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Terminal velocity

Terminal velocity is the constant maximum speed reached by an object falling through a viscous fluid, such as air or water, when the downward gravitational force is exactly balanced by the upward drag force, resulting in zero net force and no further acceleration. This phenomenon occurs after the object has accelerated from rest and the drag force increases with speed until equilibrium is achieved. The force opposing the motion typically follows a dependence on for high-speed falls, given by F_d = \frac{1}{2} \rho A [C_d](/page/Drag_coefficient) v^2, where \rho is the fluid density, A is the object's cross-sectional area, C_d is the , and v is the . At terminal v_t, this equals the weight mg, yielding the formula v_t = \sqrt{\frac{2mg}{\rho A C_d}}, with m as and g as . For lower speeds where is linear in (F_d = b v), the terminal simplifies to v_t = \frac{mg}{b}. Terminal velocity varies widely depending on the object's , , , and the medium; for example, a skydiver in freefall reaches about 60 m/s (134 mi/hr) with a around 0.5 and of 0.7 m², while a achieves roughly 33 m/s (74 mi/hr). In contrast, typical raindrops, with diameters of 1–5 mm, attain terminal speeds of 6–10 m/s due to their small and lower relative to . These values highlight the role of terminal velocity in natural phenomena like and in applications such as parachuting, where deploying a canopy increases to reduce speed to 5–7 m/s for safe landing.

Fundamentals

Definition

Terminal velocity is the maximum constant speed that an object achieves when falling through a viscous fluid, such as air or water, under the influence of gravity, at which point the net force acting on the object becomes zero and it no longer accelerates. This occurs when the downward gravitational force is exactly balanced by the upward forces of drag and buoyancy (if significant), resulting in zero net acceleration and a steady-state velocity. Qualitatively, an object dropped in a initially accelerates due to the unbalanced gravitational , but as its speed increases, the opposing drag grows proportionally to the square of the velocity for high-speed motion, eventually equaling the effective weight (gravitational minus ). At this equilibrium, the object continues falling at a constant terminal velocity without further speeding up. In air, is often negligible for dense objects like humans, but it plays a more prominent role in denser fluids like . Terminal velocity is expressed in units of meters per second (m/s) in the (). For a typical skydiver in a spread-eagle position without a , this value is approximately 53 m/s (about 120 or 193 /h), illustrating the scale for human-sized objects in Earth's atmosphere.

Historical Development

The concept of terminal velocity emerged from early philosophical and experimental inquiries into the motion of falling bodies. In the 4th century BCE, proposed that heavier objects fall faster than lighter ones due to their inherent natural tendency to seek the Earth's center, with speed proportional to weight, while assuming motion ceases without continuous force. This view dominated for nearly two millennia. In the late 16th and early 17th centuries, challenged these ideas through experiments, demonstrating that, in the absence of air resistance, all objects accelerate uniformly under at the same rate regardless of mass; he used inclined planes to measure distances proportional to the square of time and reportedly dropped objects from heights like the to show near-simultaneous falls. Isaac Newton's (1687) advanced the understanding by incorporating air resistance as a key factor in resisting media, devoting Book II to and experiments showing forces proportional to for rarefied media or to the square of for denser fluids, laying groundwork for balancing gravitational and resistive forces. By the , formalized these concepts; George Gabriel Stokes' 1851 paper on the internal friction of fluids derived a for low-speed viscous on spheres (F_d = 3πμdv, where μ is , d is diameter, and v is ), enabling calculations of terminal velocity as the point where equals gravitational for small particles at low Reynolds numbers. The saw terminal velocity integrated into and , particularly during , when precise calculations for trajectories accounted for drag-limited maximum speeds to improve accuracy; for instance, German big guns like the achieved terminal velocities exceeding 3,000 mph in their descent phase, influencing wartime engineering. This era marked the shift from theoretical milestones to practical applications in high-speed flight and munitions design.

Underlying Physics

Forces Involved

The primary force driving an object toward terminal velocity is the gravitational force, which acts downward and is given by F_g = m g, where m is the of the object and g is the , approximately 9.8 m/s² near Earth's surface. This force remains constant regardless of the object's speed and is responsible for the initial of a falling body. Opposing the gravitational force is the drag force, a resistive force that arises due to the object's motion through a fluid medium like air or water and increases with velocity. For objects moving at relatively low speeds or in viscous fluids, where the Reynolds number—a dimensionless quantity defined as \mathrm{Re} = \frac{\rho v L}{\eta}, where L is the object's characteristic length scale (e.g., diameter)—is small, the drag force follows Stokes' law and is linear in velocity: F_d = 6\pi \eta r v, with \eta as the fluid's dynamic viscosity, r as the object's radius, and v as its speed. At higher speeds, typical for larger objects or less viscous fluids, drag becomes quadratic in velocity: F_d = \frac{1}{2} C_d \rho A v^2, where C_d is the drag coefficient, \rho is the fluid density, and A is the object's cross-sectional area perpendicular to the motion. In fluids, an additional upward force is the buoyant force, which equals the weight of the displaced fluid according to and acts to reduce the effective downward pull on the object. This force is generally smaller than gravity for objects denser than the surrounding fluid but contributes to the overall balance. Terminal velocity occurs when the on the object is zero, meaning the downward gravitational force is exactly balanced by the opposing drag and buoyant forces: F_{net} = m g - F_d - F_b = 0. At this point, the object's acceleration ceases, and it maintains a velocity.

Derivation of the Formula

The derivation of terminal velocity begins with Newton's second law of motion, which relates the net force on an object to its mass and acceleration: F_{\text{net}} = m a, where a = \frac{dv}{dt} and v is the velocity. For an object falling through a fluid under gravity, the primary forces are the downward gravitational force m g (where m is mass and g is the acceleration due to gravity) and the upward drag force F_d, which opposes the motion and depends on the object's velocity. Assuming downward as the positive direction and neglecting other forces such as buoyancy, the equation of motion is m \frac{dv}{dt} = m g - F_d. Terminal velocity v_t occurs when the acceleration a = \frac{dv}{dt} = 0, meaning the is zero and the gravitational balances the drag : m g = F_d. The form of F_d varies with the regime, determined by the ; for high speeds (high ), drag is quadratic in velocity, while for low speeds (low ), it is linear. For quadratic drag, typical of macroscopic objects at high speeds, the drag force is given by F_d = \frac{1}{2} C_d \rho A v^2, where C_d is the , \rho is the fluid density, and A is the cross-sectional area perpendicular to the . Setting m g = \frac{1}{2} C_d \rho A v_t^2 and solving for v_t yields the terminal velocity: v_t = \sqrt{\frac{2 m g}{C_d \rho A}}. This expression assumes constant g and uniform fluid properties. For linear drag, applicable at low Reynolds numbers such as for small particles in viscous fluids, the drag force is F_d = b v, where b is the linear drag coefficient (e.g., from , b = 6\pi \eta r for a of r in a fluid of \eta). At terminal velocity, m g = b v_t, so v_t = \frac{m g}{b}. To find the approach to terminal velocity, solve the m \frac{dv}{dt} = m g - b v, or equivalently \frac{dv}{dt} = g - \frac{b}{m} v. Separating variables gives \frac{dv}{g - (b/m) v} = dt. Integrating from initial velocity v(0) = 0 to v(t) over time t results in v(t) = v_t \left(1 - e^{-t/\tau}\right), where \tau = m/b is the , representing the characteristic time for the velocity to approach v_t (reaching about 63% of v_t after one \tau). As t \to \infty, v(t) \to v_t, confirming the steady-state balance. This exponential approach illustrates that terminal velocity is asymptotically reached, not instantaneously. The velocity versus time graph for the linear drag case shows an initial rapid increase following v \approx g t ( without ), transitioning to a flattening curve that asymptotes to the horizontal line at v_t. These derivations assume constant g, no additional forces, and a one-dimensional motion, providing the foundational formulas under idealized conditions.

Influencing Factors

Object Characteristics

The terminal velocity of a falling object is influenced by its intrinsic properties, primarily through their roles in the balance between gravitational force and aerodynamic drag, as captured in the general where terminal velocity v_t = \sqrt{\frac{2mg}{\rho C_d A}}. These characteristics determine how quickly the object reaches a constant speed in a medium. The mass m of the object is directly proportional to its terminal velocity in regimes dominated by quadratic drag, with v_t \propto \sqrt{m}, meaning heavier objects achieve higher terminal speeds because their greater weight requires more drag force to balance. For instance, a denser will fall faster than a lighter one of similar shape and size due to this scaling. The cross-sectional area A presented to the flow inversely affects terminal velocity, as v_t \propto 1/\sqrt{A}; larger areas experience greater for a given speed, reducing the equilibrium velocity. This parameter is measured perpendicular to the direction of motion and scales with the object's overall size. The C_d is a that quantifies the object's aerodynamic , typically ranging from 0.45 for a smooth to 0.70–1.0 for a human body depending on orientation. It inversely influences terminal velocity via v_t \propto 1/\sqrt{C_d}, with higher values indicating poorer streamlining and thus lower speeds at equilibrium. Shape profoundly impacts the , as streamlined forms minimize and compared to blunt bodies, which can have C_d values up to twice as high. For example, an object's —such as a skydiver transitioning from a feet-first (lower C_d \approx 0.70) to a spread-eagle position (higher C_d \approx 1.0)—can significantly alter and thereby terminal velocity. Material primarily affects terminal velocity indirectly by contributing to the object's total , serving as a secondary factor since the explicit dependence is on m rather than alone. For objects of fixed , higher increases and thus elevates v_t, but this is subsumed under the primary .

Fluid and Environmental Properties

The terminal velocity of an object falling through a fluid is inversely proportional to the square root of the fluid's density, as derived from the balance between gravitational force and quadratic drag force, where drag F_d = \frac{1}{2} C_d \rho A v^2 and terminal velocity occurs when mg = F_d, yielding v_t = \sqrt{\frac{2mg}{C_d \rho A}}. Higher fluid density increases drag, reducing v_t; for instance, an object reaches a much lower terminal velocity in water (density ≈ 1000 kg/m³) compared to air (density ≈ 1.2 kg/m³ at sea level), with speeds roughly 29 times higher in air due to the density ratio. Viscosity plays a key role primarily in low-speed, linear drag regimes, where the drag force follows , F_d = 6\pi r \eta v, and terminal velocity is v_t = \frac{mg}{6\pi r \eta}, making v_t inversely proportional to \eta. Higher , as in more resistant fluids like oils, increases the linear coefficient and lowers v_t, though its influence diminishes in high-speed quadratic drag where form drag dominates. Altitude and significantly affect terminal velocity through changes in air density; lower density at higher altitudes reduces , increasing v_t. At the summit of (≈8848 m), air density is approximately one-third that at (≈0.4 kg/m³ vs. 1.225 kg/m³), resulting in a terminal velocity about 73% higher than at , calculated as \sqrt{3} \approx 1.73 times greater. Temperature influences terminal velocity indirectly by altering both air and ; warmer temperatures decrease (following the , \rho \propto 1/T), which reduces and increases v_t, while also lowering , further promoting higher speeds in linear regimes. For example, in atmospheric conditions, a 10°C temperature rise can decrease air by about 3-4%, leading to a slight increase in v_t of roughly 2%. The , Re = \frac{\rho v L}{\eta}, governs the transition between drag regimes and thus the applicable terminal velocity formula, with low Re (typically < 1-10 for spheres) favoring linear viscous drag and high Re (> 1000) shifting to quadratic inertial drag. This dimensionless parameter incorporates fluid properties ( and ) alongside velocity and object size, determining whether terminal velocity calculations use or the quadratic form, with transitional behavior around Re \approx 10^3.

Variations and Extensions

Effects of Buoyancy

The buoyancy force acts upward on an object immersed in a fluid, equal to the weight of the displaced fluid, and is expressed as F_b = \rho_f V g, where \rho_f is the density of the fluid, V is the volume of the displaced fluid (equal to the object's volume for full submersion), and g is the acceleration due to gravity. This force effectively reduces the net downward gravitational force on the object by opposing its weight. At terminal velocity, the drag force F_d balances this net downward force, yielding (m - \rho_f V) g = F_d, where m is the object's ; this represents a reduction in effective compared to the buoyancy-free case. For quadratic drag, where F_d = \frac{1}{2} C_d \rho_f A v_t^2 with C_d as the and A as the cross-sectional area perpendicular to the flow, the terminal velocity becomes v_t = \sqrt{ \frac{2 (m - \rho_f V) g}{C_d \rho_f A} }. This adjusted formula accounts for the diminished net force due to buoyancy, resulting in a lower v_t than without it. The derivation follows by substituting the (m - \rho_f V) g into the condition from the standard quadratic balance mg = \frac{1}{2} C_d \rho_f A v_t^2, then solving for v_t by isolating the term and taking the . In air (\rho_f \approx 1.2 kg/m³), is negligible for most objects because \rho_f V \ll m, so the formula reduces to the standard form without the \rho_f V term. In denser fluids like (\rho_f = 1000 kg/m³), significantly lowers v_t; for instance, a with average of 945–1020 kg/m³ (varying with ) achieves near-zero terminal velocity when nearly neutrally buoyant.

Terminal Velocity in Non-Uniform Conditions

In real-world scenarios, terminal velocity deviates from the steady-state value derived for uniform fluids due to spatial or temporal variations in flow conditions. winds, for instance, introduce lateral forces that alter the of a falling object, while potentially increasing the effective through enhanced components. In gusty conditions, such as transverse gusts, the descent temporarily decreases in the transient , preventing a constant terminal velocity and sometimes even uplifting the object, with the effect intensifying for higher gust ratios, Galileo numbers, and nondimensional masses. Density gradients in the atmosphere, where air decreases with altitude, lead to non-uniform . For objects falling from high altitudes, the terminal velocity is initially higher in the thinner air but decreases as the object descends into denser air near the surface, causing deceleration if the speed exceeds the local terminal velocity. For a skydiver of 80 , terminal velocity is approximately 77 m/s at 10 altitude ( 0.42 /m³) but decreases to 45 m/s at ( 1.22 /m³), reflecting the inverse square-root dependence on in the . This variation is particularly evident in high-altitude jumps, where reduced allows higher speeds initially, but increasing slows the descent lower down. Turbulence and unsteady flows induce oscillations around the mean terminal velocity, affecting through mechanisms like nonlinear , vortex trapping, and preferential sweeping. In isotropic , these effects can either enhance or reduce speed depending on and intensity, with nonspherical particles exhibiting orientation oscillations that further destabilize the path. For particles with diameters comparable to the Kolmogorov scale, small-scale eddies predominantly slow the fall, while larger particles may experience net enhancement via fast tracking along vortex edges. In multi-phase media, such as transitions from air to , falling objects undergo transient adjustments due to abrupt changes in and . Upon entry, the higher in causes rapid deceleration, often accompanied by wake and upward formation, which can reverse the motion in a bouncing for low Reynolds numbers (below ~30-46). This adjustment phase involves enhanced from the interface and cavity formation, stabilizing to a new, lower terminal velocity in the denser medium. Numerical simulations, such as direct numerical solutions of equations in cellular , model these complex cases by accounting for and unsteadiness, revealing variations in terminal velocity with and pulsation frequency. For small in 2D , can increase initially before decreasing, providing insights into non-uniform effects without analytical solutions.

Real-World Examples

Human and Animal Falls

In skydiving, a in a spread-eagle position reaches a terminal velocity of approximately 53 m/s (120 ) after about 12 seconds of freefall, as balances with aerodynamic . Adopting a head-down reduces the effective cross-sectional area, increasing terminal velocity to around 89 m/s () and allowing faster descent rates for advanced maneuvers. Upon deployment, typically at altitudes around 1,000 meters, the canopy's large surface area and high dramatically lower terminal velocity to 5-8 m/s (11-18 ), enabling safe landings by extending the descent time and reducing impact forces. Smaller animals like squirrels and cats demonstrate remarkable resilience to falls due to their low terminal velocities, which result from favorable mass-to-drag ratios. Squirrels achieve a terminal velocity of roughly 10-24 m/s, aided by their fur and skin providing a high drag coefficient relative to body mass, allowing survival from effectively any height without injury upon impact. Cats, with their lightweight frames and ability to splay limbs for increased drag, reach a terminal velocity of about 100 km/h (60 mph), which, while potentially injurious, often permits survival through instinctive righting and relaxed landing postures. For humans, however, falls exceeding 60 meters are generally fatal without protective aids, as this height suffices to attain near-terminal speeds of over 50 m/s, resulting in impact forces beyond physiological tolerance and causing severe trauma or death. A notable exception occurred in 1972 when flight attendant Vesna Vulović survived a 10 km descent after her plane exploded; she remained pinned within a section of fuselage that acted as a protective enclosure, cushioning the impact in snow. Biological adaptations in further illustrate how terminal velocity influences survival and mobility during descent. , pneumatized bones reduce overall , lowering the gravitational and thus the resulting terminal velocity compared to denser mammalian skeletons of similar size. Feathers serve a dual role, providing and during flight while allowing birds to flare wings and tail for increased during controlled glides or landings, preventing excessive speeds and enabling precise aerial maneuvers without . These traits highlight evolutionary optimizations for aerial lifestyles, where managing and weight ensures safe interaction with gravitational forces.

Natural and Everyday Phenomena

Terminal velocity manifests in various natural and atmospheric phenomena, where falling objects reach a constant speed due to the balance between gravitational force and aerodynamic . For raindrops, a typical 2 mm drop attains a terminal velocity of approximately 9 m/s, at which point the force, influenced by the drop's shape and , prevents further without causing breakup. This equilibrium ensures raindrops fall steadily, contributing to the gentle patter observed during showers rather than destructive impacts. Snowflakes exemplify low terminal velocities due to their delicate structure, low , and high from irregular, feathery shapes. Aggregate snowflakes typically achieve speeds less than 1 m/s, resulting in their slow, gentle descent that allows for widespread, soft accumulation on the ground. In contrast, hailstones, formed from larger aggregates in thunderstorms, experience higher terminal velocities of 20-50 m/s owing to their increased and , which overcome more effectively. These speeds enable to cause significant damage to crops, vehicles, and structures upon impact. Meteorites entering Earth's atmosphere initially travel at hypersonic speeds exceeding 1000 m/s in the thin upper layers, where drag is minimal, but decelerate rapidly as they descend into denser air near the surface. In the 2013 Chelyabinsk event, the meteoroid fragmented at high altitude, with surviving pieces reaching terminal velocities of 18-33 m/s before ground impact, sufficient to penetrate ice but not cause explosive craters. Similarly, falling leaves demonstrate variable low terminal velocities of 0.5-2 m/s, dictated by their irregular, broad shapes that maximize drag relative to mass, leading to a fluttering, prolonged descent during autumn.

Practical Applications

Engineering and Design

In parachute engineering, designers optimize the canopy area and to achieve a controlled terminal velocity that ensures safe landing speeds, typically targeting 4.6 to 6.1 m/s (15 to 20 ft/s) for personnel systems at . This involves selecting canopy shapes like flat circular or triconical configurations with drag coefficients ranging from 0.6 to 1.2, allowing the drag force to balance gravitational pull at desired velocities. Materials such as ripstop fabric (e.g., 1.1 oz/yd² per MIL-C-7020 standards) are prioritized for their strength, flexibility, and durability under repeated deployments, while high-tenacity or lines reduce overall system weight by up to 40%. Drogue parachutes serve as initial stabilizers in multi-stage recovery systems, deployed from high-speed vehicles to reduce velocity and provide a stable platform for main parachute inflation, often limiting descent to intermediate speeds before full terminal velocity is reached with the primary canopy. In aerospace applications, such as crew capsules, drogue designs like ribbon or conical types with diameters of 3.3 to 5 m are engineered for rapid deceleration from post-reentry speeds exceeding 100 m/s. Vehicle safety features like and airbags indirectly incorporate terminal velocity principles by managing impact dynamics to limit deceleration forces, extending the collision duration from high-speed crashes (often 50-100 km/h) and reducing g-forces on occupants to survivable levels below 50g. , typically at the front and rear, deform controllably to absorb , effectively capping the rate of change akin to a drag-balanced limit. In engineering, terminal velocity calculations predict the maximum range and descent speed of projectiles like bullets or bombs, where proportional to the square of eventually balances for falling objects at 90-180 m/s depending on mass and shape. from imparts gyroscopic stability to projectiles, preventing tumbling that would increase the effective and thus lower terminal velocity; optimal spin rates maintain a streamlined , minimizing aerodynamic losses during flight. Fall arrest systems in bridge and building construction are designed to halt descents at velocities well below lethal terminal speeds (around 53 m/s for humans), with personal systems limiting free fall to 1.8 m (resulting in ~5.9 m/s impact) and safety nets positioned no more than 9.1 m below to ensure arrest forces do not exceed 8 (1,800 lbf). Nets, with mesh openings limited to 36 square inches, decelerate falls from typical heights of 1.2-9.1 m without allowing terminal velocity attainment, prioritizing . Aerospace reentry vehicles, such as capsules from , begin descent at hypersonic speeds of approximately 28,000 km/h (17,500 mph), far exceeding atmospheric terminal velocities, and rely on blunt-body shapes to generate peak drag during deceleration. Heat shields, often ablative materials like phenolic resins or reusable tiles (e.g., silicon-based for the ), protect against frictional heating peaking at 1,650°C during this phase, enabling safe reduction to terminal speeds before deployment.

Scientific and Biological Contexts

Wind tunnel testing serves as a cornerstone in aerodynamic research for measuring the C_d and terminal velocity v_t of various models, enabling precise validation of theoretical models across scales from small to full-sized . For instance, experiments on wings in low-speed s have quantified s around 0.40 at Reynolds numbers exceeding $10^4, informing bio-inspired designs and . Similarly, studies on impacts with surfaces have assessed C_d variations to mitigate bird-strike risks, with tests spanning velocities up to terminal conditions. At larger scales, s evaluate models to determine v_t under controlled conditions, as demonstrated in tests yielding C_d values that predict safe descent profiles for unmanned aerial systems. In biological modeling, terminal velocity plays a pivotal role in simulating dispersal patterns, particularly for and , where it governs horizontal and vertical . For wind-dispersed , mechanistic models like the Wald analytical long-distance dispersal (WALD) framework incorporate v_t alongside release height and wind turbulence to predict spread rates, with lower v_t values enhancing long-distance events in forests. Studies on seed traits emphasize that v_t variations, often measured via free-fall experiments, explain up to 70% of intraspecific dispersal potential in plant populations. For aquatic systems, dynamically scaled 3D-printed models of planktonic have been used to estimate sinking v_t, revealing values around 0.1–1 cm/s that influence vertical distribution and carbon flux in ocean layers. These models highlight how v_t determines ecological , such as in biophysical simulations of larval dispersal across protected areas. Atmospheric science leverages terminal velocity to understand rain formation and cloud physics, where it critically affects collision efficiencies and precipitation processes. In warm clouds, v_t differences between droplets—ranging from 1 cm/s for cloud-sized particles to 9 m/s for raindrops—drive coalescence via collision and collection, with higher efficiencies occurring when relative v_t exceeds 1 m/s. Theoretical analyses of drop shapes aloft confirm that v_t stabilizes at values balancing drag and gravity, influencing precipitation rates in diverse regimes from laminar to turbulent flows. For example, in cloud microphysics models, accurate v_t parameterization improves simulations of rainfall efficiency, as deviations can alter predicted precipitation by up to 20%. These insights stem from wind tunnel validations and field observations, underscoring v_t's role in global water cycle dynamics. Medical research examines terminal velocity in the context of fall injury thresholds, focusing on impact velocities that exceed human tolerance limits. Federal Aviation Administration studies on free-fall survivability indicate that impacts near human v_t of approximately 53 m/s (120 mph) result in severe injuries or fatality, with feet-first orientations offering marginal survival chances up to 80% of v_t due to reduced deceleration forces. High-velocity water entry experiments further reveal that buttocks-first impacts at near-v_t speeds cause spinal and pelvic fractures, establishing velocity thresholds for trauma prediction in forensic and emergency medicine. Regarding microgravity, spaceflight-induced musculoskeletal atrophy reduces bone density by 1–2% per month, potentially lowering post-mission tolerance to falls at Earth-equivalent v_t, as evidenced by astronaut recovery studies emphasizing increased fracture risk. In , terminal velocity varies significantly across worlds due to atmospheric , informing mission designs like rover drop tests. On Mars, the thin atmosphere (about 1% of Earth's ) yields higher v_t for objects—up to 100 m/s for entry vehicles—necessitating advanced parachutes and retro-rockets, as analyzed in Exploration Rover entry, descent, and landing simulations. Drop tests from high-altitude balloons mimic these conditions, validating v_t reductions via drag devices to achieve safe velocities below 20 m/s for rover deployment. These studies contrast with Earth's denser air, where v_t is lower, and extend to other bodies like , where thicker atmospheres enable slower descents for probe landings.