---
title: Orbital Motion
module: Gravitation and Matter
moduleNumber: 6
lessonNumber: 6
order: 606
summary: >
  A circular orbit is nothing but free fall with enough sideways speed to keep
  missing the ground, and setting gravity equal to the centripetal requirement fixes
  that speed and the period once and for all. From the same energy bookkeeping we
  read off escape speed, sort orbits into bound, parabolic, and hyperbolic by the
  sign of their specific energy, and see why a tangential burn is the efficient way
  to change an orbit. We build the Hohmann transfer and its launch window, work the
  numbers for a geostationary orbit and an escape burn, and mark where finite thrust,
  perturbations, and an uncertain initial state pull a real trajectory off the ideal.
topics: [Gravitation and Matter]
draft: false
sources:
  - book: Tipler & Mosca
    ref: "Ch. 11 — Gravity; §§11-4–11-5"
---

## Circular orbits and launch energy

A circular orbit is free fall with a particular transverse speed. The only force
on an ideal satellite of mass $m$ is gravitational attraction toward a spherical
central mass $M$. For an orbit of radius $r$, gravity supplies the required
centripetal acceleration:

$$
\frac{GMm}{r^2}=\frac{mv^2}{r}.
$$

The satellite mass cancels, leaving the circular-orbit speed

$$
v_{\rm circ}=\sqrt{\frac{GM}{r}}.
$$

The speed depends on the central mass and distance from its centre. A low circular
orbit has a larger speed than a high circular orbit because gravity is stronger at
the smaller radius. Velocity is tangential, while acceleration and gravitational
force are radial. A forward-directed gravity arrow describes a propelled motion,
rather than a circular gravitational orbit.

The result assumes a spherical mass distribution, negligible atmospheric drag,
and a satellite whose mass does not alter the central body. A real Earth orbit is
perturbed by nonsphericity, atmospheric drag, solar radiation pressure, and other
bodies. The circular formula remains the reference scale against which those
perturbations are measured.

$$
% caption: Velocity is tangential to a circular orbit while gravity points inward to the central mass; setting gravity equal to the centripetal requirement $GMm/r^2=mv^2/r$ fixes the orbit speed.
\begin{tikzpicture}[>=Latex,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[fill=acc!12,draw=acc] (0,0) circle (0.45) node {$M$};
\draw[dashed,black] (0,0) circle (1.9);
\coordinate (S) at (55:1.9);
\draw[dashed,black] (0,0)--(S);
\node[black] at (0.28,0.98) {$r$};
\draw[fill=white,draw=black] (S) circle (1.9pt);
\node[above right] at (S) {satellite};
\draw[->,black,thick] (S)--($(S)+(145:1.05)$) node[above left] {$v$};
\draw[->,acc,thick] (S)--($(S)+(235:0.85)$);
\node[acc] at (0.98,0.6) {gravity};
\end{tikzpicture}
$$

**Launch paths and energy accounting.**

Orbital launch is often described as "going upward," but a low circular orbit
requires mostly horizontal speed. Reaching a high altitude with little tangential
speed produces a steep trajectory that falls back toward the central body. A
circular orbit at that altitude requires the specific kinetic energy
$GM/(2r)$ in addition to its negative potential energy. Launch vehicles therefore
turn gradually from vertical ascent toward horizontal acceleration after leaving
the densest atmosphere.

Mechanical energy tracks the final orbital state and differs from the chemical
energy carried by a rocket. During powered flight, thrust does work on the
vehicle while mass is expelled. Some energy becomes kinetic and gravitational
potential energy of the vehicle; some remains in exhaust kinetic energy; some is
lost to drag, heating, and steering. The ideal orbital expressions ignore these
processes and specify only the final mechanical state required in the central
gravity model.

An escape trajectory has no sharp outer boundary. At the threshold, speed falls
continuously toward zero as radius increases without limit. A vehicle may be on an
escape trajectory from a planet while still orbiting its star, because the energy
classification depends on which central body and reference frame are used. Mission
analysis commonly compares a spacecraft's energy relative to several bodies, then
uses a patched approximation or a full multi-body calculation when their gravity
fields overlap appreciably.

A direct speed change at the initial radius seldom minimizes the cost of transfer
between circular orbits. An intermediate ellipse places burns where velocity changes
produce the required changes in energy and angular momentum. Circular- and
escape-speed relations constrain the local states; transfer direction, burn timing,
and propellant demand require the complete trajectory.

Numerical checks help distinguish the quantities. At fixed radius,
$v_{\rm esc}$ must exceed $v_{\rm circ}$ by a factor $\sqrt2$. At larger circular
radius, speed decreases while total energy becomes less negative. A result that
predicts a higher circular speed farther from an isolated spherical mass, or a
positive energy for a circular bound orbit, has mixed a sign, radius, or reference
convention.

**Period and radius scaling.**

One orbital period is the circumference divided by the circular speed. Substituting
the speed relation gives

$$
T=\frac{2\pi r}{v_{\rm circ}}
=2\pi\sqrt{\frac{r^3}{GM}},
\qquad
T^2=\frac{4\pi^2}{GM}r^3.
$$

This is the circular-orbit form of Kepler's third law. Satellites orbiting the
same central body have $T^2/r^3$ equal to the same constant. The relation is not
a statement that every orbit is circular; elliptical-orbit period depends on the
semimajor axis instead of one fixed orbital radius. For a circular reference case,
however, it gives the correct scaling immediately: doubling radius multiplies
period by $2^{3/2}$, not by two.

Altitude is frequently confused with radius. A satellite at altitude $h$ above a
planet of radius $R$ has $r=R+h$. Using $h$ in the circular-orbit formula gives a
serious error whenever altitude is not much larger than the planet radius. A
geostationary orbit is determined by setting its period equal to the planet's
rotation period and solving for $r$; its altitude is the resulting radius minus
the planet radius.

The period relation also determines central mass. If an orbit radius and
period are observed, the central gravitational parameter is
$GM=4\pi^2r^3/T^2$. Astronomy often determines $GM$ more accurately than $G$ and
$M$ separately because orbital timing and distance measurements constrain their
product directly.

$$
% caption: With radius $r/R$ and period $T/T_0$ on the axes, the circular-orbit period grows as $r^{3/2}$: doubling the radius multiplies the period by $2^{3/2}\approx2.83$, not by two.
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\draw[acc,thick] plot[domain=0.3:3.1,samples=120] (\x,{0.5*\x^(1.5)});
\draw[dashed,black] (1.0,0)--(1.0,0.5)--(0,0.5);
\draw[dashed,black] (2.0,0)--(2.0,1.414)--(0,1.414);
\node[below] at (1.0,0) {$1$};
\node[below] at (2.0,0) {$2$};
\end{tikzpicture}
$$

> **Worked example.** A geostationary satellite has an orbital period equal to
> Earth's sidereal rotation, $T=8.616\times10^4\ \mathrm s$. With
> $GM=3.986\times10^{14}\ \mathrm{m^3\,s^{-2}}$ and Earth radius
> $R=6.37\times10^6\ \mathrm m$, find its orbital radius and altitude.
>
> Solving $T^2=4\pi^2 r^3/GM$ for the radius,
>
> $$
> r=\left(\frac{GM\,T^2}{4\pi^2}\right)^{1/3}
> =\left(\frac{(3.986\times10^{14})(8.616\times10^4)^2}{4\pi^2}\right)^{1/3}
> =4.22\times10^7\ \mathrm m.
> $$
>
> The altitude is $h=r-R=4.22\times10^7-6.37\times10^6=3.58\times10^7\ \mathrm m$,
> about $35{,}800\ \mathrm{km}$. This is a radius, measured from Earth's centre; the
> altitude above the surface is smaller by one Earth radius.

**Orbital energy and bound motion.**

Gravitational potential energy with zero at infinity is
$U=-GMm/r$. A circular orbit has kinetic energy
$K=mv_{\rm circ}^2/2=GMm/(2r)$. The total mechanical energy is therefore

$$
E=K+U=-\frac{GMm}{2r}.
$$

The negative sign identifies a bound orbit: the satellite does not have enough
mechanical energy to reach infinite separation with nonzero speed. Kinetic energy
is positive, but it is only half the magnitude of the negative potential energy.
Adding energy to a circular orbit raises the orbit's characteristic size and makes
the total energy less negative. Removing energy lowers it, although the immediate
motion after a brief tangential impulse is generally an ellipse rather than a new
circular orbit at the same location.

Specific mechanical energy, $\varepsilon=E/m$, avoids carrying satellite mass:

$$
\varepsilon=\frac{v^2}{2}-\frac{GM}{r}.
$$

In a circular orbit it becomes $-GM/(2r)$. This form applies at any point in an
ideal two-body orbit. Its sign classifies the orbit: negative for bound elliptical
motion, zero for the parabolic escape threshold, and positive for unbound
hyperbolic motion. The classification is relative to the chosen isolated central
body; a spacecraft escaping a planet can remain gravitationally bound to the Sun.

$$
% caption: A circular orbit has positive kinetic energy $K=GMm/2r$ and negative potential energy, summing to a negative total $E=-GMm/2r$ that lies below the zero-energy escape threshold, marking the orbit as bound.
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\node[acc] at (1.5,0.55) {$E$};
\node[black,below] at (4.8,1.53) {bound};
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$$

## Escape, angular momentum, and effective potentials

Escape speed is the speed at radius $r$ for which the final speed at infinity is
just zero. Setting specific mechanical energy to zero gives

$$
v_{\rm esc}=\sqrt{\frac{2GM}{r}}=\sqrt{2}\,v_{\rm circ}.
$$

The phrase "escape speed" does not mean that a rocket must instantaneously reach
that speed at ground level. It is an energy threshold in an ideal gravity-only
model. A rocket can gain energy gradually while climbing, can exploit planetary
rotation, and must also overcome atmospheric drag and gravity losses during its
finite burn. Conversely, an object given the ideal escape speed at one radius has
zero excess speed only at infinity; any additional dissipation can prevent escape.

The energy needed to move a mass from rest at radius $r$ to the ideal escape
threshold is $GMm/r$. A spacecraft already in a circular orbit has energy
$-GMm/(2r)$, so it needs only $GMm/(2r)$ of additional mechanical energy to reach
the parabolic threshold. If an impulsive tangential burn is used directly from a
circular orbit, the speed change is

$$
\Delta v=(\sqrt2-1)v_{\rm circ}.
$$

This is a velocity increment, not a statement of rocket fuel requirement. Rocket
mass changes during a burn and propulsion efficiency depends on exhaust speed; the
rocket equation gives that additional connection. The orbital-energy result
sets the ideal mechanical target against which launch and trajectory losses are
interpreted.

> **Worked example.** A spacecraft is in a low circular orbit at altitude
> $400\ \mathrm{km}$, so $r=6.77\times10^6\ \mathrm m$ with
> $GM=3.986\times10^{14}\ \mathrm{m^3\,s^{-2}}$. What single prograde impulse puts it
> on an escape trajectory?
>
> Its circular speed is
>
> $$
> v_{\rm circ}=\sqrt{\frac{GM}{r}}
> =\sqrt{\frac{3.986\times10^{14}}{6.77\times10^6}}=7.67\ \mathrm{km\,s^{-1}}.
> $$
>
> Escape needs $v_{\rm esc}=\sqrt2\,v_{\rm circ}$, so the impulsive increment is
>
> $$
> \Delta v=(\sqrt2-1)\,v_{\rm circ}=(0.414)(7.67)=3.18\ \mathrm{km\,s^{-1}}.
> $$
>
> Escaping from orbit costs far less than the $v_{\rm circ}$ already invested,
> because the orbit is already halfway (in energy) to the escape threshold.

**Launch direction and orbital angular momentum.**

An orbital burn changes both the magnitude and direction of velocity. In a
circular orbit, a prograde tangential burn increases both mechanical energy and
angular momentum. The immediate path is an ellipse with the burn point at its
pericentre; the opposite side of the new orbit lies at larger radius. A retrograde
burn decreases energy and angular momentum, placing the burn point at apocentre
of an ellipse whose pericentre is lower. These statements follow from the fact
that velocity is tangential at a circular orbit and gravity is central.

Radial and tangential impulses have different effects. A purely radial impulse
changes the direction of the velocity and produces an ellipse without initially
changing angular momentum about the central body. A tangential impulse changes
angular momentum most efficiently because it acts perpendicular to the radius.
Real launches combine vertical ascent through dense atmosphere with a gradual
turn toward the local horizontal. Once orbital altitude is approached, the needed
state is primarily transverse speed, not continued upward motion.

Angular momentum per unit mass is $h=|\vec r\times\vec v|$. In a central
gravitational field it is conserved between burns. At the nearer and farther ends
of an elliptical orbit, velocity is perpendicular to radius, so $r_pv_p=r_av_a$.
The speed is larger at pericentre and smaller at apocentre. This relation checks
transfer-orbit sketches: a raised apocentre following a prograde burn
does not imply that the spacecraft is slow immediately after the burn; it becomes
slow only after climbing outward.

Launch site also affects the initial state. A rotating planet gives an eastward
surface velocity that can contribute to an eastward launch, reducing the required
vehicle-provided inertial velocity. The contribution depends on latitude and
desired orbit plane. This is an energy and angular-momentum contribution from the
planet's rotation; it does not change the gravity-only circular-orbit speed at a
given radius.

$$
% caption: A prograde tangential burn at one point of a circular orbit turns it into an ellipse whose opposite side (apocentre) is raised; the burn increases both energy and angular momentum.
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\node[black,below] at (0,-1.4) {circular orbit};
\node[black,left] at (-2.2,0) {apocentre};
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$$

**Effective potential and radial turning points.**

Central-force motion can be separated into radial and transverse parts. With
angular momentum $L$ fixed, the energy can be written

$$
E=\frac12m\dot r^2+U_{\rm eff}(r),
\qquad
U_{\rm eff}(r)=\frac{L^2}{2mr^2}-\frac{GMm}{r}.
$$

The first term in the effective potential is associated with transverse motion.
It rises sharply at small radius when angular momentum is nonzero, while the
gravitational term is negative and falls in magnitude as radius increases. A
circular orbit occurs at the minimum of $U_{\rm eff}$ for the chosen $L$.
Small radial displacements around that minimum oscillate between a pericentre and
an apocentre, producing a bound elliptical orbit in the ideal inverse-square
model.

Turning points occur where $E=U_{\rm eff}$ and radial speed is zero. An energy
line above the minimum intersects the curve twice for a bound orbit. At the inner
intersection the spacecraft is at pericentre; at the outer intersection it is at
apocentre. A horizontal energy line at zero is not the same as the escape
threshold on an effective-potential plot unless the angular momentum contribution
has also been retained. The full energy and angular momentum together specify the
orbit geometry.

The effective potential is a mathematical reduction of the radial motion. It maps
the radial part of a two-dimensional orbit onto a one-dimensional energy diagram.
It also clarifies why a satellite with nonzero angular momentum does not fall
radially into the central mass under the ideal point-mass model: the allowed radial
motion is constrained by the angular-momentum term.

$$
% caption: The effective potential $U_{\rm eff}=L^2/2mr^2-GMm/r$ has a minimum at the circular orbit; a bound energy line $E$ meets it at the pericentre $r_p$ and apocentre $r_a$ that bound the radial motion.
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\node[below] at (3.72,0.3) {$r_a$};
\end{tikzpicture}
$$

## Transfers, periods, and orbital measurement

A Hohmann transfer is an ideal two-impulse change between coplanar circular orbits
of radii $r_1$ and $r_2$ around the same central mass. In an outward transfer, the
first prograde burn changes the initial circle into an ellipse tangent to both
the initial and final circles. Its semimajor axis is

$$
a_t=\frac{r_1+r_2}{2}.
$$

The transfer ellipse has specific energy $-GM/(2a_t)$. The speed at each end can
be found from the vis-viva relation
$v^2=GM(2/r-1/a_t)$. At the lower-radius end, the transfer speed exceeds the
initial circular speed; the first burn is prograde. At the higher-radius end, the
transfer speed is less than the final circular speed; a second prograde burn makes
the orbit circular. Reversing the order gives an inward transfer with retrograde
burns.

The transfer time is half the period of the transfer ellipse:

$$
t_t=\pi\sqrt{\frac{a_t^3}{GM}}.
$$

The final body must be at the arrival point when the spacecraft reaches apocentre.
Transfer geometry requires both the energy change and the correct initial phase. A transfer
burn aimed at the final orbit's current location generally misses because the
target continues moving during the transfer time.

Hohmann transfers are idealized benchmarks. They neglect inclination changes,
atmospheric drag, noncircular starting orbits, other gravitational bodies, and
finite burn duration. They are efficient for many coplanar radius changes because
the burns occur at the turning points of the transfer ellipse, where tangential
speed changes modify orbital energy and angular momentum. The resulting $\Delta v$
is a benchmark for those assumptions; mission time and propellant use change when
the constraints differ.

$$
% caption: A Hohmann transfer is an ellipse tangent to both circular orbits; the first prograde burn at radius $r_1$ raises the apocentre to $r_2$, and the second burn there circularizes the orbit.
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\node[black] at (0,-0.72) {$r_1$};
\node[black] at (0,-1.72) {$r_2$};
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$$

**Elliptical periods and orbital measurement.**

An ideal bound orbit is an ellipse with the central mass at one focus. Its
semimajor axis $a=(r_p+r_a)/2$ sets the period,

$$
T=2\pi\sqrt{\frac{a^3}{GM}}.
$$

The period therefore depends on the semimajor axis, not separately on eccentricity
when the central mass is fixed. A highly elongated ellipse and a circular orbit
with the same semimajor axis have the same period in the ideal two-body model.
Their speeds and distances vary very differently during the orbit, however, so
their observations require position and timing information rather than period
alone.

Orbit determination combines measurements of direction, range, range rate, or
timing with a gravitational model. Repeated position observations constrain the
orbital plane and apparent ellipse; timing constrains mean motion; Doppler shift
or range-rate data constrain velocity along the line of sight. The inferred state
is then propagated forward and compared with later observations. Residuals can
reveal atmospheric drag, nonspherical gravity, manoeuvres, or measurement bias.

Uncertainty grows when observations cover only a short arc or have similar viewing
geometry. Several different ellipses can fit a small cluster of direction-only
measurements. A long time baseline, varied viewing locations, calibrated timing,
and independent range information reduce this ambiguity. The final orbit estimate
should state its reference frame, epoch, central-body parameter, and whether it is
a two-body fit or includes perturbations.

### Limits of an ideal orbital estimate

The two-body orbit is a baseline, not a complete tracking model for every
satellite. A low orbit can lose energy to atmospheric drag, producing a gradually
shrinking semimajor axis. A nonspherical central body changes the orientation of
the orbital plane and line of apsides. Third-body gravity, radiation pressure, and
planned manoeuvres can add measurable departures. The relevant question is not
whether these effects exist, but whether their accumulated change over the
observation interval exceeds the required position, timing, or velocity accuracy.

An orbit fit should not absorb a systematic effect silently into its initial
conditions. If residuals show a repeated trend with altitude, solar orientation,
or elapsed time, a perturbation term or empirical acceleration may be needed.
Conversely, adding many adjustable terms to a short data arc can fit noise without
improving prediction. A withheld observation interval or an independent tracking
method tests whether the inferred orbit has predictive value.

The measurement frame must also be explicit. Ground-based directions are tied to
the rotating Earth, whereas the central-force equations use an inertial reference
frame to a high approximation. Time tags, station coordinates, and transformation
conventions are therefore part of an orbit result. With those quantities stated,
period, energy, and trajectory parameters can be compared across observers.

For every numerical prediction, retain enough significant digits in intermediate
state vectors and report the final uncertainty with the chosen reference epoch.

## Hyperbolic escape and orbit classes

An object with positive specific energy relative to a central body follows an
unbound hyperbolic trajectory in the ideal two-body model. Far from the body, the
gravitational potential approaches zero and the remaining speed approaches the
hyperbolic excess speed $v_\infty$. The specific energy is then
$\varepsilon=v_\infty^2/2$. A parabolic escape is the limiting case
$v_\infty=0$, with just enough energy to reach infinite separation at zero speed.

This distinction matters for arrival as well as departure. A spacecraft arriving
at a planet with nonzero $v_\infty$ is not captured by the planet merely because
it passes nearby. A burn, atmospheric interaction, or gravitational interaction
with another body must remove enough planet-relative energy to make the specific
energy negative. A close approach can bend the hyperbolic path substantially while
leaving its planet-relative energy unchanged in the ideal gravity-only model.

The speed at radius $r$ on an arrival or escape path follows the vis-viva form

$$
v^2=v_\infty^2+\frac{2GM}{r}.
$$

Near the planet, gravitational acceleration raises speed above its distant value.
This gravitational focusing makes a target cross section larger than its physical
area for objects arriving from far away. It also means that a capture burn near
pericentre can change orbital energy strongly for a given velocity change, because
the spacecraft is moving rapidly there. The effect is a consequence of the energy
and angular-momentum geometry, not a violation of conservation of energy.

The central-body reference must remain explicit. A spacecraft may have positive
energy relative to a planet and negative energy relative to the Sun. Mission
design changes between these regimes by choosing encounter geometry, burn timing,
and the reference state used to quote $v_\infty$. Confusing local escape speed
with interplanetary excess speed produces incorrect arrival and capture estimates.

$$
% caption: A hyperbolic path has positive specific energy and leaves with a nonzero excess speed $v_\infty$ far from the central mass; the parabolic path is the zero-excess limiting case that just barely escapes.
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$$

**Specific energy, speed, and orbit classes.**

Specific mechanical energy compares trajectories at the
same radius. At fixed $r$, increasing speed raises
$\varepsilon=v^2/2-GM/r$. Three speeds organize the possibilities. A speed below
circular speed at a given point can still belong to an ellipse if that point is
near apocentre. Circular speed gives the unique circle at that radius. Escape speed
sets zero energy. Speeds above escape correspond to positive-energy hyperbolic
motion in the ideal central-field model.

The labels apply to the complete state, not to speed alone without position. A
satellite at a large radius can move more slowly than one at a small radius while
having higher total energy because its potential energy is less negative. Likewise,
two elliptical orbits can cross at one radius with different speeds and different
angular momenta. Energy and angular momentum together determine the conic shape;
one scalar energy value does not determine the orientation or eccentricity.

Specific energy removes spacecraft mass from the orbital part of a calculation. A propulsion system supplies a
velocity change, but the associated energy change depends on the velocity at which
the burn occurs. For a small tangential impulse $\Delta v$, the leading energy
change per unit mass is $v\Delta v$. A burn at high speed near pericentre can
therefore alter the far side of an orbit more strongly than the same impulse near
apocentre. This is often called the Oberth effect in mission analysis; it follows
directly from the kinetic-energy term.

Mission design also requires angular momentum, orbital-plane, phase, and arrival-
direction constraints. A trajectory with favourable energy can miss a target when
its phase or arrival direction is wrong. Energy diagrams belong alongside geometry
and time-of-flight relations.

## Turning points and trajectory targeting

Radial turning points are locations where radial speed is zero, not locations
where the full velocity vanishes. At pericentre and apocentre, an ideal orbital
velocity is tangential. The radial coordinate reverses direction, while the
spacecraft continues around the central mass. This distinction matters when a
mission description says that a spacecraft "stops rising" at apocentre: its
outward radial motion stops, but its tangential speed and angular momentum remain.

At fixed specific angular momentum $h$, the radial equation is

$$
\frac12\dot r^2+\frac{h^2}{2r^2}-\frac{GM}{r}=\varepsilon.
$$

The intersections of the energy line with the effective potential identify the
allowed radial interval. A targeting burn changes both $\varepsilon$ and usually
$h$, moving the turning points. A small error in burn magnitude or direction can
therefore produce a position error that grows over the coast interval. The growth
is especially evident near a distant apocentre, where a small change in orbital
period changes arrival time and target phase.

Trajectory correction uses later measurements to estimate this state error and
apply a smaller burn before it becomes expensive. The correction is not chosen
from position error alone. Velocity error, time to encounter, measurement
uncertainty, and remaining propulsion all enter. A correction made early often
requires less velocity change because it has more time to alter the eventual
trajectory; a late correction may need a large impulse to move the same arrival
point.

The ideal two-body turning-point picture gives an error check. A bound ellipse
has two positive radii with $r_p<r_a$. A calculated transfer that
has an apocentre lower than its pericentre after a stated prograde raising burn has
reversed a sign, selected the wrong burn direction, or mixed reference radii.

**Mission errors and measurement limits.**

Mission estimates begin with imperfect observations. A tracking station measures
some combination of direction, range, range rate, or timing, each with its own
noise, bias, and reference-frame transformation. A direction measurement alone
constrains a line of sight but not distance. A range measurement constrains one
dimension strongly but needs timing and station position. Combining varied
measurements over time reduces ambiguity because orbital motion changes the
viewing geometry.

Burn execution also has errors. Thrust magnitude, burn duration, pointing, vehicle
mass, and timing determine the achieved velocity increment. A small pointing error
splits an intended prograde burn into tangential, radial, and normal components.
The tangential component changes energy, the radial component shifts orbital phase
and shape, and the normal component tilts the orbital plane. Recording the actual
burn estimate supports later trajectory reconstruction.

An uncertainty ellipse or covariance matrix represents coupled position and
velocity uncertainty. Propagating it forward usually stretches the uncertainty
along the direction of motion, but the exact shape depends on orbit geometry and
measurement schedule. A predicted close approach should report a time window and
position uncertainty with its nominal trajectory. When a correction
burn is planned, its expected effect must be compared with this uncertainty; a
burn smaller than the state-estimation error may not improve the mission outcome.

Measurement limits are also model limits. Ground observations require atmospheric
refraction and timing corrections at high precision. A two-body fit can be
inadequate over long arcs because drag or nonspherical gravity accumulates. An
instrument residual may identify a physical perturbation, a calibration bias, or
a data-processing mistake. Independent tracking methods and withheld observations
separate these possibilities more reliably than repeated fitting of the same data.

### Scope of an arrival prediction

An arrival prediction is conditional on the force model and the time span over
which it is propagated. A close encounter quoted months in advance can be changed
by a small unmodelled acceleration that would be irrelevant over one orbit. The
prediction should therefore identify the central bodies included, the reference
epoch, the last tracking update, and the uncertainty interval used for the arrival
state. A nominal path without these conditions is only a visual summary, not a
complete mission estimate.

Trajectory estimates also require operational margins. A planned correction burn
needs time for navigation, command transmission, execution, and verification. The
available margin is reduced by pointing uncertainty, finite thrust duration, and
remaining propellant. Separating physical trajectory uncertainty from operational
execution uncertainty makes a predicted correction both testable and actionable.

State estimates should retain position, velocity, and their covariance together;
rounding one component independently can create a trajectory inconsistent with the
reported energy and angular momentum.

Every published estimate should include units, epoch, and stated confidence limits.

## Launch windows and navigation corrections

A coplanar circular transfer changes orbit radius while keeping the central mass
and orbital plane fixed. The standard two-impulse construction uses an ellipse
tangent to the departure circle at one end and the arrival circle at the other.
In an outward transfer, the departure burn is prograde and occurs at the lower
radius. It raises the opposite turning point to the destination radius. The
arrival burn, also prograde, occurs at the transfer apocentre and raises the local
speed from the transfer value to the final circular speed. Inward transfers reverse
the signs: the first and second burns are retrograde.

The transfer geometry is fixed by the two circular radii. Its semimajor axis is
$a_t=(r_1+r_2)/2$, and its coast time is half an elliptical period,

$$
t_t=\pi\sqrt{\frac{a_t^3}{GM}}.
$$

During the coast, the destination body advances around its own circular orbit. The
transfer time therefore determines the launch window: the departure time at which the
target's initial angular lead has the value required for both bodies to reach the
common arrival point together. For an outward transfer, the outer target normally
begins ahead of the departure point; for an inward transfer, the required phase
relation has the opposite sense.

The phase angle follows from angular rates and transfer time. If the initial and
target circular angular speeds are $n_1$ and $n_2$, a coplanar calculation sets
the target's initial phase so that its angle after $t_t$ equals the arrival
longitude of the transfer ellipse. The calculation must use inertial angles in a
consistent direction. Drawing both planets at their current positions and aiming
at the target's present location ignores the target motion and generally produces
an incorrect departure date.

Each delta-v component is obtained from local speed differences, not from a change
in orbital radius alone. The circular speeds are $\sqrt{GM/r_1}$ and
$\sqrt{GM/r_2}$. The transfer speeds at the two tangency points follow from the
vis-viva equation. Thus the outward components are

$$
\Delta v_1=v_{t,p}-v_{c,1},
\qquad
\Delta v_2=v_{c,2}-v_{t,a}.
$$

Both are positive magnitudes for the ideal outward case. Their sum is the ideal
coplanar impulsive delta-v. It is not the propellant mass, which additionally
depends on spacecraft mass change and effective exhaust speed.

The first burn adds energy at the lower radius, where speed is large; the second
burn changes the orbit shape into the final circle at the higher radius. The two
components need not have the same size. A transfer to a much higher orbit may have
a substantial first burn and a smaller circularization burn, whereas a modest
radius change can have components of similar scale. The energy change, angular
momentum change, and phase requirement should all agree with the geometry before
numerical values are accepted.

The phase relation is often more restrictive operationally than the two burns.
If a mission misses the ideal departure date, it may wait for the next alignment,
use a transfer with a different time of flight, or spend additional delta-v to
change the trajectory. A faster transfer can require more energy; a slower
transfer can require less propulsive effort in some multi-body settings but may
increase exposure to perturbations or operational constraints. The ideal Hohmann
solution is therefore a benchmark, not an automatic schedule.

$$
% caption: The launch window is set by the target's phase lead: while the craft coasts half the transfer ellipse from departure to arrival, the target advances along its own orbit to the shared arrival point.
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\draw[dashed,black] (0,0) circle (2.05);
\draw[acc,thick] (-0.5,0) ellipse (1.55 and 1.45);
\draw[fill=white,draw=black] (1.05,0) circle (1.9pt);
\node[black,below right] at (1.05,0) {departure};
\draw[fill=white,draw=black] (-2.05,0) circle (2.0pt);
\node[left] at (-2.05,0) {arrival};
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\node[black] at (-1.15,2.28) {target now};
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$$

The construction assumes instantaneous burns, circular coplanar starting and
ending orbits, one dominant central mass, and negligible drag. An inclination
change introduces a separate velocity-vector cost. A finite-duration engine burn
spreads the impulse over changing position and velocity. Nearby bodies, planetary
atmospheres, and nonspherical gravity modify the coast path and phasing. These
effects set the corrections required when benchmark accuracy is insufficient for a
mission.

For numerical work, state the radii from the central mass, the gravitational
parameter, the reference epoch, the sense of motion, and whether delta-v values
are signed or reported as positive magnitudes. Check that the transfer time grows
with the chosen semimajor axis, that the outer target advances more slowly than an
inner departure body, and that an outward first burn is prograde. Those checks
catch most sign, phase, and radius mistakes before they propagate into a launch
window estimate.

**Navigation errors and correction maneuvers.**

An orbital prediction begins with a state vector containing position and velocity
at a stated epoch in a stated reference frame. Tracking data determine an estimate
together with uncertainty and correlation between its
components. A position estimate without its associated velocity uncertainty is
incomplete, because even a small velocity error changes the predicted position
after a coast. The uncertainty is best treated as a coupled state quantity rather
than as independent error bars placed separately on a plotted orbit.

Error growth has distinct directions. An along-track error changes where the
spacecraft is expected to be along its orbit. It is often driven by a small error
in orbital energy or period and can accumulate steadily with time. A radial error
changes the distance from the central body and can alter the local gravitational
acceleration and observed angular rate. A cross-track error changes the orbital
plane or the position normal to it, commonly through a pointing error or an
unmodelled normal acceleration. The directions can couple as the orbit evolves,
but separating them first helps diagnose the source of a navigation residual.

Covariance describes the shape and orientation of the uncertain state region. A
long narrow uncertainty region can arise when range is well measured but range
rate is not, or when repeated observations have nearly the same viewing geometry.
Forward propagation produces an arrival corridor. The corridor can rotate and
stretch because the orbital dynamics map an
initial velocity difference into a growing position difference. Reporting only a
nominal future trajectory hides this dependence on the observation history.

| uncertainty direction | strongest tracking information | principal propagated effect | correction component |
| --- | --- | --- | --- |
| along track | timing and range rate | arrival-time and orbital-phase error | tangential $\Delta v$ |
| radial | range and local angular-rate geometry | energy and orbit-shape error | radial $\Delta v$ |
| cross track | angular observations from varied stations | plane, node, or encounter-plane error | normal $\Delta v$ |
| burn execution | post-burn Doppler and range | bias in every commanded component | state update from new tracking |

The components are coupled by propagation, but the decomposition identifies which
measurement and maneuver direction address the dominant arrival error.

A correction maneuver changes velocity, and its effect depends on where and when
it is applied. An early small burn can alter a distant encounter substantially
because the resulting state difference acts throughout the remaining coast. A late
burn may need much larger delta-v to shift the same arrival point. The direction
also matters: a tangential component primarily changes energy and timing, a radial
component alters orbital shape and phase, and a normal component changes plane.
Correction design uses a sensitivity calculation that maps each candidate burn
component into a predicted change at the target condition.

Burn sensitivity is limited by execution uncertainty. Thrust magnitude, duration,
pointing, spacecraft mass, and attitude knowledge determine the achieved impulse.
A commanded burn can be known less accurately than the tracking state it is meant
to correct. Navigation teams therefore estimate the post-burn state from new
tracking rather than assuming the command was executed perfectly. A correction
that is too small relative to burn execution error may add uncertainty instead of
reducing it.

Tracking cadence follows the rate at which uncertainty grows and the time needed to
act on new data. Dense measurements are warranted near a close approach, after a
burn, or when atmospheric drag and other perturbations change rapidly.
They are less valuable when successive observations have nearly identical geometry
or share a common calibration bias. Range, Doppler, angle, and optical data have
different strengths; combining them can constrain state components that any one
method leaves ambiguous. The interval between observations must be recorded with
the measurement precision and the station or sensor geometry.

Two-body predictions are limited by the force model. Atmospheric drag changes low
orbits, planetary oblateness rotates orbital elements, and third-body gravity can
matter during long coasts or near an encounter. Solar radiation pressure, attitude
changes, and small thrust leaks can produce accelerations below a single tracking
measurement yet large enough to matter after propagation. A residual pattern that
grows systematically with time calls for a revised dynamical model or an estimated
additional acceleration. Enlarging a position-error bar alone does not represent
the cause.

| operational decision | quantitative trigger | retained evidence |
| --- | --- | --- |
| obtain another tracking arc | projected corridor exceeds a target-plane tolerance | station geometry, time tags, range, Doppler, and angles |
| update the force model | structured residuals persist across calibrated observations | residual components, candidate acceleration, fit comparison |
| execute a correction | projected target error exceeds the available execution margin | state covariance, maneuver sensitivity, remaining $\Delta v$ |
| verify a maneuver | post-burn state differs from the commanded state | thrust telemetry, range rate, attitude, epoch |

The trigger is defined at the mission quantity of interest, such as arrival time,
plane crossing, or encounter distance. A scalar position error alone does not set a
correction threshold.

Operational constraints set another limit. A planned burn needs communication
time, attitude preparation, safe thrust conditions, and a reserve for later
correction. The available delta-v is finite, so a late solution that is physically
possible may still be operationally unacceptable. A navigation product should
therefore state the state epoch, force model, tracking data used, uncertainty
representation, correction sensitivity, and the time horizon over which the
prediction is valid. These details permit review of an arrival estimate against the
trajectory data and assumptions.

State estimation should distinguish random measurement scatter from persistent
bias. A clock offset, station-coordinate error, or uncorrected atmospheric delay
can shift many observations in the same direction and persist through ordinary averaging.
Filtering methods combine a prior propagated state with new measurements, but their
reported covariance is credible only when the assumed measurement and force-model
errors represent the actual system. Comparing independent tracking sources and
examining innovations after each update provide practical checks on that claim.

Correction decisions are made against a target plane, encounter time, or orbital
element tolerance rather than against an abstract position error. A radial error
may be harmless at one epoch and critical at another when projected onto the
arrival condition. This projection, together with remaining delta-v and execution
margin, determines whether to correct immediately, obtain more tracking, or accept
the predicted dispersion.