Parachute Recovery — Theory & documentation

Overview

Recovering every piece of a student rocket intact is harder than it sounds. The usual scheme deploys a drogue high up to bring the dynamic pressure down, then a main parachute low down. The scheme analysed here, suggested by Adam Vore, uses a single parachute instead.

Just below apogee the rocket separates into a heavy, solid nose (payload or ballast) and a lighter aft body that carries the electronics. A small charge pushes the parachute out; the nose pulls out a tether coiled round an arbor, and the tether extracts the parachute. The parachute deploys reefed — held partly closed by a cord round its skirt — so the rocket descends fast and drifts little. Near the ground the reefing line is cut, the canopy opens fully and the descent slows. At landing the heavy nose touches down first; the tether goes slack and the aft body, now carried by the whole parachute, slows further before it lands.

Descent through the troposphere

In steady descent the drag balances the weight of everything under the parachute, M=m_1+m_2:

Mg=\tfrac12C_D\rho SV^2,\qquad V=-\frac{dh}{dt}=\sqrt{\frac{2Mg}{C_D\rho S}}

where S=\pi D_0^2/4 is the flat area of the canopy. In the troposphere the temperature falls linearly, T=T_0\tau with \tau=1+\alpha h/T_0, and the density follows

\rho=\rho_0\tau^m,\qquad m=-\frac{g}{\alpha R}-1\approx4.256

so V=V_0\tau^{-m/2}, where V_0 is the sea-level descent speed. Integrating dt=-dh/V between two altitudes gives the time in closed form:

t=\frac{T_0\left(\tau_a^{\,e}-\tau_b^{\,e}\right)}{\alpha V_0\,e},\qquad e=\frac m2+1

With reefing the descent has two parts: from deployment to disreef with the reefed drag, then from disreef to the ground with the full drag. The total time sets the wind drift, x_w=w\,t for a mean wind w. Altitudes in the model are above sea level, so the landing area’s elevation matters: thinner air at a high desert site means a faster landing.

Reefing

An open flat circular canopy has a mouth about 0.62 of its flat diameter. Constricting it with a reefing line of diameter D_R reduces the drag as Knacke’s measurements show (the “Reefed drag” plot, digitised from the reefing memo). A larger reefing ratio means more drag, a slower descent and more drift; a smaller one means a faster descent but a higher speed at disreef. For small parachutes keep D_R/D_0 above about 0.2 (0.06 for canopies over 4 m), or the canopy may be unstable.

The reefing line must hold the canopy at deployment. Dimensional analysis and Knacke’s data for a 28 ft canopy suggest

S_R\ge q_D\,\frac\pi4D_0^2\,C_S,\qquad C_S\approx0.02\ \ (D_R\ge0.06D_0)

where q_D is the dynamic pressure just before deployment.

Landing

The nose lands at the steady descent speed at ground level, V_I. From then on the parachute carries only the aft body, which decelerates:

m_2\frac{dV}{dt}=m_2g-\tfrac12C_D\rho SV^2\quad\Rightarrow\quad-\tfrac12m_2\frac{dV^2}{dh}=m_2g-\tfrac12C_D\rho SV^2

Starting from V_I when the aft body is a tether length L above the ground, it lands at

V_{aft}^2=\frac{2m_2g}{C_D\rho S}+\frac{2m_1g}{C_D\rho S}\exp\left(-\frac{C_D\rho SL}{m_2}\right)

The speed falls from the nose speed towards \sqrt{2m_2g/(C_D\rho S)} over a distance of order m_2/(C_D\rho S); a tether several times longer than that gains little more. The memo’s example uses a 33 ft tether, which a 2 in spool of 1/8 in nylon cord can hold.

Wind drift

The parachute drifts with the wind for the whole descent, and the drift grows with the descent time. That is why reefing high up, to come down fast, helps keep the rocket on the range. There are two ways to give the wind.

One average wind w: the drift is x_w=w\,t. Winds aloft are usually stronger than at the ground, so average the forecast winds over the heights of the descent.

Wind layers: a table of height, speed and the direction the wind blows from (the meteorological convention: 270° comes from the west and carries the rocket east). Between layers the east and north components are interpolated linearly; above and below the table the end layers hold. The calculator follows the rocket through the descent, adding up the drift in each time step:

x=\int u(h)\,dt,\quad y=\int v(h)\,dt,\quad dt=\frac{dh}{V(h)}

Here u and v are the east and north wind components at the rocket’s height. The result is the landing point: its distance \sqrt{x^2+y^2}, which the verdict compares with the recovery area, and its bearing. A uniform layer profile reproduces the single-wind drift exactly; the tests check this.

Forecast winds. “Load winds from Open-Meteo” fetches the forecast for the latitude, longitude and hour you give. It takes the 10 m wind and the winds on 14 pressure levels from 1000 to 200 hPa, each at its forecast geopotential height minus the site elevation, and keeps the levels that are above the ground. It also sets the ground elevation. The request goes from your browser straight to Open-Meteo; nothing else is sent. Weather data by Open-Meteo.com, licensed under CC BY 4.0, free for non-commercial use. Open-Meteo’s documentation does not state its direction convention; it is taken as the meteorological “from” direction.

Verdict and what to change

You give two limits: the landing speed you accept for the aft body, V_T, and the radius of your recovery area, R_f. The calculator does not impose either value; they depend on your hardware and your launch site.

Verdict	Landing speed	Drift
SAFE	V_{aft}\le V_T	x_w\le0.8\,R_f
MARGINAL	up to 1.25\,V_T	up to R_f
UNSAFE	above 1.25\,V_T	beyond R_f

The 1.25 and 0.8 bands are this project’s choice (DECISIONS D25), a margin for the uncertainty in wind and drag coefficient, not published limits. When the design misses a limit, the calculator solves for:

the parachute diameter that gives exactly V_T, by bisection on the landing-speed formula;
the tether length that gives V_T, in closed form from the impact formula: L=-\frac{m_2}{C_D\rho_GS}\ln\frac{V_T^2-2m_2g/(C_D\rho_GS)}{2m_1g/(C_D\rho_GS)}. There is none if V_T is below the endless-tether speed;
the disreef height that keeps the drift inside R_f, by bisection. With reefing off, it reports this for a 0.25 reefing ratio.

The Choosing the parachute size plot shows the trade-off directly. A bigger canopy lowers the landing speed but raises the drift; the sizes that satisfy both limits lie between the two crossings.

Verification

The tests compare the closed-form descent time with direct numerical integration of dt=dh/V, the landing formula with an integration of the aft body’s equation of motion over the tether, and the constant-density case with the 25 000 ft dart-descent table. The field tool’s cells are reproduced exactly when its own density exponent is used (see below). The reefing curve is checked to be monotone and to give the memo’s reefed descent speed (≈ 61 ft/s at sea level) within 2 ft/s.

Notes on the method

Density exponent. The original field tool writes the troposphere density as \rho_0\tau^{-1/(\alpha R)-1} with R=1716\ \mathrm{ft^2\,s^{-2}\,{}^\circ R^{-1}}, which omits g (the exponent needs R/g=53.3\ \mathrm{ft\,{}^\circ R^{-1}}). Its density then barely changes with altitude, and its descent speed even falls with height. The calculator uses the standard exponent -g/(\alpha R)-1. For the tool’s example (25 000 ft to 3 400 ft) the descent takes 257 s instead of the tool’s 334 s, and the landing speeds use the actual density at the ground.
Steady descent. Deployment transients and opening shock are not modelled; use an opening-shock tool (such as OSCALC) for the loads.
Troposphere. Deployment must be below the tropopause (about 36 000 ft).

References

C. P. Hoult, “Dual Purpose Parachute Recovery System (rev. 2),” RS&T, 2 May 2013. View in library
C. P. Hoult, “Parachute Reefing,” RS&T, 25 July 2013. View in library
T. W. Knacke, “Reefing of Parachutes,” U.S. Air Force Aeronautical Systems Division, AD-A036 497, 1976.
G. Peek and J. Potvin, “OSCALC — Opening Shock Calculator, Version 1.01, User’s Manual,” Parks College Parachute Research Group, Saint Louis University.

Parachute Recovery — theory