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Gravity assists require precise navigational support when executed, and despite the additional complexity in the planning stage they can make otherwise impossible missions viable. Examples of multiple flyby missions were those of the Pioneer and Voyager spacecraft, and more recently the Cassini and Rosetta missions. What are not shown in Figure 2.5 are the speed boosts gained at each planetary encounter, which in total were equivalent to a heliocentric speed increase of 21.4kms_1.

The present low cost of computing power allows the interested reader to examine such transfer schemes easily, and visualization/programming languages such as MATLAB or IDL are well-suited to such work.

2.2.3 Transfer orbits: continuous thrust

Spacecraft that use low-thrust propulsion systems to initiate interplanetary transfer necessarily must run their drives for long periods, and their trajectories cannot be modelled with the foregoing simple analysis. However, for co-planar transfers approximations exist (Fearn and Martin, 1995) that allow at least a firstcut to be made of 1V requirements. For continuous thrust spiral orbits about a body of mass M, a craft starting with an initial mass of mi has a final mass mf and develops an exhaust of speed ve. Here af and ai are the semi-major axes of the final and initial orbits.

4 The plane intersecting the target body's centre which is perpendicular to the inbound asymptote of the craft's path.





Venus flyby






Venus flyby



Earth flyby



Jupiter flyby



Saturn arrival


Figure 2.5. A cartoon of the Cassini spacecraft's path to Saturn.

Figure 2.5. A cartoon of the Cassini spacecraft's path to Saturn.

Ve VVOf aM.

Low, near-continuous thrust trajectories can have significant knock-on consequences for the mission as a result of the protracted transfer. Electric propulsion systems vary in their Isp and in the mechanisms employed to heat the working fluid, which is generally a compressed gas. For example, the Mu10 engines of the Hayabusa craft delivered an Isp of 4000 s, and used a microwave system to heat and ionize xenon gas, but gave a peak thrust of only 20 mN. A measure of efficiency for an electric propulsion system is its specific thrust per unit of power, and modern systems can have values around 20 to 30mNkW~1. Photovoltaic cells, at least for the immediate term, are the only practical choice for the multi-kW needs of electric propulsion systems, and over long periods suffer from efficiency degradation at a rate of a few percent per year. Thus the peak thrust levels of electric propulsion systems generally varies with solar distance by somewhat more than an inverse square.

The demands of electric propulsion on a spacecraft's attitude control system are no more taxing than those arising from the use of impulsive manoeuvres, which would also require accurate attitude control for communication and, perhaps, solar power-raising needs. However, EM interference of communication frequencies by electric propulsion systems and the thermal regulation and accommodation of power-conditioning electronics are extant problems to be solved on a case-by-case basis (Jankovsky et al., 2002).

2.2.4 Transfer orbits: chaotic transfers

A spacecraft's path under the influence of three bodies, such as the Earth, Moon and Sun, can show great sensitivity to small changes in its initial trajectory. In 1982 the spacecraft ISEE-3 was directed, using these non-linear dependencies, first towards the Sun-Earth L2 point, and then towards comets Giacobini-Zinner and Halley with the use of relatively small manoeuvres. Put simply, the Solar System is threaded by a complex web of trajectories based on the presence of Lagrange points associated with each pair of planetary bodies. Relatively small manoeuvres are needed to pass from orbits about one Lagrange point, in say, the Earth-Moon system, to a Lagrange point in the Sun-Earth system; this non-intuitive process description is explained by Koon et al. (2000). This scheme was used by the Genesis mission, which orbited the Sun-Earth L1 point and returned to Earth via the Sun-Earth L2 point, a considerable orbital change brought about by little more than appropriate use of the weakly bound orbit about the Sun-Earth L1 point. However, these methods can result in long transport times, and such schemes are not robust to changes in launch date or manoeuvre underperformance.

2.3 Arrival strategies

A mission can require that a lander be delivered to a planet's surface directly from an interplanetary trajectory. The Luna sample return spacecraft and the Genesis, Stardust and Hayabusa missions, all used such a direct arrival scheme. Earth, the common target body for these craft, offers a relatively dense and deep atmosphere for aerodynamic braking to be an efficient way of slowing the returning craft. Arrival at an airless body requires one manoeuvre to slow the craft from its hyperbolic path so that it is captured by the target body. Further manoeuvres are used to lower the apoapsis of this new orbit, and to control other aspects of the orbit such as its inclination and periapsis position.

2.3.1 Aerocapture and aerobraking

For missions to atmosphere-bearing planets the presence of an atmosphere provides the mission designer with the opportunity to remove the hyperbolic excess of the inbound spacecraft. Aerocapture refers to the passage of a spacecraft with excess hyperbolic speed through a planet's atmosphere in order to reduce its speed, and to thus achieve an orbit. Aerobraking is the more general word used to describe the use of an atmosphere to reduce a craft's speed. These techniques, especially aerocapture, require that the target atmosphere's density profile is well known before the encounter. Long-term studies by spacecraft around Mars show that that planet's thermosphere has daily, seasonal and dust-storm-induced density variations that can be as large as 200% (Keating et al., 1998) and which are not easily predicted. For aerocapture missions a spacecraft might require autonomous trajectory control via aerodynamic or reaction-control systems.

Entering atmospheres

The entry of a spacecraft into a planetary atmosphere has led to an iconic image of the space age; that of a capsule being roasted in a fireball streaking across the sky. The second familiar image is that of a pilot experiencing progressively heavier 'g' loads and these processes are common for all objects entering a planetary atmosphere. A planetary mission designer has to understand how these phenomena vary with characteristics of the entry craft and the target atmosphere. This section will illustrate these relationships and the engineering solutions that may be adopted for atmospheric entry.

3.1 Entry dynamics

A useful simplification is to disregard the spherical nature of the target planet; a reasonable premise because the atmosphere is often crossed in a short span of time and space; roughly a few minutes in length and a small fraction of the body's radius in extent. Similarly, the atmosphere can initially be treated as being non-rotating, isothermal, chemically homogeneous, and in hydrostatic equilibrium at some temperature T. In this case, the density at some height above a reference surface has the familiar exponential form:

Here H is the density scale height, and is defined as

mg where m is the mean molecular mass of the atmosphere in a uniform gravitational field, g, and k is Boltzmann's constant. For massive planet atmospheres this assumption of constant gravitational 'g' is a fair approximation. For example, during its entry into Jupiter the gravitational acceleration experienced by the Galileo probe varied by ~0.1%, although the probe experienced far higher aerodynamic decelerations. A counter-example is found in lighter planets with larger scale heights; in the region where the Huygens Titan probe encountered aerodynamic decelerations greater than 1 m s-2, its gravitational weight varied by over 40%, a consequence of the extensive atmosphere of Titan and the inverse-square law of gravity.

Current entry vehicles are both rotationally symmetric and passive, in the sense that they are not capable of making deliberate changes to their trajectory. We will also assume that drag is the only aerodynamic force applied to the craft, which acts parallel to its velocity vector at all times. An entry craft of mass m, and effective cross-sectional area S, passing through an atmosphere of density p, then experiences an acceleration that can be modelled by dV = -pSCD V2

dt 2m 13'3J

where CD is the drag coefficient. In the Newtonian or free-molecular flow regime at very high altitude, this is typically ~2.1 for all shapes, but falls to lower values that depend on the Mach and Reynolds numbers and the exact shape and angle of attack in the denser parts of the atmosphere. The instantaneous deceleration can be written as dV <34>

where the new parameter • is pjzmaD

Note, the absence of gravity in this model leads to a straight trajectory, and the flight path angle y is therefore constant, thus dh

dt which, after re-writing the dimensionless 'altitude' parameter • as

p0SHCd exp z

2m allows us to re-write Equation 3.3, using the identity dV _dVdh d^ dh d^

V sin y

Integration of Equation 3.8 with respect to • yields

Finally, this may be substituted back into Equation 3.4 to give the deceleration

exp sin y

Clearly, the peak value for deceleration occurs when the exponential term is a maximum, which happens when has a value of sin(y). Substitution into Equation 3.8 shows that the peak deceleration of a spacecraft in this simple model depends only on the scale height of the atmosphere, not on the drag coefficient or mass of the spacecraft. In Table 3.1 the peak deceleration is calculated for vehicles entering atmosphere-bearing targets in the Solar System.

The foregoing discussion describes a naive view of an entry craft's trajectory. In reality there are a number of important factors to be considered. Firstly, atmospheres are neither static nor isothermal. However, the exploratory nature of current spacecraft necessarily means that they will encounter atmospheres that are not well understood in terms of their spatial and temporal variability. It is then a challenge to size the decelerator system so that the craft experiences sufficient deceleration for the chosen type of mission. In spite of the increasing availability of computing power, many critical aspects of entry processes (such as transonic stability tests, the behaviour of catalytic surfaces in reactive gas flows, etc.) are best suited to experimental analysis. Thus, the bottle-neck in

Table 3.1. The peak decelerations at various flight path angles for entry at escape speed and lift-to-drag of 0

Peak deceleration (m s 2)

Table 3.1. The peak decelerations at various flight path angles for entry at escape speed and lift-to-drag of 0

Peak deceleration (m s 2)


y o = 2.5°

yo = 5°

y o = 15°

































development generally is one of performing an adequate number and range of tests that validate a given entry capsule design, often under conditions of high airflow speed and density that are reproducible only in shock-tube facilities that are expensive to operate and have long lead-times for tests. For this reason entry-capsule design tends to be conservative although aerothermal modelling is not the only area in which experimental testing is unavoidable. In lower-speed portions of the entry process, decelerators such as parachutes or airbags may have to be deployed, and used heat shields jettisoned. These events generally happen well after the peak heating and deceleration phases of entry and will not be discussed further.

3.2 Thermodynamics of entry

The kinetic energy of an entering spacecraft is mostly dissipated as heat. In Table 3.2 the specific energies associated with two types of entry path are calculated for atmosphere-bearing bodies of the Solar System. The first value in each row shows the energy associated with entry from a hypothetical circular parking orbit,5 and the second shows the energy for arrival from outside that body's gravitational influence. In this last case the speed of entry is taken as the escape speed of that body.

The heat generated during entry is more than sufficient to destroy any object if all of the heat were to be absorbed by the spacecraft. To show this, Table 3.3 lists the heat needed to warm various compounds to their melting points, and their heat of vaporization; carbon and beryllium clearly are excellent theoretical candidates for a thermal protection system (TPS). Carbon's high melting point and low comparative toxicity and cost make it the practical choice of these two elements. The role of vaporization will be discussed later.

Table 3.2. The energy per unit mass associated with arrival at a planet's surface


Arrival from orbit (MJ kg 1)

Interplanetary arrival

























5 With an orbital radius 1.5 times that of the particular body's radius.

The production of heat from kinetic energy depends on the environment of the spacecraft. At the edge of an atmosphere the gas density around the entering craft is so low that the gas flow around the vehicle is ballistic at the molecular scale; this is often referred to as free-molecular flow. Parts of the craft are shadowed from the oncoming gas flow and forward-facing faces experience direct molecular collisions. In a denser gas, the molecular mean free path will be shorter than scales characteristic of the craft and a different flow type emerges. Here, the air ahead of the craft is slowed, compressed and heated. If the craft exceeds the local speed of sound then a shock field develops around the front of the vehicle. Air moving through the shock is rapidly heated and compressed. The strength of this shock field is dictated by, amongst other things, the geometry of the entry body. Narrow spear-like objects tend to have relatively weak and sharply pointed angular shock fields draped downstream from their noses. Blunt objects develop stronger and broader shocks that in turn influence larger masses of air by virtue of their large cross-sectional area. A frequently used parameter for describing the relative aerodynamic load experienced by an entry craft is the ballistic coefficient,6 which will be used later in Chapter 4.

At a qualitative level, it can be seen that the heat load experienced by a hypersonic object can be lessened if its energy of entry is dissipated into a larger mass of air. Therefore, the large enthalpy change across a blunt object's shock reduces the energy that is absorbed by the object. Conversely, a slender object, shrouded in a weaker shock field, experiences an air flow that has been slowed comparatively little and so absorbs a larger fraction of the entry energy. However, the designer of an entry heat shield cannot pick a given geometry with impunity. A working spaceprobe must be accommodated within the envelope of the entry shell, stowed robustly in some manner. If a non-spherical entry shell is used, the centre-of-mass of this configuration must lie adequately below the aeroshell's centre-of-pressure, which in turn moves the spacecraft and its dense components (batteries, etc.) closer to the leading face of the entry shell. If the offset between the centres of mass

Table 3.3. The heat needed to warm materials from ^300K to their melting point, and the enthalpy required to vaporize those substances


Melting point (K)

AHwarm (MJ kg"1)

AHVaporize (MJ kg 1)

















6 Defined as M/(SCD), a measure of the craft's areal density.

6 Defined as M/(SCD), a measure of the craft's areal density.

and aerodynamic pressure is made too small, then the craft may be unstable to disturbances and make large pitching movements, exposing non-shielded parts to the energetic airflow. As an example, a craft's transition from supersonic to subsonic speed causes changes in the wake flow which in turn can be coupled to the craft, destabilizing it. Some entry craft, such as that of the Genesis sample-return mission, are designed to deploy small drogue parachutes at supersonic speeds to provide extra stability through the transonic region (Desai and Lyons, 2005).

It is also worth mentioning here that atmospheric density profiles can be derived from entry accelerometry, using Equation 4.3, an assumption of hydrostatic equilibrium, and integrating the acceleration with appropriate boundary conditions to obtain velocity and altitude. Temperature and pressure profiles can also be derived using the ideal gas law and knowledge of the mean molecular mass of the gas. Accelerometry is usually included in entry vehicles anyway for engineering purposes, to provide a 'g-switch' to initiate the parachute descent sequence. The first use for atmospheric science was on Venera 8 (Cheremukhina et al., 1974), but high sensitivity accelerometry was pioneered on Viking (Seiff and Kirk, 1977) and has also been implemented on Pioneer Venus (Seiff et al., 1980), Galileo (Seiff et al., 1998), Pathfinder (Seiff et al., 1997) and Huygens (Colombatti et al., 2006), among others. The detailed processing of the data must take into account a number of error sources and perturbations, as well as the three-dimensional nature of the problem (e.g. Withers et al., 2003).

Simple models for the heating rates experienced by entry craft will necessarily neglect many important phenomena. In Table 3.4 only convective heating is considered, and topics such as the variation of the heating rate with the flow regime (turbulent or laminar) and real-gas properties of the atmosphere are not examined.

At subsonic and low-Mach numbers the gas ahead of the craft is primarily heated by being rapidly compressed by the craft. The vehicle is then immersed in a hot flow of gas and absorbs heat by convection. This form of heating is not applied uniformly to the craft, but is a function of the local geometry and the

Table 3.4. Summarizing the principal features of this simple model for ballistic entry

Value Speed at peak (m s_1) Altitude at peak (m)

Peak heating

Vo exp h J 3P0SHCR)

Peak deceleration

2He sin yo

Vo exp

nature (laminar or turbulent) of the flow. In contrast, at hypersonic speeds the atmosphere interacts with, and is heated by, a shock field some distance ahead of the craft, rather than by the craft itself.

At high entry speeds the temperature rise in the shock wave around the craft may be sufficiently intense for radiant heating from the hot gas to be equivalent to the convective heating rate. For Earth entry this occurs at speeds above 10 km s-1 for bluff objects, as is shown in Figure 3.1, adapted from Sherman (1971). Note that the same equivalence in the heating processes occurs at higher speeds for objects with smaller radii, but for such craft the temperatures in the shock would be far higher, potentially compromising the temperature limits of the TPS.

The brevity of the heating process can be seen in aerothermal models and in experimental data. The modelled stagnation heat flux for a Martian entry craft with a ballistic coefficient of 150 kg m-2 is shown in Figure 3.2; note that the peak heating occurs somewhat before the instant of peak deceleration as predicted in Table 3.4.

3.2.1 Flow chemistry

Entry trajectories leading to air temperatures of up to 2000 K cause molecular excitation, dissociation, and partial ionization of the gas ahead of the vehicle.

100 r

5 10 15 20

Figure 3.1. Here radiative and convective heating rates are compared for two spheres of different radii entering the Earth's atmosphere.

5 10 15 20

Figure 3.1. Here radiative and convective heating rates are compared for two spheres of different radii entering the Earth's atmosphere.

TPS technologies

TPS technologies

Figure 3.2. Predicted heating history for a cone-sphere Mars entry vehicle with 60° half-angle forebody.

Hypersonic entry craft, in general, produce flow fields that are described by a Damkohler number7 close to unity, and so the chemistry in the air around the craft varies as a function of distance along the flow. From the TPS designer's viewpoint the chemical species in the flow are of no interest except for their potential to deliver heat to the TPS through recombination. Ionized gas species can recombine at cool (<1800K) material surfaces and dump the enthalpy of molecular formation into the TPS. The degree of catalytic behaviour depends on the material temperature and gas species, but fully catalytic materials can experience heating rates several times higher than non-catalysing surfaces for the commonly encountered species of CO, C and O (Marraffa and Smith, 1998).

3.3 TPS technologies

The great value of a spacecraft lander and the highly energetic entry path that it takes often leads to a conservative TPS design that can accommodate uncertainties in atmospheric structure or mission performance. Prior to the flights of the

7 A ratio of characteristic timescales for chemical reactions in the flow, and the duration of the flow itself in crossing the vehicle length.

American Space Shuttles, there was no need for reusability in spacecraft TPS and simplicity was the watchword. While purely heatsink-like shields have been proposed in pre-production studies, actual planetary missions have all used ablating materials. When heated sufficiently, such substances vaporize with the gas escaping from the underlying solid. As was seen in Table 3.3, carbon has many advantages as the main material component in such a system because of its high enthalpy of vaporization and high melting point.

Ablating TPS materials also lead to a reduction in the heat absorbed by the craft through convection because the ablation products are blown into the boundary layer around the craft, buffering the hotter incoming gas that has passed through the shock field. The Apollo and Luna programmes were the earliest missions to employ sample-return capsules and both systems used an ablating material that formed a crust of carbon-rich charred material, in itself a poor thermal conductor. Since those early missions, ablator-based protection systems have become widespread. Their efficiency and simplicity suggest that they will continue to be a preferred system for high-speed atmospheric entry. In Table 3.5 a number of entry-craft thermal-protection systems are described, with their composition detailed in the second column where possible.

3.4 Practicalities

Mechanical factors are also pertinent to the design of a TPS. Vibration during launch, exposure to hard vacuum during cruise, and aerodynamic loading during entry are just some of the hazards that a TPS must pass through without its design margins being compromised. To an extent, many of these points can be simulated in ground-based thermal vacuum chambers, but some processes are more difficult. For example, the ablating materials used for heat shields are composites of some sort, with phenolic resins or epoxies providing a matrix in which refractory particles or fibres can be embedded. Bulk thermophysical properties of the finished TPS can be tuned by controlling the recipe used in its manufacture, but the internal structure is also of concern. During ablation the gas released by pyrolysis has to escape from the TPS, otherwise the material could spall or delaminate, allowing hot gas to impinge on the entry craft's structure. A schematic of the Hayabusa return capsule is shown in Figure 3.3, adapted from Yamada et al. (2002), showing how each layer of resin-impregnated carbon fibre is slitted to give gas paths throughout the TPS.

Aerothermal research is still seeking to improve the efficiency of a TPS either through the use of novel materials, improved modelling, or different techniques.

Table 3.5. Some TPS systems used in planetary probes. Where known, the manufacturer and marque of the TPS material is listed. Here, mass fraction is the ratio of TPS mass to that of the entry body. It has been noted that there is an approximately linear relation between TPS mass fraction and peak heat flux

Table 3.5. Some TPS systems used in planetary probes. Where known, the manufacturer and marque of the TPS material is listed. Here, mass fraction is the ratio of TPS mass to that of the entry body. It has been noted that there is an approximately linear relation between TPS mass fraction and peak heat flux

Heat shield

Forward heat shield

Peak heat

mass kg/kg

Entry speed


TPS materials

flux (MW m~2)


(km s-1)


Phenolic honeycomb filled with mixture of silica micro spheres, cork, and silica fibres [SLA-561V]: Martin Marietta





Carbon phenolic

69 (large probe)




composite General Electric Co.

106 (small probes)



Carbon phenolic composite Hughes Aircraft Company





Asbestos composite over honeycomb NPO Lavochkin


-900/ -1500



Asbestos composite over honeycomb NPO Lavochkin





Phenolic honeycomb





filled with mixture of silica micro spheres, cork, and silica fibres [SLA-561V]: Lockheed Martin

Beagle 2

Cork particles bonded in phenolic resin [NORCOAT Liege]: EADS





Phenolic honeycomb filled with mixture of silica micro spheres, cork, and silica fibres [SLA-561V]: Lockheed Martin





Silica fibres in phenolic resin [AQ60]: EADS





Segmented carbon





phenolic composite ISAS

Virgin material

Pyrolysis zone

Gas blow off

Figure 3.3. The lay-up style of carbon fibre reinforced epoxy in the Hayabusa sample capsule.

Transverse slits in pre-preg CFRP sheets.

Gas blow off

Virgin material

Pyrolysis zone


Transverse slits in pre-preg CFRP sheets.

Figure 3.3. The lay-up style of carbon fibre reinforced epoxy in the Hayabusa sample capsule.

Examples of studies in these fields are:

• Aerobraking in planetary atmospheres is an attractive alternative to propulsive orbital capture, and leads to the partial 're-use' of a TPS during each atmospheric pass. Materials such as C/SiC, which have high infrared emissivity and reject much of the radiative heat-load, can act as coatings or main TPS components but are challenging to produce in complex shapes.

• Modelling of high heat fluxes and the associated radiation fields is almost exclusively performed in one dimension. The absorption and emission balance in the multi-species and spatially varied flow around a vehicle is still a non-trivial computational process.

Figure 3.4. Cross-section of the VeGa entry assembly, showing the mass for oscillation damping mounted near the apex of the entry shell.

• Mechanically complex TPS materials may be required by missions that do not involve passively delivered surface-impacting landers. Such materials may arise in the form of hinged control surfaces capable of withstanding and interacting with hot, dense flows for autonomously targeted entry craft, or in extendable/inflatable structures that are too large to be launched as a solid entity.

The dynamics of atmospheric entry may in some cases (particularly non-spinning probes) stimulate oscillations of the probe attitude. While these are not always large enough to cause concern, several probes have carried internal damping masses held in a flexible mounting structure. Figure 3.4 shows a cross-section of the VeGa entry assembly, with the damping mass visible at the lowest point of the probe.

Descent through an atmosphere

4.1 Overview and fundamentals

The descent through the atmosphere is often the only part of a planetary probe mission, as for example the Pioneer Venus and Galileo probes; on other missions it is just the last stage of a long journey prior to surface operations. The key parameters are the altitude of deployment - usually the altitude at which the vehicle ends its entry phase, as defined by some Mach number threshold - and the required duration of descent.

The duration of descent for an atmospheric probe is often dictated by an external constraint on the mission duration, such as the visibility window of a flyby spacecraft that is to act as a communications relay. This imposes an upper limit on the descent duration - it may be that (as for the Huygens probe) some part of that mission window is desired to be spent on the surface.

The instantaneous rate of descent (and thus the total duration) is determined at steady state by the balance between weight and drag. The former is simply mass times gravity; the latter depends on ambient air density, the drag area of the vehicle and any drag-enhancement device such as a parachute or ballute. The drag area is usually expressed as a reference area and a drag coefficient. Often these parameters and the mass are lumped together into the so-called ballistic coefficient fl.

Often the dynamic pressure of descent is used to force ambient air into sampling instruments such as gas chromatographs. In steady descent, the dynamic pressure can be equated to the ballistic coefficient times ambient gravity.

4.2 Extreme ballistic coefficients

There are situations where it may be desired to maximise fl, and thus the descent rate. Example applications are balloon-dropped microprobes on Venus, payload delivery penetrators and probes for the deep atmospheres of the outer planets.

Considering the first, the desire is to reach the surface with a miniature probe before the balloon that released it has drifted out of sight. Because the probe is small, its ballistic coefficient is low (for a given mass density, mass increases with size more steeply than area, so small things have low mass/area ratios). Further, to protect the probe from the very hot lower Venusian atmosphere it may be coated in a layer of (low density) insulation that increases the cross-sectional area.

The natural tendency is to lengthen the probe such that its ballistic coefficient is increased. However, this increases the total surface area and thus increases the heat input into the probe. For a given descent duration, this increases the amount of thermal ballast needed, the insulation performance required, or the maximum temperature that can be tolerated by the equipment.

This underscores the issue, highlighted in a different context by the DS-2 Mars microprobes, that achieving high packaging densities is often important in small vehicles. Meeting a volume constraint with small vehicles is often more challenging than meeting the mass constraint.

Most of the descent takes place at terminal velocity, which can be considered steady-state (strictly, since the air density increases with decreasing altitude, the terminal velocity drops with time, although typically rather slowly). The terminal velocity Vt can be computed thus

g can often be considered constant (except on Titan, where the scale height is not negligible compared with the planetary radius). Often the drag coefficient (usually of order 1), the mass and the reference area S are lumped in a single parameter fl, which equals 2M/(SCd) (NB: sometimes the factor of 2 is not included in the definition - care!). This parameter has dimensions of mass per unit area, and values of 10-100 kgm-2 are typical.

Substituting this parameter, then we have

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