds of millions of kilometres away, the spacecraft was aimed at exactly the right point in a plane that passes through the centre of Mars.

Impact parameter. On the arrival hyperbola, the incoming asymptote is a straight line that misses the planet's centre by a distance

b

— the impact parameter (the semi-minor axis of the hyperbola). Angular momentum gives the link to the periapsis radius,

h = b\,v_\infty = r_p v_p

b = r_p\sqrt{1 + \frac{2\mu}{r_p v_\infty^2}} \qquad\Longleftrightarrow\qquad r_p = \frac{\mu}{v_\infty^2}\left[\sqrt{1 + \left(\frac{b v_\infty^2}{\mu}\right)^2} - 1\right]

Aim with b, get r_p. The relation is how a navigation team converts a periapsis altitude into a target in space.

💡Gravitational focusing: the target is bigger than the planet

Set

r_p = R

, the planet's radius: every asymptote with

b < b_{cap} = R\sqrt{1 + 2\mu/(R v_\infty^2)}

hits the planet. At

v_\infty = 2.65

km/s Mars presents a target 2.14 Mars radii across; Earth, 4.3; Jupiter, 22.5. Slower arrivals are focused more. This matters for aiming, for impact probability (asteroid hazard studies use exactly this), and for understanding why so much of the incoming debris that reaches Jupiter hits it.

The B-plane. Introduced by William Kizner at JPL in 1961, it is the plane through the target's centre perpendicular to the incoming asymptote. Three unit vectors define it:

$\hat S$ along the incoming $\vec v_\infty$ ;
$\hat T = \hat S \times \hat k / |\hat S \times \hat k|$ , parallel to a reference plane (with pole $\hat k$ — the ecliptic, Earth's equator, or the target's equator; always state which);
$\hat R = \hat S \times \hat T$ .

The B vector runs from the planet's centre to where the incoming asymptote pierces the plane. Its length is

b

; its direction is

\hat B = \hat S\times\hat h

. Targets are written as two numbers,

B\cdot T

and

B\cdot R

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💡Why navigators love the B-plane

It turns three dimensions into two. The time of closest approach is a third coordinate, but it is nearly decoupled; the two B-plane numbers carry periapsis altitude and orbit orientation. It is nearly linear. Small trajectory-correction manoeuvres move the aim point almost linearly, so corrections can be designed with a 2×3 sensitivity matrix. It separates altitude from plane.

|B|

sets

r_p

; the angle

\theta = \operatorname{atan2}(B\cdot R, B\cdot T)

rotates the orbit plane about

\hat S

. In the lab, moving

\theta

from 0° to 90° to 180° at fixed

|B|

keeps the periapsis at 300 km while the inclination goes from 16° through 90° to 164°. Uncertainty becomes an ellipse. Navigation errors are drawn as a dispersion ellipse in

(B\cdot T, B\cdot R)

— the picture a flight team checks before every correction.

📐Worked Example: how a few hundred kilometres becomes fatal

Mars at

v_\infty = 2.65

km/s. A 226 km periapsis needs

b = 7570

km; a 57 km periapsis needs

b = 7352

km. A 218 km error in the aim point — about 3% of

b

— drops the spacecraft 169 km lower, into air dense enough to destroy it. Near 226 km, every kilometre of periapsis altitude corresponds to 1.28 km in

b

. Those two altitudes are the planned and the reconstructed periapsis of Mars Climate Orbiter in September 1999 (met in the unit on orbital elements). The units error in the thruster data accumulated, month by month, as a drift of the aim point across the B-plane.

Orbit insertion. At periapsis a retrograde burn turns the hyperbola into a closed orbit of eccentricity

e

\Delta v = \sqrt{v_\infty^2 + \frac{2\mu}{r_p}} - \sqrt{\frac{\mu(1+e)}{r_p}}

e = 0 is a circular orbit at r_p; e → 1 is the cheapest possible capture, barely bound.

💡The optimal periapsis

For a target eccentricity

e

, set

d(\Delta v)/dr_p = 0

r_p^* = \frac{2\mu}{v_\infty^2}\,\frac{1-e}{1+e},\qquad \Delta v_{\min} = v_\infty\sqrt{\frac{1-e}{2}}

At Mars with

v_\infty = 2.56

km/s, a circular capture is cheapest at

r_p^* = 13{,}070

km (1,810 m/s) — nearly 10,000 km up, usually not where the science is. For

e = 0.9

the formula's optimum is inside Mars; then the real optimum is the lowest safe periapsis. Check feasibility before trusting a stationary point.

Target orbit	Eccentricity	Apoapsis altitude	Insertion Δv
300 km circular	0	300 km	2,048 m/s
1-sol ellipse (24.6 h)	0.819	33,800 km	861 m/s
4-sol ellipse	0.928	96,000 km	726 m/s
Barely bound	0.98	363,000 km	663 m/s

Aerobraking closes the gap. Capture into the 1-sol orbit (861 m/s), lower periapsis into the upper atmosphere with a small apoapsis burn (11 m/s), let drag shrink the apoapsis over hundreds of passes, then lift periapsis out of the atmosphere (45 m/s). Total 917 m/s instead of 2,048 — a saving of 1.13 km/s, paid for in months of careful operations and a spacecraft designed to survive the heating. Magellan demonstrated it at Venus in 1993; Mars Global Surveyor, Mars Odyssey and MRO relied on it.

💡Capture, aerocapture, or straight in

Three ways to deal with arrival

v_\infty

Propulsive capture — the burn above, optionally followed by aerobraking.
Aerocapture — a single deep pass through the atmosphere inside a heat shield removes all the excess energy at once. Studied for decades, demanding guidance, not yet used by a planetary mission.
Direct entry — landers such as Perseverance never enter orbit at all. They aim the B-plane at an entry corridor and hit the atmosphere straight from the hyperbola. At $v_\infty = 2.56$ km/s the inertial speed at 125 km altitude is $\sqrt{v_\infty^2 + 2\mu/r} \approx 5.6$ km/s; the heat shield does the rest.

↓But there is a problem

The B-plane decides everything about an encounter: the periapsis altitude, the orbit plane, whether the spacecraft is captured, lands, or burns up. There is a fourth outcome. Aim the B vector so that the hyperbola misses the atmosphere and fire no engine at all. The spacecraft whips round the planet and leaves with exactly the speed it arrived with — relative to the planet. Relative to the Sun, its speed can change by kilometres per second. The lesson on gravity assists and Lagrange points explained why. This lesson's tools let us calculate how much, in which direction, and how to string several of them together.