** What I know about orbital mechanics is restricted to
the fundamentals of elliptic orbits as simple two-body problems, out of a topic that is far wider.** These can be computed from a few equations
rather easily. Solutions can be combined
with an “energy trick” to estimate interplanetary delta-vee requirements for
vehicle design,

__but this is most definitely inadequate for navigation purposes__.

The two-body problem involves a “primary” body whose mass
far-and-away overwhelms the mass of the “secondary” body, by multiple orders of magnitude. An example would be satellite moving about
the Earth, or a moon moving about its
planet. Those are closed orbital paths.

The shape of the closed orbital path is the ellipse from
high school geometry, with the primary
located at one focus, and the other
focus unoccupied. The secondary body is
located somewhere along that ellipse, at
a radius vector r from center-of-primary to center-of-secondary. See Figure 1.

Figure 1 – Basic Elliptic Orbit Geometry

The longer semi-major axis length is “a”, the shorter semi-minor axis length is
“b”, and the focal distance from ellipse
center to focus is “c”, just as in high
school geometry. The ratio of c to a is
the eccentricity “e” of the ellipse,
which cannot quite reach 1 and still be an ellipse, and which becomes zero as the path becomes a
circle, with both foci located exactly
at the center. The figure is symmetrical in both directions. There is a relationship among these variables
that holds for this geometry:

a^{2} = b^{2} + c^{2}

At nonzero eccentricity e,
the ellipse is located off-center,
with the primary at one focus.
Thus there is a larger radius “r_{apo}” to the “apoapsis”
(farthest point) of the ellipse from the primary, and a smaller radius “r_{per}” to the
“periapsis” (nearest point) of the ellipse from the primary. These are related to known values of a and e
as follows:

r_{apo} = a(1 + e)

r_{per} = a(1 – e)

If instead you know the max and min orbit radii, you can find the a and e values as follows:

a = (r_{apo} + r_{per})/2

e = (r_{apo} – r_{per})/(r_{apo} + r_{per})

Note also that altitude h of the secondary body above the
primary body surface is the radius length to the secondary, minus the radius of the primary itself. For an orbit in (or very near) the equatorial
plane of the primary, you would use its
equatorial radius. For a strongly
inclined orbit, the average radius would
be more representative.

h = r – R_{primary}

The speed V of the secondary body (with respect to the
primary) varies around this orbit, with
the radius distance, in a well-defined
way. The orbit also has a definite and
repeatable period T, meaning the time
required to go fully around the orbit once.
Knowing that G is the universal gravitation constant, and M is the mass of the primary body, those quantities (using SI metric units) are
governed by:

V^{2} = GM (2/r – 1/a)

T = 2 pi (a^{3}/GM)^{0.5} where pi is the familiar approximate
3.141592654 value

Position around the orbit as a function of time is __not__
an easy thing to calculate. The
fundamental notion dates to Kepler’s time:
** equal areas are swept out by the radius vector r in equal
intervals of time**. In the old
pencil-and-paper days, we would draw the
orbit to scale on paper, and measure the
desired swept-out area with a planimeter.
We would calculate the whole ellipse area (or measure it with the planimeter). The ratio of swept-out area to ellipse area, multiplied by the period, is the time to the location of the radius
vector, usually measured from the
periapsis or apoapsis point.

Today one would use the equation of the ellipse in
appropriate coordinates, and integrate
the appropriate areas under the curve, plus subtracting the appropriate triangle area, to accomplish precisely the same thing as was
done in the old days with a planimeter.
The equation of an ellipse, with
its major axis aligned with the coordinate system x axis, is known from high school geometry to be:

[(x-h)/a]^{2} + [(y-k)/b]^{2} = 1 where (h,k) is the set of coordinates of the
center of the ellipse

The other two rather-useful things also easily calculated
are the circular orbit (about the primary) and escape velocity speeds, evaluated at any desired radius r from
center-of-primary. The speeds differ by
a factor of square root of two. Those governing
equations use SI metric units, and are
as follows:

Vcirc^{2} = GM/r

Vesc^{2} = 2 GM/r

For reference, here
is the SI-units value of G: 6.6732 x 10^{-11}
N-m^{2}/kg^{2}

**Terminology**

Noting that apoapsis is the farthest point from the primary
(as denoted by the “apo”) and periapsis is the nearest point to the primary (as
denoted by “peri”), be aware that the
nomenclature “apsis” is generic. For
orbits about the Earth, we customarily
use “gee” for “apsis”, resulting in
“apogee” for the farthest point and “perigee” for the nearest point. For orbits about the sun we use “helion” for
“apsis”, resulting in “apohelion” and
“perihelion”. For orbits about the
moon, we use “lune” for “apsis”, resulting in “apolune” and “perilune”. I have seen “jove” used for “apsis” for
orbits about Jupiter. But the generic
“apsis” cannot be misunderstood,
regardless of context.

**Changing the Orbit You Are In**

There are four things of general interest: (1) changing the shape and size of the
ellipse you are in, (2) changing the
plane of your orbit, (3) deorbiting from
an elliptical orbit, and (4) deorbiting
from an already-circular orbit. The
first 3 are covered by Figure 2, and the
last one by Figure 3.

If you are in an elliptical orbit, the best (most efficient) places to do a burn
in order to change the shape or size of the orbit, are the apoapsis or periapsis. ** The radius to the point where you make
the burn does not change, the other one
does, as illustrated in the figure.** In general,
you know the changes in apoapsis or periapsis radii that you
desire, so you refigure the apoapsis and
periapsis speeds for the new orbit from the known and new radius values, as appropriate. The

__change__in speed at the location of your burn, is the unfactored kinematic delta-vee (dV) for making that orbital change.

Figure 2 – Changing Your Elliptic Orbit

__To
emphasize__: if you want to raise or lower your
apoapsis, you make the orbit-changing
burn at the periapsis of your current orbit.
If you want to raise or lower your periapsis, you make the orbit-changing burn at the
apoapsis of your current orbit.

If you
burn posigrade (in the direction of current motion), you will raise the altitude of the other end
of the orbit. If you burn retrograde
(against the direction of current motion),
you will lower the altitude of the other end of the orbit.

You have new r_{apo} and r_{per} values (one
unchanged where you burn, the other the
desired new value). You refigure “a” and
“e” from these r_{apo} and r_{per} values using the equations
already given. Then you refigure the
speeds at apoapsis and periapsis, using
the velocity equation already given.
Then you refigure period, using the
equation already given. The difference
in velocity values at your burn point is the delta-vee (dV) for the burn, assuming it is “impulsive” (which for early 21^{st}
century electric propulsion, it is not!!!). If the burn is “impulsive”, then it does not require factoring up for any
gravity and drag losses. __That
impulsive-burn presumption is the default for this article__.

** Orbital plane changes** are a bit
different. This is merely a vector
problem such that speed is unchanged,
but

__direction is changed__. The delta-vee (dV) calculated for a plane change angle of Δi depends directly upon the magnitude of the orbital velocity at the point where you make the burn. Therefore, this minimizes when your orbital velocity is minimum. That happens at apoapsis, so

*the best, most efficient location to make a plane change in an elliptic orbit is an impulsive burn at apoapsis.*The formula is given in the figure, where the illustration shows initial and
final velocities of equal magnitude, and
an angle bisector that is perpendicular to the dV vector. Whether you measure angles in degrees or
radians is irrelevant, as long as you
are consistent with how you figure the sine function of the indicated
half-angle. Be aware that plane changes
through significant angles always cost large dV.

** Deorbiting from an elliptical orbit** may (or
may not) match up very well with a desired location for landing.

**It does not have to be exactly a surface-grazing transfer orbit, just close. If the primary has an atmosphere, your transfer periapsis needs to be at, or under, about half the accepted “interface altitude” for entry. If the primary is airless, it needs to be at, or under, the surface-grazing point.**

*What you are doing is lowering the periapsis altitude to one that more-or-less grazes the surface of the primary.*The calculation procedure is the same as the other in-plane
orbital changes. You are going to make
your burn at apoapsis, with the
surface-graze point at the new periapsis.
You know the original apoapsis radius,
which will not change. The new
periapsis radius is the surface graze point,
or thereabouts, as
applicable. You refigure the new “a” and
“e” values, then the new V_{apo}
and V_{per} values, using the
equations already given. The change in
apoapsis velocities is the deorbit burn dV value, which should need no factoring, if an “impulsive” burn.

The only way to make the desired location of the landing
match up, is to wait until the location
of the apoapsis of you initial elliptic orbit matches up with the rotating
geometry of the primary body beneath,
such that the desired landing location will rotate under your arrival
trajectory (your periapsis), just as you
get there. This is a window-timing
problem that __does not exist with circular orbits__, which do not have a definite apoapsis or
periapsis. From circular, you just time the burn to hit the target.

** Deorbiting from a circular orbit** is far
simpler. Because the velocity is constant
around the orbit, you can easily time
your deorbit burn to hit exactly what target you desire on the surface, there is no need to wait for the apoapsis to
be in the “right place”. The transfer
orbit is merely an approximately-surface-grazing ellipse from your circular
orbit, as shown in Figure 3.

Figure 3 – Deorbiting From a Circular Orbit

If the primary has an atmosphere, your transfer orbit periapsis needs to be at
or under about half the accepted entry interface altitude. If the primary is airless, the transfer orbit periapsis needs to be at
or at most slightly under the surface-grazing ellipse periapsis. The transfer orbit apoapsis is the distance
at which your circular orbit already exists.

Again, with known
apoapsis and periapsis radii for the transfer orbit, you use the equations already given to figure
“a” and “e”, then V_{apo} and V_{per}, and the period T. The difference between your circular orbit
speed and the transfer orbit V_{apo} is the unfactored dV for your
deorbit burn. If “impulsive”, that dV needs no factoring.

Half the period of the surface-grazing transfer orbit is a
pretty good estimate of the time from deorbit burn to landing. You simply time your deorbit burn to make the
surface graze point your desired landing location. If the body has an atmosphere, your trajectory will decelerate-and-“droop”
to landing a bit early.

**Interplanetary Transfer Orbits**

There are two basic classes of interplanetary transfer
orbits: min-energy Hohmann ellipses, and faster transfer ellipses. The delta-vee requirements are lower with
min-energy Hohmann, and higher with
faster transfer ellipses. ** Min-energy
Hohmann transfer ellipses are tangent to both the departure planetary
orbit, and to the arrival planetary
orbit**. That tangency assures that
planetary motion and spacecraft motion are parallel, such that arithmetic calculations suffice to estimate
delta-vees.

** The faster trajectories to outer planets all have
perihelion points located at the Earth’s orbit**, and so are tangent there, for an arithmetic calculation of departure delta-vee. The transfer aphelion points are beyond the
orbit of the destination planet, so that

__full vector subtraction__is necessary to determine delta-vees at arrival. This involves determination of both speed and direction at arrival.

The point of a faster transfer trajectory is to shorten
travel time. Therefore it makes little
sense to put the outer planet transfer trajectory’s perihelion inward of
Earth, as that puts some of the fastest
travel speeds out of consideration.

** The faster trajectories to inner planets all have
perihelion points located at the orbit of the destination planet.** The transfer apohelion is located outward of
Earth’s orbit. That puts all the higher available
speeds on the transfer orbit, for minimum
travel time. Arrival delta-vees become
arithmetic calculations due to the tangency,
while departure delta-vees become full vector calculations because of
the lack of tangency.

*Min-energy**Hohmann Transfer*

The fundamental idea for planetary destinations outward of
Earth is to put the transfer orbit perihelion at the distance of the Earth from
the sun, and to put the transfer orbit
apohelion at the destination planet’s distance from the sun. For destinations inward of Earth (those being
Venus and Mercury), one puts the
transfer apohelion at Earth’s distance,
and its perihelion distance at the destination planet’s distance. ** The orbit always has Earth-at-departure
on one side of the transfer orbit, and
arrival-at-destination on the other side of the transfer orbit.** That is inherent to Hohmann transfer. Thus the one-way trip time is

__always__half the transfer orbit period. See Figure 4.

Figure 4 – Min Energy Hohmann Transfer Orbits

While simple enough in concept, there is a complication: planetary orbits about the sun are
elliptical, not circular. Some are more eccentric ellipses than
others. Accordingly, there are minimum and maximum distances from
the sun, for Earth, and for any destination planet. The higher delta-vee requirements are
generally associated with the larger transfer orbits, in turn associated with max distances of
Earth and destination planet from the sun.

Minimum delta-vee is associated with min planetary distances
from the sun. “Average” conditions
prevail when both planets are near their average distances from the sun, that being “a” for their orbits. There is no guarantee that Earth will be at
its min distance, when the destination
planet is also at its min or max distance.
** All you can do is bound the delta-vee problem between the max
values at max planetary distances from the sun,
and min values at min planetary distances from the sun.** Orbital calculations made at average
planetary distances are merely “typical values” between these bounds.

For any given distances of Earth and destination planet from
the sun, you figure “a” from the sum of
the two planetary distances, and “e”
from the difference divided by the sum,
as per the equations already given.
From these, you can figure
everything else, including transfer
orbit perihelion and apohelion speeds,
and the period, per the equations
already given.

__Bear in mind that these orbital speeds for spacecraft and
planets are measured with respect to the sun__. For the departure, the difference between the spacecraft speed
and Earth’s speed is effectively the spacecraft speed with respect to the
Earth, something required of the
spacecraft when “far” from Earth,
meaning outside the Earth’s gravitational influence. That speed is tangent to both Earth’s orbit
and the transfer orbit, which is why an arithmetic difference is adequate.

For the arrival, the
difference between spacecraft speed and the destination planet’s speed is the
spacecraft speed with respect to the destination planet, something required of the spacecraft when “far”
from the destination planet, meaning
outside its gravitational influence.
That speed is tangent to the destination planet’s orbit, and the transfer orbit, which is why an arithmetic difference is
adequate.

For destinations outward of Earth, the departure speed is larger than Earth’s
orbital speed, requiring a posigrade-direction
burn. For destinations inward of
Earth, the departure speed is smaller
than Earth’s orbital speed, requiring a
retrograde-direction burn.

For destinations outward of Earth, the arrival speed is less than the
destination’s orbital speed, so that the
destination planet wants to “run over” the spacecraft “from behind”. For destinations inward of Earth, the arrival speed is greater than the
destination’s orbital speed, so that the
spacecraft overtakes the destination,
and wants to “run into it from behind”.

*Faster
Ellipse Trajectories*

The situation for faster ellipse trajectories is similar to
Hohmann, except that the transfer
ellipse is only tangent to the inward planet.
The case of Earth to Mars is illustrated in Figure 5, including the area-under-the-curve concept
for calculating 1-way transfer time (1-way trip time). For the example shown, the difference between transfer perihelion
speed and Earth orbital speed can be computed arithmetically, precisely because of the parallel
vectors, which is due to the
tangency. That value is the velocity
“far” from Earth (after escape), which
is required in order to be on the transfer trajectory.

Again for the example shown,
the apohelion of the transfer orbit lies far beyond the orbit of
Mars. ** Arrival is at the point
where the two orbits cross.** The
spacecraft velocity vector with respect to the sun is not parallel to the
velocity vector of Mars with respect to the sun. The velocity of the spacecraft with respect
to Mars is its velocity with respect to the sun, minus the velocity of Mars with respect to
the sun, done as a vector calculation.

Because of the nonparallel geometry, this subtraction is inherently a vector
subtraction, as indicated in the
figure. The angle between the two vectors
is the difference of the angles of the tangent lines of the two orbits. Because it is a vector subtraction, the negative of the Mars velocity is added
graphically to the spacecraft velocity.
This can be done by the method of resolved components, in any convenient coordinates, such as those which are radial and tangent to
Mars’s orbit.

That vector velocity with respect to Mars is the speed and
direction of the spacecraft “far” from Mars,
before Mars’s gravity can act to pull the spacecraft both closer and
faster.

Figure 5 – Faster Ellipse Trajectories (Mars as Example)

Besides the faster trip time, there is a second possibly-beneficial
characteristic of the faster orbit. If
its semi-major axis length is correctly chosen (solve the period equation for “a”, given a desired period value), the period of the transfer orbit becomes an
integer multiple of Earth’s 1-year orbital period. ** What that means is the transfer orbit
can serve as an abort orbit, taking the
spacecraft back to Earth, if the Mars
arrival is not made.** The period

__must be an integer multiple of 1 year__, so that Earth will actually be there, as the spacecraft arrives at perihelion.

There is another difference between this abort-orbit
situation and a simple Hohmann transfer:
it is possible to support a shorter stay upon (or near) Mars. The time available in the example is the
transfer period minus two of the 1-way trip times. With Hohmann at average conditions, there is a little over a year’s stay on Mars
required, before the orbits are “right”
for the Hohmann trip home to Earth.

**Getting to “Near” Velocities from “Far” Velocities, and Back **

** The following “trick” is a way to correct spacecraft
velocities with respect to a planet which were computed “far” from a planet
(per above), to the velocities “near”
the planet, after its gravity has pulled
the spacecraft to faster speeds.**
This is based on conservation of mechanical energy, and works well for figuring delta-vee values
for spacecraft design.

**For that, you must run a real 3-body problem in a computer program. Here is the estimated correction:**

*It is totally inadequate for navigation!*V_{near}^{2} = V_{far}^{2} +
V_{esc}^{2} where V_{esc}
is figured at the appropriate distance r from center-of-planet

For departure, V_{near}
is the vehicle burnout speed (with respect to the planet) that must be
reached, which is larger than V_{esc}, such that “far” from the planet, it will still traveling at the lower value V_{far}
with respect to the planet, which when
corrected to be with respect to the sun,
is the transfer trajectory velocity with respect to the sun that is required
at departure.

For arrival,
spacecraft velocity with respect to the sun gets refigured by vector
subtraction to velocity with respect to the planet, but __before the planet’s gravity has effect__, here denoted as V_{far}. This gets corrected to an appropriate V_{near}, which could then be the direct atmospheric
entry speed, the speed which must be
“killed” to land directly on an airless planet,
or the speed which must be reduced by a suitable delta-vee, in order to enter orbit about the planet.

**What Delta-Vee Values Will You Need?**

There are at least three practical things to consider, sometimes four: (1) departure delta-vee, (2) correcting the inclination of the
transfer orbit plane, (3) other course
corrections (some are always required for a precision approach), and (4) the delta-vee or delta-vees required
to enter orbit or to land directly. Be
aware that the same basic considerations apply for return trips, although the detailed numerical circumstances
can be quite different.

*Departure
Delta-Vee*

Departure delta-vee has already been addressed in the sense
that there is a “near” velocity greater-than-escape (with respect to the
planet) that the spacecraft must attain.
This could be from orbit, or from
the surface. The details of calculating
delta-vee from that information vary with exactly how you ascend and depart.

Regardless, there is
a “far” velocity larger than escape that the vehicle must achieve, oriented parallel to the Earth orbit velocity
if Hohmann, and also parallel to the
Earth orbit velocity if a faster ellipse headed outward of Earth. If a faster trajectory headed inward of
Earth, these velocities are not parallel, requiring vector addition processes.

Figure 6 shows what must happen, for departures from low Earth orbit onto
Hohmann trajectories to other planets.
The transfers outward of Earth require V_{far} values exceeding
Earth escape, directed in the same
direction as Earth’s orbital velocity about the sun, and oriented parallel to the Earth orbital
velocity. This parallelism is true
whether Hohmann or faster. Arithmetic
calculations work.

Transfers inward of Earth require V_{far} values
exceeding Earth escape, directed
opposite to the Earth’s orbital velocity about the sun. The direction is parallel to Earth’s orbital
velocity if Hohmann transfer, and not
parallel if a faster trajectory. Not
parallel requires vector addition processes.

Estimates of this type are adequate for the delta-vee sizing
of vehicles, but they are totally
inadequate for navigation. That requires
a 3-body trajectory program. The
departure delta-vee is the difference between V_{near} and the circular
orbit velocity. If the burn is
impulsive, no factoring-up is needed:

dV_{from orbit} = V_{near }– V_{circ} for factor = 1.00 if impulsive

Figure 6 – Departure From Low Earth Orbit

Departures from the Earth’s surface can be either to Earth
orbit, or direct to escape onto the
transfer trajectory. Either way, the initial sharply-rising portion of the
trajectory is subject to both gravity and drag losses. The nearly-horizontal but exoatmospheric
portion is subject to a minimal gravity loss,
but no drag loss. That gets you
to low circular orbit.

You can try to model the higher losses on the early portion
of the ascent if you want to, __but
just using 5% each for gravity and drag losses against low orbit velocity works
just as well as any other estimate__.
Which is why I generally just do that simpler easier estimate. See Figure 7,
which shows this.

Departure can then be direct, or from circular orbit, but either way, that portion of the delta-vee needs no
factoring, because from orbit, there are no drag and gravity losses for
impulsive burns.

These numbers and recommendations are for eastward launches
from Earth. The desired orbital velocity
has both kinetic and potential energy values associated, so that the “more representative” orbit velocity
in this book-keeping would actually be surface circular.

For polar launches,
you would need to add to that ideal velocity requirement the tangential
velocity of your launch point as a function of its latitude. The eastward velocity of a point on the
Earth’s surface at its equator is 1670 km/hr,
or 0.463 km/s. Multiply that by
the cosine of latitude for sites off the equator. If launching retrograde, add two of those tangential velocities to the
ideal surface circular orbit velocity.

For launches from other worlds of different atmospheres and
surface gravities __using the simpler factored orbital velocity method__, reduce the 0.05 values by the ratio of
surface density and gravity to Earth standard values, as indicated in the figure. Such is adequate for vehicle design delta-vee
values, but totally inadequate for
navigation estimates.

Figure 7 – Factored Delta-Vee Recommendations for Earth
Ascent and Elsewhere

Navigation-grade estimates require a full trajectory program
incorporating appropriate models for gravity and atmosphere, and for all the vehicle characteristics. 3
degrees of freedom is adequate, but 6
degrees of freedom is better, although
it costs a lot more effort to set a 6-dof model up.

*Correcting
to the Inclination of the Transfer Orbit at Departure*

Up to this point, the
entire solar system has been treated as if co-planar. __It is not__. Each planet has an orbit slightly
differently-inclined (and oriented) than the Earth’s orbital plane, which is customarily used as the reference plane
for describing Solar System orbit inclinations.
The difference in inclination between the destination planet’s orbit and
the Earth’s orbit is the maximum plane change angle magnitude that you need to
“cover” in your design. It can even be
zero.

This needs to be done as you depart, so that the error does not build-up into
something out-of-reach later. It can be
book-kept as part of the general course correction budget, but it really needs to be part of the
departure burn process.

If done after escape,
this gets figured with the plane change equation, at the spacecraft transfer orbit departure
velocity with respect to the sun:
i.e., V = V_{per} for the
transfer orbit to Mars. That is because
you are adjusting the plane of __an orbit about the sun__. That equation is:

dV = 2 V sin(Δi/2) ~ V Δi for small-angle Δi
measured in radians (“SRT rule”)

** One should understand that this equation applies to a
2-body problem, being originally
developed for satellite orbits about the Earth.** For Δi < 10 degrees, the “SRT rule” applies, such that the sine and tangent values for Δi
are numerically the same as the value of Δi itself measured in
radians.

*The speed V is the current speed in the two-body orbit at the point you make the burn to change the inclination.*__Whatever it is you are orbiting__,

__is what you measure V “with respect to”__.

The change in angle Δi for our transfer orbit relative to
the plane of Earth’s orbit about the sun,
is the difference in planetary inclinations, as a maximum value, usually in the neighborhood of a few degrees
(the inclination of Mars’s orbit is 1.85 degrees, Earth as a reference is zero by definition).

For direct surface launch onto the interplanetary transfer
trajectory, you simply aim your ascent
trajectory such that its final post-escape residual “V_{far}” is
aligned for the transfer trajectory. You
make this aiming __well before you escape__,
** while the 2-body problem with Earth as primary still applies!** That makes V in the inclination equation the
relatively small number of your flight speed relative to the Earth. The customary factoring-up for gravity and
drag losses during the ascent covers all the mostly-eastward launches well
enough, including this case.

*When estimated this way, no additional departure delta-vee for inclination correction is needed.*If you launch into low Earth orbit, and then depart onto the transfer trajectory
from there, __you want the inclination
of that orbit about the Earth to match the inclination of your transfer
trajectory__. If your Earth orbit is
elliptical, you also want its
orientation to match your transfer trajectory.
If circular, that second
requirement is deleted. No additional
departure delta-vee for inclination correction is needed, since you already corrected it early in your
ascent.

Many mission concepts have called for a space station in
orbit about the Earth, at which to
assemble vehicles, and from which to
depart to other planets. Such a station
should be in a circular orbit whose inclination matches that of Earth’s orbit
about the sun, not equatorial about the
Earth.

From there, you
correct to the transfer orbit inclination with a plane change burn, while still in orbit about the Earth! Then you burn for departure, with no further course correction required
until later. For this case, the worst-case angle change for Mars is 1.85
degrees, and your velocity in orbit
about the Earth is about 7.9 km/s. The
resulting inclination-correcting dV is about 0.26 km/s, worst case.

If instead you wait until after escape, when you are already on the interplanetary
trajectory, you are now orbiting the
sun, at around 30+ km/s. The max angle change is still 1.85
degrees, and so the dV for the
correction burn is now 0.97 km/s. That’s
a big difference, and very
unfavorable! It is far better to correct
the inclination while still in orbit about the Earth, __or even earlier__, during direct ascents. This is because the values of velocities
relative to the planet are far lower than those relative to the sun.

*Midcourse
and Terminal Course Corrections*

These always take place while on the interplanetary trajectory, but involve very tiny angle changes. These are easier estimated as cross range
distance divided by range-to-target,
which is essentially the angle change in radians. As a credible guess, the midcourse correction for Mars might be
about 10 planet diameters (at roughly 4200 miles diameter), when the range is close to the average
distance of Mars from the sun (close to 150 million miles). The resulting angle change is near 42,000
miles/150 million miles = 0.00028 radians.
Speeds will still be in the 25-30 km/s class, for a dV in the 0.008 km/s class. This rises rapidly if you wait to later in
the trajectory. But you do need enough
travel to successfully determine where you are versus where you really need to
be. That’s why it is called a
“midcourse” correction!

The terminal correction can be estimated similarly. For a Mars mission, you may be wanting to “hit” a narrow “window”
for direct entry or orbital entry. The
cross-range distance correction might be on the order of 100 miles, when you are two or three planet diameters
(4200 miles diameter) away. Velocity
will be near 25 km/s with respect to the sun,
so the dV will be a bit larger, at something like 0.24 km/s.

Add these up and then double it for a safety factor. For the Mars mission example, that would be 2 x 0.248 km/s ~ 0.5 km/s
budget for midcourse and terminal course corrections, excluding any initial inclination adjustment
(which should be taken care of before escape onto the interplanetary
trajectory).

*Delta-Vees
for Landings or for Entering Orbit*

Upon arriving at Mars or any other destination, there are only two things you can do. ** You can either enter orbit about the
destination, or you can try to land directly
upon it.** The calculated arrival
“V

_{near}” is your relative velocity to the destination, at very close distances, so time is very short to impact! Your location and orientation relative to the planet is critical, so the navigation problem is critical to success. That is not the topic here, but delta-vee for sizing vehicles is.

** To enter orbit**, there is an orbit speed, usually presumed to be low circular, with respect to the planet. Your “V

_{near}” value is also with respect to the planet. For destinations outward of Earth that rotate in more-or-less the same direction as they revolve about the sun, you would like to orbit in that same direction, if you intend to land later. That is to take advantage of the rotation to reduce any delta-vee taking off. See Figure 8.

On the sunward side of the planet, the orbital velocity reduces the burn
requirement, while increasing it on the
anti-sunward side, as indicated in the
figure. The ideal delta-vee to enter
such a sunward-side orbit is V_{near} – V_{orbit}, and circumstances may require vector addition
instead of arithmetic computation. If impulsive,
this delta-vee needs no factoring. (On the anti-sunward side, the delta-vee is far higher at V_{near }+
V_{orbit}.)

** How you might land directly upon land upon the
destination depends upon whether or not it has an atmosphere substantial enough
to use for aerobraking.** See
Figure 9.

** If airless**,
you must “kill” all of V

_{near}(as figured at the surface with the surface V

_{esc}) with retropropulsion, and that must be factored for gravity losses, and a budget added for any terminal zone hover or divert effects avoiding obstructions. Just as a rule-of-thumb, that terminal budget should be something like the last 0.5 km/s of V

_{near}, factored up by at least 1.5. Thus your airless delta-vee estimate to land might be: dV = V

_{near}* (1 + grav loss) + 0.5 km/s *(1 + 50% or more).

** If there is an atmosphere substantial enough for aerobraking**, you need to do an entry estimate to find out
what your

__altitude__,

__angle-below-horizontal__, and

__speed__might be at the end of braking hypersonics. Such entry analyses are not covered here, see Refs. 1 and 2 for that. That end-of-hypersonics speed would be the velocity corresponding to local Mach 3 at the destination. On Mars, because the atmosphere is so thin, the corresponding altitudes are invariably quite low, compared to those experienced here in the thicker atmosphere at Earth.

** Your choices from that point are twofold: (1) continued aero-deceleration followed by a
final retropropulsive touchdown, or
retropropulsion-to-touchdown without any further aero-deceleration.** Your ballistic coefficient and resulting
end-of-hypersonics altitude will decide between those choices at any given
destination, including the Earth. That topic is not covered here, see Refs. 2 and 3 for that.

Either way, for
delta-vee estimating purposes, there is
a speed-at-some-altitude which must be “killed” propulsively. Calculate the kinetic and potential energies
at that point, add them, and convert it to a surface speed “V_{surf}”, using V_{surf}^{2}/2 = V^{2}/2
+ g*h, where g is the __local__
acceleration of gravity. That is what
you have to “kill” with your propulsion.
It requires factoring for hover/divert considerations: dV = V_{surf} *(1 + at least 50%)

Figure 8 – How to Enter Orbit

Figure 9 – Paths to Landing Directly

**What You Do With All These Delta-Vees**

Any given stage or vehicle must deliver its share of the
total delta-vee to perform the mission. Since these are appropriately-factored
delta-vees, they are “mass
ratio-effective”. Thus you can use the
rocket equation in reverse to determine required stage or vehicle mass ratios
from the factored delta-vees, given
appropriate exhaust velocities for the engines and propellants you are
presuming. That topic is also out of
scope here. See Ref. 4 for the “how-to”
of estimating those things.

**References (all located at
http://exrocketman.blogspot.com):**

#1. 30 June 2012 Atmosphere
Models for Earth, Mars, and Titan

#2. 14 July 2012 “Back
of the Envelope” Entry Model

#3. 5 August 2012 Ballistic
Entry From Low Mars Orbit

#4. 23 August 2018 Back-of-the-Envelope
Rocket Propulsion Analysis

**An Example One-Way Cargo Mission to Mars**

As indicated earlier,
this kind of orbital mechanics analysis is suitable for sizing
vehicles, __not for navigation__. The entire analysis is oriented toward
determining realistic delta-vees for all the necessary things that must be
done. The concept for a one-way cargo
delivery to Mars is shown in Figure 10.
Upper and lower bounds and an average for the transfer orbit are given
in Figure 11. Departure delta-vees and
arrival entry interface speeds are given in Figure 12. Ascent and landing estimates are given in Figure
13. Figure 14 indicates the appropriate
uses for this kind of data.

Figure 10 – Mission Concept and Assumptions

Figure 11 – Average and Bounds on Transfer Trajectory

Figure 12 – Departure Delta-Vees and Arrival Entry Interface
Speeds

Figure 13 – Ascent and Landing Delta-Vees

Figure 14 – Appropriate End Uses for These Data

Dear Mr. Johnson,

ReplyDelete“The faster trajectories to outer planets all have perihelion points located at the Earth’s orbit, and so are tangent there, for an arithmetic calculation of departure delta-vee.”

I saw a similar statement in Bate (Fundamentals of Astrodynamics, page 362). Checking empirically, using

rTerra =149600000000 m, rMars = 227900000000 m, and Msol*G = 1.3274E20, I found

Perihelion 149600000000 m 1480000000 m 149600000000 m 148800000000 m

Aphelion 247000000000 m 2377250000 m 237122000000 m 232992000000 m

Transit time

rT to rM 180 days 180 days 200 days 200 days

total delta V 8419 m/s 7486 m/s 7161 m/s 6836 m/s

showing that for a required transit time, there is a savings in delta vee if perihelion is smaller than rTerra. Have I misunderstood the situation?

I tried to check my calculations against your Figure 10. Assuming a circular orbit and deriving

Msol*G = VE2 * rE = (29771 m/s)^2 *149570000000 m = 1.3257E20 m^3/s^2 ,

then perihelion speed for the Holmann orbit would be 32716 m/s, not 33378 m/s. Again, have I misunderstood?

Thank you, MBMelcon

I'm not sure how to respond to your comment or question. My reasoning was very simple: orbital velocities are higher nearer periapsis than near apoapsis. Higher velocities over the same distance means shorter transit times. I was looking at shortest transit time, not min delta-vee for a fixed transit time (I have no way to do that). As for the difference in numbers, your input numbers are not identical to mine. So you should not be surprised when you answer differs from mine. -- GW

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