The Determination of the Altitude of Aircraft*

W. G. Brombacher

doi:10.1364/JOSA.7.000719

Journal of the Optical Society of America
Vol. 7,
Issue 9,
pp. 719-774
(1923)
•https://doi.org/10.1364/JOSA.7.000719

The Determination of the Altitude of Aircraft

W. G. Brombacher

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W. G. Brombacher, "The Determination of the Altitude of Aircraft*," J. Opt. Soc. Am. 7, 719-774 (1923)

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Abstract

SYNOPSIS

The determination of the altitude of bodies above the surface of the earth has interested physicists, meteorologists, balloonists, mountain climbers, and aviators. The methods used have been similar and rest on facts which have been well known for a long time. This paper is confined to the theory and methods of the accurate determination of altitude of aircraft above the earth’s surface and brings together all of the well known facts and methods in addition to some new material.

It is here emphasized that for aviation purposes, the altitude is determined by observations of pressure and temperature. A relation of pressure and temperature to altitude is derived. This expression is:

h = 62900 log \frac{P_{o}}{P} (1 + \frac{t_{m} - 10}{283})

in which h is the altitude in feet, P₀ is the atmospheric pressure at the ground level, P is the pressure at the upper level, and t_m is the mean temperature of vertical column of air between the two levels.

A more complete expression for the altitude is also derived in which the factors of slight practical importance are included. These factors are those of humidity and gravity.

The observations of pressure and temperature necessary to determine an altitude are discussed. The instruments used to make these observations are briefly described.

The method of computation of the altitude from these observations is outlined and illustrated by a sample computation.

The paper concludes with tables of upper air data, aircraft performance standards and altimeter calibration standards for the United States and other countries.

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1	2	3	4	5	6

Altitude 1000 ft.	Pressure mm Hg	Temperature °C	Vapor Pressures mm Hg	Density kg/m³	Density Relative to 1.225 kg/m³

0	760.0	19.1	10.54	1.202	0.981
1	736.4	17.1	9.27	1.172	.956
2	710.2	15.1	8.05	1.140	.930
3	685.0	13.1	7.01	1.108	.904
4	659.9	11.7	6.15	1.073	.876
5	637.0	10.4	5.41	1.040	.849
6	614.2	9.3	4.78	1.006	.821
7	592.1	7.5	4.17	0.977	.798
8	570.5	5.7	3.58	.948	.774
9	549.4	3.9	3.07	.919	.750
10	528.3	+ 2.1	2.64	.890	.727
11	507.8	+ 0.2	2.26	.862	.704
12	488.7	− 1.5	1.93	.835	.682
13	469.9	− 3.4	1.65	.808	.660
14	452.1	− 5.4	1.40	.783	.639
15	435.1	− 7.2	1.17	.758	.619
16	418.9	− 9.0	0.99	.736	.601
17	402.3	−11.0	.81	.712	.582
18	387.1	−13.0	.69	.690	.564
19	371.6	−15.0	.56	.668	.545
20	356.1	−17.0	.48	.645	.527
21	341.9	−19.0	.41	.624	.510
22	327.9	−21.1	.33	.605	.494
23	313.9	−23.2	.28	.584	.477
24	301.5	−25.2	.23	.565	.461
25	289.3	−27.2	.20	.546	.446
26	277.4	−29.2	.18	.528	.431
27	265.7	−31.2	.15	.510	.416
28	254.3	−33.1	.13	.492	.402
29	243.6	−35.0	.10	.476	.389
30	233.7	−36.8	.08	.459	.375
31	225	−38.7		.444	.362
32	216	−40.3		.428	.349
33	206	−41.9		.413	.337
34	197	−43.5		.398	.325
35	188	−45.0	.03	.383	.313
36	179	−46.7		.369	.301
37	171	−47.9		.355	.290
38	164	−49.0		.340	.278
39	158	−50.0		.327	.267
40	151	−50.8	.009	.313	.256
45	119	−54.0	.006	.251	.205
50	95	−55.3	.004	.199	.162

1	2	3	4

Altitude km	Pressure mm Hg	Temperature °C	Density kg/m³

0	762.8	9.0	1.258
1	677.3	5.3	1.134
2	598.5	+ 1.8	1.011
3	527.3	− 3.4	0.905
4	463.5	− 8.9	.815
5	407.3	−15.3	.733
6	356.3	−22.1	.662
7	309.8	−29.5	.592
8	268.5	−37.1	.528
9	231.8	−43.7	.470
10	199.5	−49.8	.415
11	171.0	−53.7	.365
12	146.2	−56.8	.314
13	125.2	−59.0	.268
14	106.5	−59.5	.233
15	90.0	−62.0	.198

1	2	3	4

Altitude km	Pressure mm Hg	Temperature °C	Density kg/m³

0	760.5	9.0	1.253
1	675.1	5.0	1.128
2	596.3	+ 0.2	1.014
3	524.3	− 5.3	0.909
4	461.3	−11.3	.819
5	403.5	−18.2	.735
6	351.8	−25.2	.658
7	305.3	−32.3	.589
8	264.0	−39.4	.524
9	227.3	−45.5	.463
10	195.8	−50.8	.409
11	168.0	−53.4	.355
12	144.0	−54.2	.305
13	123.0	−54.3	.261
14	105.0	−54.1	.223
15	90.0	−54	.191
16	76.5	−54	.162
17	65.6	−54	.139
18	56.2	−54	.119
19	48.1	−54	.102
20	41.1	−54	.087

1	2	3	4	5

Altitude km	Pressure mm Hg	Temperature °C	Vapor Pressure mm Hg	Density kg/m³

0	762.5	14.8	10.5	1.224
0.5	718.8	13.6	9.2	1.159
1.0	677.2	11.8	7.8	1.100
1.5	637.8	9.0	6.2	1.046
2.0	600.3	6.3	5.0	0.995
2.5	564.7	3.6	4.0	.946
3	530.8	+ 1.0	3.1	.898
4	468.2	− 4.3	1.9	.808
5	411.9	− 9.9	1.1	.727
6	361.3	−16.3	0.6	.653
7	315.8	−23.3		.587
8	275.0	−30.8		.527
9	238.4	−38.3		.472
10	205.8	−44.9		.419
11	176.9	−50.1		.369
12	151.8	−52.9		.319
13	130.1	−52.8		.274
14	111.6	−52.4		.234
15	95.7	−52.0		.201
16	82.0	−51.6		.172
17	70.3	−51.1		.148
18	60.3	−50.2		.127
19	51.7	−49.6		.109
20	44.4	−49.0		.093
25	20.6			.043
30	9.6			.020
35	4.5			.009
40	2.1			.004

1	2	3	4	5

Altitude km	Pressure mm Hg	Temperature °C	Vapor Pressure mm Hg	Density kg/m³

0	763.3	+ 1.7	4.7	1.288
0.5	717.4	+ 1.2	4.3	1.212
1.0	674.1	− 0.6	3.6	1.147
1.5	633.1	− 2.3	2.9	1.084
2.0	594.4	− 4.2	2.3	1.025
2.5	557.7	− 6.5	1.7	0.970
3	523.0	− 9.1	1.3	.920
4	458.9	−15.3	0.7	.827
5	401.3	−22.2		.743
6	349.6	−29.3		.666
7	303.3	−36.6		.596
8	261.9	−43.6		.530
9	225.4	−49.6		.469
10	193.2	−54.3		.410
11	165.2	−56.8		.355
12	141.1	−57.2		.303
13	120.5	−56.3		.259
14	103.0	−56.5		.221
15	88.0	−57.1		.189
16	75.2	−57.3		.162
17	64.2	−57.6		.138
18	54.9	−57.6		.118
19	46.9	−57.6		.101
20	40.1	−57.6		.086
25	18.3			.039
30	8.3			.018
35	3.8			.008
40	1.7			.004

1	2	3	4	5	6

Altitude km	Pressure mm Hg Altimeters	Pressure mm Hg Observations	Temperature °C	Density kg/m³	Density Relative 1.20585 to kg/m³

.0	762	761.8	15.5	1.223	1.0139
.5	717	717.2	13.2	1.160	.9622
1.0	675	675.1	10.7	1.102	.9137
1.5	635	634.7	8.0	1.046	.8678
2.0	597	596.9	5.1	.994	.8243
2.5	561	560.7	+ 2.1	.944	.7829
3.0	527	526.7	− 0.7	.897	.7435
3.5	495	494.7	− 3.6	.851	.7056
4.0	464	463.5	− 6.8	.807	.6692
4.5	435	434.8	− 9.9	.765	.6347
5.0	407	407.2	−13.3	.726	.6024
5.5	380	381.0	−16.7	.689	.5714
6.0	355	356.3	−20.2	.653	.5416
6.5	332	333.2	−23.9	.620	.5140
7.0	310	310.6	−27.7	.587	.4867
7.5	289	289.5	−31.5	.555	.4604
8.0	269	269.5	−35.2	.525	.4354
8.5	…	250.9	−38.7	.496	.4113
9.0	…	232.7	−42.0	.467	.3873
9.5	…	215.6	−45.3	.439	.3637
10.0	…	199.5	−48.3	.411	.3406

1	2	3	4	5	6

Altitude 1000 ft.	Pressure In. Hg	Pressure mm Hg	Temperature °C	Density Kg/m³	Density Relative

0	29.92	760.0	15.0	1.225	1.000
1	28.86	733.0	12.8	1.191	0.972
2	27.83	706.9	10.7	1.156	.944
3	26.82	681.2	8.7	1.122	.916
4	25.85	656.6	6.7	1.089	.889
5	24.91	632.7	4.8	1.057	.863
6	23.99	609.3	2.9	1.025	.837
7	23.10	586.7	+ 1.1	0.993	.811
8	22.24	546.9	− 0.6	.963	.786
9	21.41	543.8	− 2.3	.932	.761
10	20.60	523.2	− 3.9	.903	.737
11	19.82	503.4	− 5.5	.873	.713
12	19.07	484.4	− 7.0	.845	.690
13	18.34	465.8	− 8.5	.817	.667
14	17.63	447.8	−10.0	.790	.645
15	16.95	430.5	−11.3	.764	.624
16	16.30	414.0	−12.7	.739	.603
17	15.66	397.8	−14.0	.713	.582
18	15.05	382.3	−15.2	.688	.562
19	14.47	367.5	−16.4	.665	.543
20	13.90	353.1	−17.5	.640	.524
21	13.35	339.1	−18.7	.619	.505
22	12.82	325.6	−19.7	.597	.487
23	12.32	312.9	−20.8	.576	.470
24	11.83	300.5	−21.8	.555	.453
25	11.36	288.5	−22.7	.535	.437
26	10.91	277.1	−23.6	.516	.421
27	10.47	265.9	−24.5	.497	.406
28	10.05	255.3	−25.4	.479	.391
29	9.65	245.1	−26.2	.461	.376
30	9.26	235.2	−27.0	.443	.362

1	2	3	4	5	6

Altitude 1000 ft.	Pressure Relative 760 mm	Pressure mm Hg	Temperature °C	Density kg/m³	Density Relative to 1.221 kg/m³

0	1.000	760	9	1.252	1.025
1	0.964	733	7.5	1.214	0.994
2	.929	706	6	1.176	.963
3	.895	680	4.5	1.138	.932
4	.861	654	3	1.103	.903
5	.829	630	+ 1.5	1.067	.874
6	.798	606	0	1.032	.845
7	.768	584	− 1.5	0.999	.818
8	.739	562	− 3	.967	.792
9	.711	540	− 4.5	.934	.765
10	.684	520	− 6	.904	.740
11	.658	500	− 8	.875	.717
12	.633	480	−10	.849	.695
13	.607	461	−12	.822	.673
14	.583	443	−14	.796	.652
15	.560	426	−16	.769	.630
16	.538	409	−18	.745	.610
17	.516	392	−20	.720	.590
18	.496	377	−22	.697	.571
19	.476	362	−24	.675	.553
20	.456	347	−26	.653	.535
21	.437	332	−28	.629	.515
22	.419	318	−29.5	.608	.498
23	.402	306	−31.5	.587	.481
24	.385	293	−33	.567	.464
25	.369	280	−35	.547	.448
26	.353	268	−37	.527	.432
27	.338	257	−38.5	.509	.417
28	.324	246	−40.5	.491	.402
29	.310	236	−42	.474	.388
30	.296	225	−44	.452	.370

1	2	3	4	5

Altitude meters	Pressure mm Hg	Constant A	Altitude meters	Density Relative

0	760
994	672.2	18756	0	1.000
2000	592.7	18522	869	0.907
3007	520.9	18286	1808	.819
4001	457.1	18116	2845	.735
5001	399.4	17895	4019	.649
5994	347.9	17681	5377	.563
6991	301.8	17452	6952	.469
7992	260.7	17217	8903	.367
8995	224.5	16994	11523	.253
10011	192.4	16761	13379	.192
11025	164.6	16557	15976	.128
12034	140.7	16382	17390	.103
13018	120.6	16260	20360	.064
14008	103.2	16145
15003	82.2	16037
15971	75.7	15973
16985	64.5	15869

Mean Temp. °C		Isothermal Altitude

		5000	10000	15000	20000	25000	30000	35000	40000

10	10	0	0	0	0	0	0	0	0
9	11	20	40	50	70	90	110	. . . .	. . . .
8	12	40	70	110	140	180	210	. . . .	. . . .
7	13	50	110	160	210	270	320	. . . .	. . . .
6	14	70	140	210	280	350	420	. . . .	. . . .

5	15	90	180	270	350	440	530	. . . .	. . . .
0	20	180	350	530	710	880	1060	1240	1410
− 5	25	270	530	800	1060	1330	1590	1850	2120
−10	30	350	710	1060	1410	1770	2120	2470	2830
−15		440	880	1330	1770	2210	2650	3090	3530

−20		530	1060	1590	2120	2650	3180	3710	4240
−25		620	1240	1860	2470	3090	3710	4330	4950
−30		. . . .	. . . .	. . . .	. . . .	. . . .	. . . .	4950	5660

1	2	3	1	2	3

Altitude 1000 ft.	Pressure Millimeters of Hg	Pressure Inches of Hg	Altitude 1000 ft.	Millimeters of Hg	Inches of Hg

0	759.6	29.90	30	253.2	9.97
1	732.4	28.83	31	244.2	9.61
2	706.0	27.79	32	235.4	9.27
3	680.6	26.79	33	227.0	8.94
4	656.2	25.83	34	218.8	8.61
5	632.6	24.90	35	211.0	8.30
6	609.8	24.00	36	203.4	8.01
7	587.8	23.14	37	196.0	7.72
8	566.8	22.31	38	189.0	7.44
9	546.4	21.51	39	182.2	7.17
10	526.8	20.73	40	175.6	6.91
11	507.8	19.99
12	489.6	19.27
13	472.0	18.58
14	455.0	17.91
15	438.6	17.27
16	422.8	16.65
17	407.6	16.05	Temperature:The temperature is +10°C at all altitudes.
18	393.0	15.47
19	378.8	14.91
20	365.2	14.38
21	352.2	13.86
22	339.4	13.36
23	327.2	12.88
24	315.4	12.42
25	304.2	11.98
26	293.2	11.54
27	282.6	11.13
28	272.4	10.73
29	262.8	10.34

1	2	3

Altitude 1000 ft.	Pressure in Hg	Pressure mm Hg

0	29.90	759.5
1	28.82	732.0
2	27.78	705.6
3	26.77	680.0
4	25.81	655.6
5	24.88	631.9
6	23.98	609.1
7	23.11	587.0
8	22.27	565.7
9	21.47	545.3
10	20.69	525.5
11	19.95	506.7
12	19.23	488.4
13	18.53	470.7
14	17.86	453.6
15	17.22	437.4
16	16.61	421.9
17	16.00	406.4
18	15.42	391.7
19	14.86	377.4
20	14.33	364.0
21	13.80	350.5
22	13.31	338.1
23	12.83	325.9
24	12.36	313.9
25	11.92	302.8
26	11.49	291.9
27	11.07	281.2
28	10.67	271.0
29	10.29	261.4
30	9.93	252.2

1	2	3

Altitude in km	Pressure mm Hg	Temperature °C

0	762	8.7
2	595	− 1.0
4	461	−11.6
6	354	−23.7
8	272	−37.0
9	240	−45.0

1	2	3	4

Altitude m	Pressure mm Hg	Temperature	Density kg/m³

0	762.0	10.0	1.250
500	717.2	7.5	1.188
1000	674.6	5.0	1.127
1500	634.2	2.5	1.069
2000	595.9	0	1.013
2500	559.7	− 2.5	0.960
3000	525.3	− 5.0	0.910
3500	492.7	− 7.5	0.862
4000	461.9	−10.0	0.815
4500	432.8	−12.5	0.770
5000	405.2	−15.0	0.729
5500	379.2	−17.5	0.689
6000	354.6	−20.0	0.651
6500	331.5	−22.5	0.614
7000	309.6	−25.0	0.579
7500	289.0	−27.5	0.546
8000	269.5	−30.0	0.515

Pressure mm Hg	Average Altitude in Feet

	Winter	Summer	Year

700	2400	2400	2400
600	6380	6800	6600
500	11150	11750	11400
400	16650	17750	17200
300	23350	24850	24150
200	32600	34500	33600
150	38800	41000	40100

1	2	3	4

Altitude km	Pressure mm Hg	Temperature °C	Density kg/m³

0	758.9	27	1.174
1	677.3	22	1.067
2	602.3	17	0.968
3	534.8	12	.871
4	474.1	+ 6	.789
5	418.5	− 1	.714
6	368.3	− 8	.645
7	322.5	−15	.581
8	282.0	−22	.522
9	245.3	−30	.469
10	212.3	−38	.419
11	183.0	−46	.374
12	156.8	−54	.331
13	133.5	−62	.294
14	114.0	−70	.261
15	96.0	−75	.225
16	80.2	−78	.191
17	67.5	−80	.162

1	2	3	4	5

Altitude km	Pressure mm Hg	Temperature °C	Density kg/m³	Density Relative

0	760	15	1.225	1.000
1	715	12	1.165	0.951
2	674	9	1.110	.906
3	635	5	1.060	.865
4	597	2	1.010	.824
5	562	0	0.957	.782
6	526	− 3	.905	.739
7	493	− 6	.858	.700
8	462	− 8	.810	.662
9	434	−11	.769	.628
10	407	−13	.726	.594
11	381	−15	.686	.560
12	356	−17	.645	.526
13	333	−19	.612	.500
14	312	−20	.573	.468
15	293	−22	.542	.442
16	274	−23	.510	.416
17	257	−25	.482	.393
18	240	−27	.453	.370
19	225	−28	.427	.349
20	210	−29	.400	.327

1	2	3	4	5

Altitude km	Pressure mm Hg	Temperature °C	Density kg/m³	Density Relative

0	760.0	15	1.225	1.000
.5	715.9	11.75	1.166	0.953
1.0	674.1	8.5	1.111	.907
1.5	633.0	5.25	1.058	.864
2.0	596.1	2	1.006	.832
2.5	560.0	− 1.25	0.957	.781
3.0	525.7	− 4.25	.909	.742
3.5	493.1	− 7.75	.863	.704
4.0	462.2	−11	.819	.668
4.5	432.8	−14.25	.777	.634
5.0	405.0	−17.5	.736	.600
5.5	378.6	−20.75	.695	.568
6.0	353.7	−24	.660	.538
6.5	330.2	−27.25	.624	.509
7.0	307.8	−30.5	.589	.481
7.5	288.7	−33.75	.556	.454
8.0	266.8	−37	.525	.428
8.5	248.1	−40.25	.495	.404
9.0	230.4	−43.5	.466	.380
9.5	213.7	−46.75	.439	.358
10.0	198.1	−50	.412	.337
10.5	183.4	−53.25	.387	.316
11	169.6	−55.5	.364	.297
11.5	156.7	*	.336	.274
12.0	144.8		.311	.254
12.5	133.8		.287	.234
13.0	123.7		.265	.216
13.5	114.3		.245	.200
14.0	105.6		.226	.185
14.5	97.88		.210	.171
15.0	90.25		.194	.158

n	Isothermal Altitude interval thousand feet	Mean temperature of the interval	∑t_n	$t_{m} = \frac{Σ t_{n}}{n}$

1	0– 1	14	14	14
2	1– 2	11	25	12.5
3	2– 3	9	34	11.5
4	3– 4	8	42	10.5
5	4– 5	6	48	9.5
6	5– 6	4	52	8.5
7	6– 7	1.5	53.5	7.5
8	7– 8	−1.5	52	6.5
9	8– 9	−4	48	5.5
10	9–10	−5	43	4.5
11	10–11	−7	36	3.5

N. or S. Latitude	Altitude in 1000 feet, h

	0	5	10	15	20

0	0	+13	+27	+39	+52
10	0	+13	+25	+37	+49
20	0	+10	+21	+30	+40
30	0	+ 7	+14	+19	+26
40	0	+ 2	+ 5	+ 6	+ 8

45	0	0	0	0	0

50	0	− 2	− 4	− 7	− 9
60	0	− 6	−13	−20	−27
70	0	−10	−20	−30	−41

Abstract

Cited By

Figures (13)

Tables (22)

Equations (85)

Journal of the Optical Society of America