Three readings, and the one each answer leans on

The thermometer that heats itself followed one error through an aircraft’s air data: a total-temperature probe recovering 98 per cent of the stagnation rise, reduced as though it recovered all of it, reports the air 0.56 kelvin too cold at Mach 0.85. It followed that error into the speed of sound and the true airspeed and found it negligible there, and it ended by naming the calculation it had not done. Total pressure, static pressure and total temperature are three readings; the air’s state has three unknowns; so the reduction is exactly determined and its errors pass through a Jacobian that can be written down.

This essay writes it down. It turns out to be a small table with a structure more interesting than its entries, and two of its rows say things the probe essay’s reasoning did not. The Mach number’s dependence on the pressures is not modest at low speed; it is unbounded. And the static temperature’s dependence on those same pressures is pinned near one value at every speed, for a reason that is a cancellation rather than a coincidence.

Three readings, three unknowns #

The flush static port reads the static pressure $p_s$. The pitot tube, facing the flow, reads the pressure of air brought to rest, $p_t$, which in subsonic flow is the isentropic stagnation pressure — the series whose first term is ½ρU². The total-temperature probe reads $T_{ti}$, which is the static temperature raised by a fraction $r$ of the stagnation rise, the recovery arithmetic of a wall told nothing about its temperature.

Running those three relations backwards gives the air’s state. With $k = (\gamma-1)/2$ and the pressure ratio $\pi = p_t/p_s$,

$$M^2 = \frac{1}{k}\left(\pi^{(\gamma-1)/\gamma} - 1\right),\qquad T = \frac{T_{ti}}{1 + rkM^2},\qquad V = M\sqrt{\gamma R T},\qquad \rho = \frac{p_s}{RT}.$$

The order matters and it is fixed by what each relation contains. The Mach number comes from the pressures alone. The temperature needs the Mach number, to know how much of the probe’s reading is recovered rise. The airspeed and the density need the temperature. Errors flow downhill through that chain and never back up it: nothing about the thermometer reaches the Mach number, while everything about the pressures reaches everything.

The convenient way to write the sensitivities is logarithmically, $e = \partial\ln(\text{output})/\partial\ln(\text{input})$, so that a fractional error in a reading times $e$ is the fractional error it puts into a result. Three of the entries are then exact integers or halves for structural reasons: the temperature leans on its probe by exactly one, the airspeed by exactly one half through the square root in the sound speed, and the density by exactly minus one. The interesting entries are the others.

The Mach number is a small difference of two large pressures #

The Mach number depends on the two pressures only through their ratio, so its sensitivities to them are equal and opposite, and differentiating the first relation gives each in closed form:

$$e(M, p_t) = -e(M, p_s) = \frac{\pi^{(\gamma-1)/\gamma}}{\gamma M^2}.$$

At low speed the numerator tends to one and the denominator to zero, so the sensitivity grows as $1/\gamma M^2$ without limit. The reason is the one every differential instrument meets. At 11 km, where the static pressure is 22,632 pascals, the pitot tube reads only 1,458 pascals more than the static port at Mach 0.3, and 13,666 more at Mach 0.85. The Mach number is being inferred from a difference that is a small fraction of either reading, and an error that is a small fraction of a reading is a large fraction of that difference — the same predicament as a flowmeter inferring a flow from a pressure drop across a restriction that has been made gentle to save energy.

So the Mach number’s accuracy is a property of the speed as much as of the sensors. A pressure transducer good to a tenth of a per cent gives the Mach number to 0.81 per cent at Mach 0.3 and to 0.11 per cent at Mach 0.85, and the same transducer at a quarter of that speed would give it to about 13 per cent.

What a differential sensor does and does not buy #

The usual remedy is to measure the pitot difference $q_c = p_t - p_s$ directly with a differential sensor and keep one absolute sensor for the static pressure. It is worth being exact about what that changes, because the obvious account — it avoids subtracting two large readings — is only partly right.

Written in the new readings, the Mach number’s sensitivity to the differential reading is $e(M,\pi)\,q_c/p_t$, which tends to one half at low speed rather than growing without limit: 0.499 at Mach 0.1, 0.489 at Mach 0.3, 0.426 at Mach 0.85. The static reading’s sensitivity, with the difference held, is the same number with a minus sign. The $1/M^2$ has left the sensitivities. It has not left the problem, because it is now in the reading itself: the pitot difference at 11 km is 159 pascals at Mach 0.1 and 1,458 at Mach 0.3, and an error of fixed size is a large fraction of a small difference whichever way the sensors are arranged.

So with sensors of equal accuracy in pascals the gain is modest. Two absolute sensors good to 25 pascals give the Mach number to 1.23 per cent at Mach 0.3; a differential sensor and an absolute one, each good to 25 pascals, give it to 0.84 per cent — better by about the square root of two, and at Mach 0.1 the error is still 7.9 per cent.

The real gain is range. A pressure sensor’s error is usually proportional to its full scale, and a differential sensor only has to span the difference. Taking both as good to a twentieth of a per cent of range — an absolute sensor spanning 115 kilopascals, so ±57.5 pascals, and a differential one spanning 20, so ±10 — the two-absolute arrangement gives the Mach number to 2.82 per cent at Mach 0.3 and the differential one to 0.36 per cent: eight times better, and 3.1 against 25.6 per cent at Mach 0.1. The differential sensor wins by being a small instrument for a small quantity, not by avoiding a subtraction. Those full scales and that accuracy class are stated values; the factor of two between the two kinds of gain is not.

A sensitivity pinned near minus 0.28 #

The static temperature inherits the pressures’ error through the Mach number, and here the chain does something unexpected.

$$e(T, p_t) = -\frac{2rkM^2}{1 + rkM^2}\; e(M, p_t).$$

The first factor is the share of the probe’s reading that is recovered rise, doubled, and it grows as $M^2$. The second is the Mach number’s own sensitivity, and it falls as $1/M^2$. They cancel.

The product tends to $-(\gamma-1)r/\gamma$ as the Mach number falls, which is −0.2800 for air and a probe recovering 0.98, and it barely leaves that value across the subsonic range: −0.2801 at Mach 0.3, −0.2807 at Mach 0.85. At low speed the Mach number is badly known but hardly matters to the temperature, because almost none of the probe’s reading is rise; at high speed the Mach number matters a great deal to the temperature, but it is now well known. The two effects trade exactly.

That is the precise form of a remark the probe essay made in passing — that taking the Mach number from the pressures keeps the temperature channel from feeding back into the Mach number. The converse is now quantified: the pressure channel does feed forward into the temperature, and it does so at a fixed rate of about 0.28 per unit of fractional pressure error, whatever the speed.

The recovery-factor term behaves the ordinary way. It is the share of the reading that is rise, and it grows with $M^2$, from −0.017 at Mach 0.3 to −0.124 at Mach 0.85; it is the logarithmic form of the probe essay’s −27.8 kelvin per unit of recovery factor.

That term is where a drifting probe shows, and the table says where it shows most. The probe essay described a recovery factor that changes slowly in service, from erosion or contamination, without any reading looking wrong. A drift of 0.005 puts 0.0088 per cent into the density and 0.0044 into the true airspeed at Mach 0.3, and 0.063 and 0.032 at Mach 0.85: seven times more at the higher speed, and in both cases twice as much into the density as into the airspeed, because the density carries the temperature to the power minus one and the airspeed to the power one half. A slow calibration drift is a density error before it is an airspeed error, and a cruise error before it is an approach one.

Where the airspeed’s error comes from #

The true airspeed is the Mach number times the speed of sound — a speed set by the temperature and nothing else — so it takes the Mach number’s large pressure sensitivity whole and adds half the temperature’s. To say which reading dominates, the sensitivities need errors to multiply, and those are a stated budget: each pressure sensor good to 25 pascals, the probe to half a kelvin, the recovery factor known to 0.005, all independent and combined as a root sum of squares. Those are values of the order of air-data practice, not a specification, and the essay’s structure does not depend on them.

At 11 km and Mach 0.3 the pressure sensors put 1.20 per cent into the true airspeed and the temperature channel 0.11: the airspeed’s error is almost entirely the pressures’. By Mach 0.85 the pressure contribution has fallen to 0.13 per cent, because the Mach number’s sensitivity has fallen by a factor of seven, while the temperature channel’s has barely changed. The two are equal only at Mach 0.930 — so at cruise altitude, for this budget, the pressure sensors decide the airspeed’s accuracy over nearly the whole subsonic range and the thermometer matters only near its top.

That is not the division the obvious reasoning gives. True airspeed is a Mach number times a sound speed, and the sound speed depends only on temperature, so it is natural to picture the thermometer as the airspeed’s weak link. The table says the Mach number is the weak link, and the temperature’s error is a square root away from the answer while the pressures’ error is a $1/M^2$ away.

Altitude moves the handover #

A pressure error stated in pascals is a fraction of the pressure, and the pressure at 11 km is less than a quarter of the sea-level value, so the same sensors are worse relative sensors at altitude.

At sea level the same 25 pascals are a fraction 4.48 times smaller of the static pressure, and the pressure contribution to the airspeed error is correspondingly smaller at every Mach number. The handover moves from Mach 0.930 down to Mach 0.524: at sea level the thermometer governs the airspeed’s accuracy above about half the speed of sound, and at cruise altitude hardly at all.

The practical reading is that an air-data error budget has no single dominant term. Low and slow, the pressure transducers decide the airspeed; low and fast, the probe does; high, the pressures decide almost everything. An instrument upgrade that improves the wrong channel for the flight condition that matters buys nothing, and which channel is wrong is a function of altitude and Mach number that this table gives directly.

Density belongs to the thermometer #

The density is the static pressure over $RT$, and its sensitivities have the opposite structure to the airspeed’s.

The density leans on the temperature by exactly minus one, and on the static pressure by $1 + e(T, p_t)$, about 0.72 at every speed, because the pressure’s direct effect is partly offset by its effect on the inferred temperature. Neither contains the $1/M^2$ the Mach number carries, since the density does not need the Mach number except through the temperature, where the cancellation above removes it.

So with the stated budget the temperature channel contributes 0.23 per cent to the density at Mach 0.3 and 0.21 at Mach 0.85, against 0.08 from the pressures, and there is no Mach number at either altitude at which the two are equal. A quantity that depends on density — an engine’s corrected mass flow, a true dynamic pressure computed from true airspeed, a gust load — inherits the thermometer’s accuracy, while one that depends on Mach number inherits the pressure sensors’. The two families of calculation downstream of the same three readings have opposite weak links.

Checked by perturbing the readings #

Sixteen sensitivities written in closed form are sixteen chances to drop a factor, and a table like this is only useful if its entries are right. So each is checked by a route that shares none of its algebra: perturb one reading by a part in a million either way, run the whole reduction again, and take the difference of the logarithms.

The two routes agree to a few parts in a billion everywhere, including at low speed where the Mach number’s sensitivity exceeds eight and a dropped factor of $\gamma$ or a misplaced square would show immediately. The same check confirms the three exact entries and the equal-and-opposite pressure pairs to machine precision, and the low-speed limit of the temperature’s pressure sensitivity against the closed form $-(\gamma-1)r/\gamma$ evaluated at a thousandth of Mach one.

What the budget borrows #

The sensor errors are stated, not specified. Twenty-five pascals, half a kelvin and 0.005 on the recovery factor are values of a plausible order. Every handover Mach number in this essay moves if they do; the $1/M^2$ growth, the pinned −0.28 and the exact exponents do not.

The errors are independent. A real static port has a position error that scales with the dynamic pressure and with the aircraft’s attitude and configuration — the instrument disturbing the flow it reports, in the form every air-data calibration has to measure — so its error is correlated with the very quantity the pitot difference measures. Correlated errors do not combine as a root sum of squares, and the static-source error is often the largest single term in a real budget; nothing here models it.

Subsonic only. Above Mach one the pitot tube reads the stagnation pressure behind its own bow shock, the reduction is the Rayleigh pitot formula rather than the isentropic one, and every entry of the Mach row changes.

The probe is steady and its recovery factor constant. A real probe lags the way a heated skin does, on a shorter kernel, and its recovery factor drifts with erosion and contamination; both are errors in the temperature channel of a kind no fixed budget represents.

A perfect gas. The relations assume constant specific heats, which holds comfortably in the subsonic range but is the same assumption a probe at high speed eventually violates.

Who worked it out #

Linearised error propagation of this kind is older than air data and is the ordinary method of any measurement that is reduced through formulas: the variance of a result from the variances of its inputs, weighted by the squared partial derivatives. What air-data practice added was the discipline of splitting the budget by flight condition, and the recognition, as central air-data computers replaced separate instruments in the 1950s and 1960s, that the pressure transducers and the temperature probe matter in different parts of the envelope.

The sensitivities themselves follow from the isentropic pitot relation, which is Saint-Venant and Wantzel’s energy equation of the 1830s, and from the recovery relation the probe essay applies. Nothing in the table is new; what writing it out adds is the cancellation in the temperature’s row and the altitude-dependent handover in the airspeed’s, neither of which is visible from any single entry.

Still open: the probe as a transient #

The probe essay named two continuations, and this essay has taken one. The other is the probe’s own lag: a total-temperature probe has a thermal time constant of a second or two, so during a climb or an acceleration its reading trails the air, and the error is a convolution of the flight profile with the probe’s kernel rather than a fixed offset. Two probes of different mass on one aircraft disagree during a manoeuvre and agree in cruise, and the disagreement measures the kernel.

What this essay adds to that question is where the lag would show. Because the density leans on the temperature by exactly minus one at every speed, a lagging probe’s error passes into the inferred density undiminished, while the airspeed at altitude is barely affected until the upper end of the subsonic range. A manoeuvre at cruise altitude would show the lag in the density-derived quantities first — which is the kind of prediction that makes an unmodelled transient measurable.