899:
1803:
20:
1568:
2409:
2009:
in a thin gas disk around a central object can change the scale height of the disk. For example, if a non-perfectly conducting disk is rotating through a poloidal magnetic field (i.e., the initial magnetic field is perpendicular to the plane of the disk), then a toroidal (i.e., parallel to the disk
906:
For a disk of gas around a condensed central object, such as, for example, a protostar, one can derive a disk scale height which is somewhat analogous to the planetary scale height. We start with a disc of gas that has a mass which is small relative to the central object. We assume that the disc is
793:, which shows the air density dropping from 1200 g/m at sea level to 0.125 g/m at 70 km, a factor of 9600, indicating an average scale height of 70 / ln(9600) = 7.64 km, consistent with the indicated average air temperature over that range of close to 260 K.
2211:
2936:
2247:
1798:{\displaystyle h_{D}={\sqrt {\frac {2kTr^{3}}{GM_{*}{\bar {m}}}}}\approx 0.0306{\sqrt {\frac {\left(T/100\ {\text{K}}\right)\left(r/1{\text{ au}}\right)^{3}}{\left(M_{*}/M_{\odot }\right)\left({\bar {m}}/2{\text{ amu}}\right)}}}\ {\text{ au}}}
1421:
1505:
2773:
1023:
1209:
505:
2019:
1270:
643:
2805:
562:
149:
352:
423:
1957:
2242:
192:
1328:
1882:
1426:
2676:
2613:
2584:
2553:
2471:
1856:
1830:
1315:
1532:
2439:
3207:
International
Workshop on Planetary Probe Atmospheric Entry and Descent Trajectory Analysis and Science, Lisbon, Portugal, October 6–9, 2003, Proceedings: ESA SP-544
1131:
2800:
2671:
2504:
1995:
1563:
1101:
914:
3301:
3010:
2640:
2524:
1909:
2404:{\displaystyle \rho _{\rm {cut}}(r)=(\mu _{0}\sigma _{D}r)^{2}\left({\frac {B_{z}^{2}}{\mu _{0}}}\right)\left({\frac {\Omega _{*}}{\Omega _{K}}}-1\right)^{2}}
1136:
430:
3056:
3031:
1220:
586:
3149:
3175:
514:
104:
275:
The pressure (force per unit area) at a given altitude is a result of the weight of the overlying atmosphere. If at a height of
2970:
818:
At an altitude over 100 km, the atmosphere is no longer be well-mixed, and each chemical species has its own scale height.
308:
385:
3461:
2206:{\displaystyle \rho (r,z)=\rho _{0}(r)\exp \left(-\left({\frac {z}{h_{D}}}\right)^{2}\right)-\rho _{\text{cut}}(r)\left}
154:
3466:
2442:
821:
Here temperature and gravitational acceleration were assumed to be constant but both may vary over large distances.
366:
to be constant; the minus sign indicates that as the height increases the pressure decreases. Therefore, using the
76:
2931:{\displaystyle h_{B}=h_{D}{\sqrt {\ln \left(1+{\frac {1-1/e}{1/e+\rho _{\text{cut}}/\rho _{0}}}\right)}}\ .}
2474:
68:
1914:
1063:
1053:
1033:
3323:
3063:
911:
component of gravity from the star, where the gravity component is pointing to the midplane of the disk:
2220:
3035:
3427:
3383:
3335:
92:
3416:"A magnetic scaleheight: the effect of toroidal magnetic fields on the thickness of accretion discs"
3372:"Disc structure around strongly magnetic accretors: a full disc solution with turbulent diffusivity"
3202:
2984:
789:
These figures should be compared with the temperature and density of Earth's atmosphere plotted at
655:
44:
3250:
3153:
1865:
3228:
3179:
2591:
2562:
2531:
2449:
1839:
225:
205:
80:
3272:
3105:
3080:
2559:
of the central object (if the poloidal magnetic field is independent of the central object then
1808:
1291:
804:
laws. Therefore, density will also decrease exponentially with height from a sea level value of
3127:
1510:
902:
A schematic depiction of the force balance in a gas disk around a central object, e.g., a star.
2417:
1859:
648:
367:
72:
64:
52:
3435:
3391:
3343:
3032:"Daniel J. Jacob: "Introduction to Atmospheric Chemistry", Princeton University Press, 1999"
2988:
2619:
2556:
1959:
and that the disk increases in altitude as one moves radially away from the central object.
1885:
1110:
898:
2778:
2649:
2482:
1973:
1541:
1086:
99:. The scale height remains constant for a particular temperature. It can be calculated by
35:: At an altitude of 0, 8.5, and 17 km, the pressure is about 1000, 370, and 140
3431:
3387:
3339:
1891:
As an illustrative approximation, if we ignore the radial variation in the temperature,
2625:
2509:
2006:
1894:
1416:{\displaystyle {\frac {d\rho }{dz}}\approx -{\frac {GM_{*}{\bar {m}}\rho z}{kTr^{3}}}}
564:
which will not change unless the temperature does. Integrating the above and assuming
3455:
3440:
3415:
3396:
3371:
2946:
1500:{\displaystyle \rho =\rho _{0}\exp \left(-\left({\frac {z}{h_{D}}}\right)^{2}\right)}
1322:
1214:
676:
48:
2768:{\displaystyle H_{B}=h_{D}{\sqrt {\ln \left(1+\rho _{0}/\rho _{\rm {cut}}\right)}},}
704:. As a function of temperature, the scale height of Earth's atmosphere is therefore
865:
264:
3322:
Lovelace, R.V.E.; Mehanian, C.; Mobarry, C. M.; Sulkanen, M. E. (September 1986).
91:
For planetary atmospheres, scale height is the increase in altitude for which the
3293:
790:
234:
3002:
744:. This yields the following scale heights for representative air temperatures.
1833:
254:
1018:{\displaystyle {\frac {dP}{dz}}=-{\frac {GM_{*}\rho z}{(r^{2}+z^{2})^{3/2}}}}
2616:
801:
576:
371:
2966:
1279:= the gas temperature in the disk, where the temperature is a function of
1056:
for the distance from the center of the star or centrally condensed object
19:
830:
Approximate atmospheric scale heights for selected Solar System bodies:
879:
853:
280:
268:
1204:{\displaystyle {\frac {dP}{dz}}\approx -{\frac {GM_{*}\rho z}{r^{3}}}}
2014:
and compress the disk. In this case, the gas density of the disk is:
873:
859:
238:
3348:
2010:
plane) magnetic field will be produced within the disk, which will
500:{\displaystyle {\frac {dP}{P}}={\frac {-dz}{{k_{\text{B}}T}/{mg}}}}
1534:
is the gas mass density at the midplane of the disk at a distance
897:
885:
841:
835:
298:, equal to the weight of a layer of atmosphere of thickness
3084:
847:
3201:
Justus, C. G.; Aleta Duvall; Vernon W. Keller (1 August 2003).
1066:
for the distance from the disk midplane (or center of the star)
36:
1962:
Due to the assumption that the gas temperature in the disk,
24:
2506:
is the magnetic flux density of the poloidal field in the
3324:"Theory of Axisymmetric Magnetohydrodynamic Flows: Disks"
16:
Distance over which a quantity decreases by a factor of e
3203:"Engineering-Level Model Atmospheres For Titan and Mars"
1997:
is sometimes known as the isothermal disk scale height.
1265:{\displaystyle P={\frac {\rho k_{\text{B}}T}{\bar {m}}}}
638:{\displaystyle P=P_{0}\exp \left(-{\frac {z}{H}}\right)}
87:
Scale height used in a simple atmospheric pressure model
27:, as can be confirmed from this diagram of air pressure
3298:
The NIST Reference on
Constants, Units, and Uncertainty
3007:
The NIST Reference on
Constants, Units, and Uncertainty
3294:"2022 CODATA Value: Newtonian constant of gravitation"
290:, then moving upwards an infinitesimally small height
257:
of atmospheric particles = 0.029 kg/mol for Earth
23:
The earth atmosphere's scale height is about 8.5
3057:"Example: The scale height of the Earth's atmosphere"
2808:
2781:
2679:
2652:
2628:
2594:
2565:
2534:
2512:
2485:
2452:
2420:
2250:
2223:
2022:
1976:
1917:
1897:
1868:
1842:
1811:
1571:
1544:
1513:
1429:
1331:
1294:
1223:
1139:
1113:
1089:
917:
589:
517:
507:
which can then be incorporated with the equation for
433:
388:
311:
157:
107:
3370:Campbell, C. G.; Heptinstall, P. M. (August 1998).
2930:
2794:
2767:
2665:
2634:
2607:
2578:
2547:
2518:
2498:
2465:
2433:
2403:
2236:
2205:
1989:
1951:
1903:
1876:
1850:
1824:
1797:
1557:
1526:
1499:
1415:
1309:
1264:
1203:
1125:
1095:
1017:
637:
556:
499:
417:
346:
186:
143:
3420:Monthly Notices of the Royal Astronomical Society
3376:Monthly Notices of the Royal Astronomical Society
1325:and the hydrostatic equilibrium equation, gives:
1075:= the mass of the star/centrally condensed object
557:{\displaystyle {\frac {dP}{P}}=-{\frac {dz}{H}}}
1213:To determine the gas pressure, one can use the
358:is the acceleration due to gravity. For small
144:{\displaystyle H={\frac {k_{\text{B}}T}{mg}}}
8:
3414:Liffman, Kurt; Bardou, Anne (October 1999).
1133:and the hydrostatic equilibrium equation is
2775:while the e-folding magnetic scale height,
3409:
3407:
3439:
3395:
3347:
2906:
2897:
2891:
2876:
2863:
2851:
2832:
2826:
2813:
2807:
2786:
2780:
2742:
2741:
2732:
2726:
2703:
2697:
2684:
2678:
2657:
2651:
2627:
2599:
2593:
2570:
2564:
2539:
2533:
2511:
2490:
2484:
2457:
2451:
2425:
2419:
2395:
2376:
2366:
2360:
2342:
2332:
2327:
2321:
2311:
2298:
2288:
2256:
2255:
2249:
2228:
2222:
2187:
2175:
2166:
2121:
2103:
2091:
2082:
2048:
2021:
1981:
1975:
1939:
1935:
1922:
1916:
1896:
1869:
1867:
1843:
1841:
1816:
1810:
1790:
1773:
1765:
1754:
1753:
1737:
1728:
1722:
1705:
1695:
1687:
1668:
1657:
1645:
1624:
1623:
1617:
1602:
1585:
1576:
1570:
1549:
1543:
1518:
1512:
1486:
1474:
1465:
1440:
1428:
1404:
1375:
1374:
1368:
1358:
1332:
1330:
1296:
1295:
1293:
1250:
1240:
1230:
1222:
1193:
1176:
1166:
1140:
1138:
1112:
1088:
1002:
998:
988:
975:
954:
944:
918:
916:
620:
600:
588:
539:
518:
516:
486:
481:
471:
466:
452:
434:
432:
395:
387:
312:
310:
164:
156:
121:
114:
106:
2967:"Glossary of Meteorology - scale height"
2646:These formulae give the maximum height,
673:, the mean molecular mass of dry air is
347:{\displaystyle {\frac {dP}{dz}}=-g\rho }
59:, usually denoted by the capital letter
18:
3003:"2022 CODATA Value: Boltzmann constant"
2958:
800:Density is related to pressure by the
418:{\displaystyle \rho ={\frac {MP}{RT}}}
2001:Disk scale height in a magnetic field
1081:= the pressure of the gas in the disk
294:will decrease the pressure by amount
7:
1952:{\displaystyle h_{D}\propto r^{3/2}}
1317:= the mean molecular mass of the gas
907:in hydrostatic equilibrium with the
2749:
2746:
2743:
2596:
2567:
2536:
2373:
2363:
2263:
2260:
2257:
2237:{\displaystyle \rho _{\text{cut}}}
1103:= the gas mass density in the disk
382:, the density can be expressed as
187:{\displaystyle H={\frac {RT}{Mg}}}
14:
3441:10.1046/j.1365-8711.1999.02852.x
3397:10.1046/j.1365-8711.1998.01576.x
3328:Astrophysical Journal Supplement
1538:from the center of the star and
1107:In the thin disk approximation,
647:This translates as the pressure
427:Combining these equations gives
2971:American Meteorological Society
2308:
2281:
2275:
2269:
2133:
2127:
2060:
2054:
2038:
2026:
1759:
1629:
1565:is the disk scale height with
1380:
1301:
1255:
995:
968:
1:
2673:, of the magnetized disk as
1877:{\displaystyle {\text{amu}}}
894:Scale height for a thin disk
658:, the pressure at sea level
2608:{\displaystyle \Omega _{K}}
2579:{\displaystyle \Omega _{*}}
2548:{\displaystyle \Omega _{*}}
2466:{\displaystyle \sigma _{D}}
1851:{\displaystyle {\text{au}}}
3483:
2622:of the disk at a distance
2443:permeability of free space
1825:{\displaystyle M_{\odot }}
1310:{\displaystyle {\bar {m}}}
571:is the pressure at height
1527:{\displaystyle \rho _{0}}
579:) the pressure at height
362:it is possible to assume
247:= mean mass of a molecule
95:decreases by a factor of
75:decreases by a factor of
2642:from the central object.
2434:{\displaystyle \mu _{0}}
649:decreasing exponentially
83:, approximately 2.718).
2985:"Pressure Scale Height"
2475:electrical conductivity
1423:which has the solution
374:of mean molecular mass
271:at the current location
3178:. NASA. Archived from
3152:. NASA. Archived from
2932:
2796:
2769:
2667:
2636:
2609:
2580:
2549:
2520:
2500:
2467:
2435:
2405:
2238:
2207:
1991:
1953:
1905:
1878:
1852:
1826:
1799:
1559:
1528:
1501:
1417:
1311:
1266:
1205:
1127:
1126:{\displaystyle z\ll r}
1097:
1064:cylindrical coordinate
1062:= the height/altitude
1054:cylindrical coordinate
1034:Gravitational constant
1019:
903:
639:
558:
501:
419:
348:
188:
145:
40:
2933:
2797:
2795:{\displaystyle h_{B}}
2770:
2668:
2666:{\displaystyle H_{B}}
2637:
2610:
2581:
2550:
2521:
2501:
2499:{\displaystyle B_{z}}
2468:
2436:
2406:
2239:
2208:
1992:
1990:{\displaystyle h_{D}}
1954:
1906:
1879:
1853:
1827:
1800:
1560:
1558:{\displaystyle h_{D}}
1529:
1502:
1418:
1312:
1283:, but independent of
1267:
1206:
1128:
1098:
1096:{\displaystyle \rho }
1020:
901:
640:
559:
511:given above to give:
502:
420:
349:
189:
146:
22:
3462:Atmospheric dynamics
3251:"Neptune Fact Sheet"
3150:"Jupiter Fact Sheet"
2806:
2779:
2677:
2650:
2626:
2592:
2563:
2532:
2510:
2483:
2450:
2418:
2248:
2221:
2020:
1974:
1966:, is independent of
1915:
1895:
1866:
1840:
1809:
1569:
1542:
1511:
1427:
1329:
1292:
1221:
1137:
1111:
1087:
915:
587:
515:
431:
386:
309:
155:
105:
93:atmospheric pressure
3432:1999MNRAS.309..443L
3388:1998MNRAS.299...31C
3340:1986ApJS...62....1L
3229:"Uranus Fact Sheet"
3176:"Saturn Fact Sheet"
2586:can be set to zero)
2337:
882:: 19.1–20.3 km
813:1.2 kg⋅m
738:9.81 m⋅s
583:can be written as:
279:the atmosphere has
233:= mean atmospheric
3273:"Pluto Fact Sheet"
3156:on 13 October 2011
3106:"Earth Fact Sheet"
3081:"Venus Fact Sheet"
2928:
2792:
2765:
2663:
2632:
2605:
2576:
2555:is the rotational
2545:
2516:
2496:
2463:
2431:
2401:
2323:
2234:
2203:
1987:
1949:
1901:
1874:
1848:
1822:
1795:
1555:
1524:
1497:
1413:
1307:
1262:
1201:
1123:
1093:
1015:
904:
826:Planetary examples
656:Earth's atmosphere
635:
554:
497:
415:
344:
226:molar gas constant
206:Boltzmann constant
184:
141:
81:natural logarithms
41:
3467:Vertical position
3182:on 18 August 2011
3128:"Mars Fact Sheet"
2924:
2920:
2913:
2894:
2760:
2635:{\displaystyle r}
2519:{\displaystyle z}
2382:
2348:
2231:
2181:
2124:
2097:
1904:{\displaystyle T}
1872:
1860:astronomical unit
1846:
1793:
1789:
1785:
1784:
1776:
1762:
1698:
1671:
1667:
1637:
1636:
1632:
1480:
1411:
1383:
1350:
1304:
1260:
1258:
1243:
1199:
1158:
1013:
936:
811:roughly equal to
726:10 J⋅K
628:
575:= 0 (pressure at
552:
531:
495:
474:
447:
413:
368:equation of state
330:
241:= 250 K for Earth
182:
139:
124:
73:physical quantity
63:, is a distance (
3474:
3446:
3445:
3443:
3411:
3402:
3401:
3399:
3367:
3361:
3360:
3358:
3356:
3351:
3319:
3313:
3312:
3310:
3309:
3290:
3284:
3283:
3281:
3280:
3269:
3263:
3262:
3260:
3258:
3247:
3241:
3240:
3238:
3236:
3225:
3219:
3218:
3216:
3214:
3198:
3192:
3191:
3189:
3187:
3172:
3166:
3165:
3163:
3161:
3146:
3140:
3139:
3137:
3135:
3124:
3118:
3117:
3115:
3113:
3102:
3096:
3095:
3093:
3091:
3077:
3071:
3070:
3068:
3062:. Archived from
3061:
3053:
3047:
3046:
3044:
3043:
3034:. Archived from
3028:
3022:
3021:
3019:
3018:
2999:
2993:
2992:
2989:Wolfram Research
2981:
2975:
2974:
2963:
2937:
2935:
2934:
2929:
2922:
2921:
2919:
2915:
2914:
2912:
2911:
2910:
2901:
2896:
2895:
2892:
2880:
2871:
2867:
2852:
2833:
2831:
2830:
2818:
2817:
2801:
2799:
2798:
2793:
2791:
2790:
2774:
2772:
2771:
2766:
2761:
2759:
2755:
2754:
2753:
2752:
2736:
2731:
2730:
2704:
2702:
2701:
2689:
2688:
2672:
2670:
2669:
2664:
2662:
2661:
2641:
2639:
2638:
2633:
2620:angular velocity
2614:
2612:
2611:
2606:
2604:
2603:
2585:
2583:
2582:
2577:
2575:
2574:
2557:angular velocity
2554:
2552:
2551:
2546:
2544:
2543:
2525:
2523:
2522:
2517:
2505:
2503:
2502:
2497:
2495:
2494:
2472:
2470:
2469:
2464:
2462:
2461:
2440:
2438:
2437:
2432:
2430:
2429:
2410:
2408:
2407:
2402:
2400:
2399:
2394:
2390:
2383:
2381:
2380:
2371:
2370:
2361:
2353:
2349:
2347:
2346:
2336:
2331:
2322:
2316:
2315:
2303:
2302:
2293:
2292:
2268:
2267:
2266:
2243:
2241:
2240:
2235:
2233:
2232:
2229:
2212:
2210:
2209:
2204:
2202:
2198:
2197:
2193:
2192:
2191:
2186:
2182:
2180:
2179:
2167:
2126:
2125:
2122:
2113:
2109:
2108:
2107:
2102:
2098:
2096:
2095:
2083:
2053:
2052:
1996:
1994:
1993:
1988:
1986:
1985:
1958:
1956:
1955:
1950:
1948:
1947:
1943:
1927:
1926:
1910:
1908:
1907:
1902:
1886:atomic mass unit
1883:
1881:
1880:
1875:
1873:
1870:
1857:
1855:
1854:
1849:
1847:
1844:
1831:
1829:
1828:
1823:
1821:
1820:
1804:
1802:
1801:
1796:
1794:
1791:
1787:
1786:
1783:
1782:
1778:
1777:
1774:
1769:
1764:
1763:
1755:
1747:
1743:
1742:
1741:
1732:
1727:
1726:
1711:
1710:
1709:
1704:
1700:
1699:
1696:
1691:
1677:
1673:
1672:
1669:
1665:
1661:
1647:
1646:
1638:
1635:
1634:
1633:
1625:
1622:
1621:
1608:
1607:
1606:
1587:
1586:
1581:
1580:
1564:
1562:
1561:
1556:
1554:
1553:
1533:
1531:
1530:
1525:
1523:
1522:
1506:
1504:
1503:
1498:
1496:
1492:
1491:
1490:
1485:
1481:
1479:
1478:
1466:
1445:
1444:
1422:
1420:
1419:
1414:
1412:
1410:
1409:
1408:
1392:
1385:
1384:
1376:
1373:
1372:
1359:
1351:
1349:
1341:
1333:
1316:
1314:
1313:
1308:
1306:
1305:
1297:
1271:
1269:
1268:
1263:
1261:
1259:
1251:
1249:
1245:
1244:
1241:
1231:
1210:
1208:
1207:
1202:
1200:
1198:
1197:
1188:
1181:
1180:
1167:
1159:
1157:
1149:
1141:
1132:
1130:
1129:
1124:
1102:
1100:
1099:
1094:
1046:
1043:
1041:
1024:
1022:
1021:
1016:
1014:
1012:
1011:
1010:
1006:
993:
992:
980:
979:
966:
959:
958:
945:
937:
935:
927:
919:
814:
743:
739:
735:
733:
727:
725:
703:
701:
695:
693:
687:
679:
672:
670:
644:
642:
641:
636:
634:
630:
629:
621:
605:
604:
563:
561:
560:
555:
553:
548:
540:
532:
527:
519:
506:
504:
503:
498:
496:
494:
493:
485:
480:
476:
475:
472:
464:
453:
448:
443:
435:
424:
422:
421:
416:
414:
412:
404:
396:
353:
351:
350:
345:
331:
329:
321:
313:
218:
215:
213:
193:
191:
190:
185:
183:
181:
173:
165:
151:or equivalently
150:
148:
147:
142:
140:
138:
130:
126:
125:
122:
115:
3482:
3481:
3477:
3476:
3475:
3473:
3472:
3471:
3452:
3451:
3450:
3449:
3413:
3412:
3405:
3369:
3368:
3364:
3354:
3352:
3321:
3320:
3316:
3307:
3305:
3292:
3291:
3287:
3278:
3276:
3271:
3270:
3266:
3256:
3254:
3249:
3248:
3244:
3234:
3232:
3227:
3226:
3222:
3212:
3210:
3200:
3199:
3195:
3185:
3183:
3174:
3173:
3169:
3159:
3157:
3148:
3147:
3143:
3133:
3131:
3126:
3125:
3121:
3111:
3109:
3104:
3103:
3099:
3089:
3087:
3079:
3078:
3074:
3066:
3059:
3055:
3054:
3050:
3041:
3039:
3030:
3029:
3025:
3016:
3014:
3001:
3000:
2996:
2983:
2982:
2978:
2965:
2964:
2960:
2955:
2943:
2902:
2887:
2872:
2853:
2844:
2840:
2822:
2809:
2804:
2803:
2782:
2777:
2776:
2737:
2722:
2715:
2711:
2693:
2680:
2675:
2674:
2653:
2648:
2647:
2624:
2623:
2595:
2590:
2589:
2566:
2561:
2560:
2535:
2530:
2529:
2508:
2507:
2486:
2481:
2480:
2453:
2448:
2447:
2421:
2416:
2415:
2372:
2362:
2359:
2355:
2354:
2338:
2317:
2307:
2294:
2284:
2251:
2246:
2245:
2224:
2219:
2218:
2171:
2162:
2161:
2157:
2153:
2140:
2136:
2117:
2087:
2078:
2077:
2073:
2069:
2044:
2018:
2017:
2003:
1977:
1972:
1971:
1931:
1918:
1913:
1912:
1893:
1892:
1864:
1863:
1838:
1837:
1812:
1807:
1806:
1752:
1748:
1733:
1718:
1717:
1713:
1712:
1683:
1679:
1678:
1653:
1649:
1648:
1613:
1609:
1598:
1588:
1572:
1567:
1566:
1545:
1540:
1539:
1514:
1509:
1508:
1470:
1461:
1460:
1456:
1452:
1436:
1425:
1424:
1400:
1393:
1364:
1360:
1342:
1334:
1327:
1326:
1290:
1289:
1236:
1232:
1219:
1218:
1189:
1172:
1168:
1150:
1142:
1135:
1134:
1109:
1108:
1085:
1084:
1074:
1044:
1039:
1037:
994:
984:
971:
967:
950:
946:
928:
920:
913:
912:
896:
891:
862:: 59.5 km
828:
812:
810:
741:
737:
731:
729:
723:
721:
699:
697:
691:
689:
685:
674:
668:
666:
665:averages about
664:
616:
612:
596:
585:
584:
570:
541:
520:
513:
512:
467:
465:
454:
436:
429:
428:
405:
397:
384:
383:
378:at temperature
322:
314:
307:
306:
216:
211:
209:
203:
174:
166:
153:
152:
131:
117:
116:
103:
102:
89:
71:) over which a
39:, respectively.
17:
12:
11:
5:
3480:
3478:
3470:
3469:
3464:
3454:
3453:
3448:
3447:
3403:
3362:
3349:10.1086/191132
3314:
3285:
3264:
3242:
3220:
3193:
3167:
3141:
3119:
3097:
3072:
3069:on 2011-07-16.
3048:
3023:
2994:
2976:
2957:
2956:
2954:
2951:
2950:
2949:
2942:
2939:
2927:
2918:
2909:
2905:
2900:
2890:
2886:
2883:
2879:
2875:
2870:
2866:
2862:
2859:
2856:
2850:
2847:
2843:
2839:
2836:
2829:
2825:
2821:
2816:
2812:
2789:
2785:
2764:
2758:
2751:
2748:
2745:
2740:
2735:
2729:
2725:
2721:
2718:
2714:
2710:
2707:
2700:
2696:
2692:
2687:
2683:
2660:
2656:
2644:
2643:
2631:
2602:
2598:
2587:
2573:
2569:
2542:
2538:
2527:
2515:
2493:
2489:
2478:
2460:
2456:
2445:
2428:
2424:
2398:
2393:
2389:
2386:
2379:
2375:
2369:
2365:
2358:
2352:
2345:
2341:
2335:
2330:
2326:
2320:
2314:
2310:
2306:
2301:
2297:
2291:
2287:
2283:
2280:
2277:
2274:
2271:
2265:
2262:
2259:
2254:
2227:
2201:
2196:
2190:
2185:
2178:
2174:
2170:
2165:
2160:
2156:
2152:
2149:
2146:
2143:
2139:
2135:
2132:
2129:
2120:
2116:
2112:
2106:
2101:
2094:
2090:
2086:
2081:
2076:
2072:
2068:
2065:
2062:
2059:
2056:
2051:
2047:
2043:
2040:
2037:
2034:
2031:
2028:
2025:
2007:magnetic field
2002:
1999:
1984:
1980:
1946:
1942:
1938:
1934:
1930:
1925:
1921:
1911:, we see that
1900:
1819:
1815:
1781:
1772:
1768:
1761:
1758:
1751:
1746:
1740:
1736:
1731:
1725:
1721:
1716:
1708:
1703:
1694:
1690:
1686:
1682:
1676:
1664:
1660:
1656:
1652:
1644:
1641:
1631:
1628:
1620:
1616:
1612:
1605:
1601:
1597:
1594:
1591:
1584:
1579:
1575:
1552:
1548:
1521:
1517:
1495:
1489:
1484:
1477:
1473:
1469:
1464:
1459:
1455:
1451:
1448:
1443:
1439:
1435:
1432:
1407:
1403:
1399:
1396:
1391:
1388:
1382:
1379:
1371:
1367:
1363:
1357:
1354:
1348:
1345:
1340:
1337:
1319:
1318:
1303:
1300:
1287:
1257:
1254:
1248:
1239:
1235:
1229:
1226:
1196:
1192:
1187:
1184:
1179:
1175:
1171:
1165:
1162:
1156:
1153:
1148:
1145:
1122:
1119:
1116:
1105:
1104:
1092:
1082:
1076:
1072:
1067:
1057:
1047:
1042:10 m⋅kg⋅s
1009:
1005:
1001:
997:
991:
987:
983:
978:
974:
970:
965:
962:
957:
953:
949:
943:
940:
934:
931:
926:
923:
895:
892:
890:
889:
883:
877:
876:: 27.7 km
871:
870:
869:
857:
851:
850:: 11.1 km
845:
839:
838:: 15.9 km
832:
827:
824:
823:
822:
819:
816:
808:
787:
786:
781:= 210 K,
776:
771:= 260 K,
766:
761:= 273 K,
756:
751:= 290 K,
742:29.28 m/K
686:28.964 Da
662:
633:
627:
624:
619:
615:
611:
608:
603:
599:
595:
592:
568:
551:
547:
544:
538:
535:
530:
526:
523:
492:
489:
484:
479:
470:
463:
460:
457:
451:
446:
442:
439:
411:
408:
403:
400:
394:
391:
343:
340:
337:
334:
328:
325:
320:
317:
273:
272:
258:
248:
242:
228:
219:
201:
180:
177:
172:
169:
163:
160:
137:
134:
129:
120:
113:
110:
88:
85:
15:
13:
10:
9:
6:
4:
3:
2:
3479:
3468:
3465:
3463:
3460:
3459:
3457:
3442:
3437:
3433:
3429:
3425:
3421:
3417:
3410:
3408:
3404:
3398:
3393:
3389:
3385:
3381:
3377:
3373:
3366:
3363:
3350:
3345:
3341:
3337:
3333:
3329:
3325:
3318:
3315:
3303:
3299:
3295:
3289:
3286:
3274:
3268:
3265:
3252:
3246:
3243:
3230:
3224:
3221:
3208:
3204:
3197:
3194:
3181:
3177:
3171:
3168:
3155:
3151:
3145:
3142:
3129:
3123:
3120:
3107:
3101:
3098:
3086:
3082:
3076:
3073:
3065:
3058:
3052:
3049:
3038:on 2013-04-10
3037:
3033:
3027:
3024:
3012:
3008:
3004:
2998:
2995:
2990:
2986:
2980:
2977:
2972:
2968:
2962:
2959:
2952:
2948:
2947:Time constant
2945:
2944:
2940:
2938:
2925:
2916:
2907:
2903:
2898:
2888:
2884:
2881:
2877:
2873:
2868:
2864:
2860:
2857:
2854:
2848:
2845:
2841:
2837:
2834:
2827:
2823:
2819:
2814:
2810:
2787:
2783:
2762:
2756:
2738:
2733:
2727:
2723:
2719:
2716:
2712:
2708:
2705:
2698:
2694:
2690:
2685:
2681:
2658:
2654:
2629:
2621:
2618:
2600:
2588:
2571:
2558:
2540:
2528:
2513:
2491:
2487:
2479:
2476:
2458:
2454:
2446:
2444:
2426:
2422:
2414:
2413:
2412:
2396:
2391:
2387:
2384:
2377:
2367:
2356:
2350:
2343:
2339:
2333:
2328:
2324:
2318:
2312:
2304:
2299:
2295:
2289:
2285:
2278:
2272:
2252:
2244:has the form
2225:
2216:
2199:
2194:
2188:
2183:
2176:
2172:
2168:
2163:
2158:
2154:
2150:
2147:
2144:
2141:
2137:
2130:
2118:
2114:
2110:
2104:
2099:
2092:
2088:
2084:
2079:
2074:
2070:
2066:
2063:
2057:
2049:
2045:
2041:
2035:
2032:
2029:
2023:
2015:
2013:
2008:
2000:
1998:
1982:
1978:
1969:
1965:
1960:
1944:
1940:
1936:
1932:
1928:
1923:
1919:
1898:
1889:
1887:
1861:
1835:
1817:
1813:
1779:
1770:
1766:
1756:
1749:
1744:
1738:
1734:
1729:
1723:
1719:
1714:
1706:
1701:
1692:
1688:
1684:
1680:
1674:
1662:
1658:
1654:
1650:
1642:
1639:
1626:
1618:
1614:
1610:
1603:
1599:
1595:
1592:
1589:
1582:
1577:
1573:
1550:
1546:
1537:
1519:
1515:
1493:
1487:
1482:
1475:
1471:
1467:
1462:
1457:
1453:
1449:
1446:
1441:
1437:
1433:
1430:
1405:
1401:
1397:
1394:
1389:
1386:
1377:
1369:
1365:
1361:
1355:
1352:
1346:
1343:
1338:
1335:
1324:
1323:ideal gas law
1298:
1288:
1286:
1282:
1278:
1275:
1274:
1273:
1252:
1246:
1237:
1233:
1227:
1224:
1216:
1215:ideal gas law
1211:
1194:
1190:
1185:
1182:
1177:
1173:
1169:
1163:
1160:
1154:
1151:
1146:
1143:
1120:
1117:
1114:
1090:
1083:
1080:
1077:
1071:
1068:
1065:
1061:
1058:
1055:
1052:= the radial
1051:
1048:
1035:
1031:
1028:
1027:
1026:
1007:
1003:
999:
989:
985:
981:
976:
972:
963:
960:
955:
951:
947:
941:
938:
932:
929:
924:
921:
910:
900:
893:
888:: ~50 km
887:
884:
881:
878:
875:
872:
867:
864:
863:
861:
858:
855:
852:
849:
846:
844:: 8.5 km
843:
840:
837:
834:
833:
831:
825:
820:
817:
807:
803:
799:
798:
797:
794:
792:
785:= 6000 m
784:
780:
777:
775:= 7610 m
774:
770:
767:
765:= 8000 m
764:
760:
757:
755:= 8500 m
754:
750:
747:
746:
745:
719:
715:
711:
707:
694:10 kg/Da
683:
678:
661:
657:
652:
651:with height.
650:
645:
631:
625:
622:
617:
613:
609:
606:
601:
597:
593:
590:
582:
578:
574:
567:
549:
545:
542:
536:
533:
528:
524:
521:
510:
490:
487:
482:
477:
468:
461:
458:
455:
449:
444:
440:
437:
425:
409:
406:
401:
398:
392:
389:
381:
377:
373:
369:
365:
361:
357:
341:
338:
335:
332:
326:
323:
318:
315:
303:
301:
297:
293:
289:
286:and pressure
285:
282:
278:
270:
266:
262:
259:
256:
252:
249:
246:
243:
240:
236:
232:
229:
227:
223:
220:
207:
200:
197:
196:
195:
178:
175:
170:
167:
161:
158:
135:
132:
127:
118:
111:
108:
100:
98:
94:
86:
84:
82:
79:(the base of
78:
74:
70:
66:
62:
58:
54:
50:
46:
38:
34:
30:
26:
21:
3423:
3419:
3379:
3375:
3365:
3353:. Retrieved
3331:
3327:
3317:
3306:. Retrieved
3297:
3288:
3277:. Retrieved
3267:
3257:28 September
3255:. Retrieved
3245:
3235:28 September
3233:. Retrieved
3223:
3213:28 September
3211:. Retrieved
3206:
3196:
3186:28 September
3184:. Retrieved
3180:the original
3170:
3160:28 September
3158:. Retrieved
3154:the original
3144:
3134:28 September
3132:. Retrieved
3122:
3112:28 September
3110:. Retrieved
3100:
3090:28 September
3088:. Retrieved
3075:
3064:the original
3051:
3040:. Retrieved
3036:the original
3026:
3015:. Retrieved
3006:
2997:
2979:
2961:
2645:
2214:
2016:
2011:
2004:
1967:
1963:
1961:
1890:
1535:
1320:
1284:
1280:
1276:
1212:
1106:
1078:
1069:
1059:
1049:
1029:
908:
905:
868:: 21 km
856:: 27 km
829:
805:
795:
788:
782:
778:
772:
768:
762:
758:
752:
748:
717:
713:
709:
705:
681:
675:28.964
659:
653:
646:
580:
572:
565:
508:
426:
379:
375:
363:
359:
355:
304:
299:
295:
291:
287:
283:
276:
274:
265:acceleration
260:
250:
244:
230:
221:
198:
101:
96:
90:
60:
57:scale height
56:
55:sciences, a
42:
32:
31:by altitude
28:
2477:of the disk
791:NRLMSISE-00
235:temperature
214:10 J⋅K
45:atmospheric
3456:Categories
3426:(2): 443.
3355:26 January
3308:2024-05-18
3304:. May 2024
3279:2020-09-28
3042:2013-04-18
3017:2024-05-18
3013:. May 2024
2953:References
2213:where the
1834:solar mass
1321:Using the
734:10 kg
702:10 kg
680:and hence
671:10 Pa
255:molar mass
3382:(1): 31.
2904:ρ
2889:ρ
2858:−
2838:
2739:ρ
2724:ρ
2709:
2617:keplerian
2597:Ω
2572:∗
2568:Ω
2541:∗
2537:Ω
2526:direction
2455:σ
2423:μ
2385:−
2374:Ω
2368:∗
2364:Ω
2340:μ
2296:σ
2286:μ
2253:ρ
2226:ρ
2159:−
2151:
2145:−
2119:ρ
2115:−
2075:−
2067:
2046:ρ
2024:ρ
1929:∝
1818:⊙
1775: amu
1760:¯
1739:⊙
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1640:≈
1630:¯
1619:∗
1516:ρ
1458:−
1450:
1438:ρ
1431:ρ
1387:ρ
1381:¯
1370:∗
1356:−
1353:≈
1339:ρ
1302:¯
1256:¯
1234:ρ
1183:ρ
1178:∗
1164:−
1161:≈
1118:≪
1091:ρ
961:ρ
956:∗
942:−
802:ideal gas
618:−
610:
577:sea level
537:−
456:−
390:ρ
372:ideal gas
342:ρ
336:−
53:planetary
2941:See also
2217:density
1792: au
1697: au
65:vertical
3428:Bibcode
3384:Bibcode
3336:Bibcode
2615:is the
2473:is the
2441:is the
2215:cut-off
1045:
1025:where:
880:Neptune
854:Jupiter
370:for an
281:density
269:gravity
267:due to
253:= mean
239:kelvins
217:
194:where:
3275:. NASA
3253:. NASA
3231:. NASA
3130:. NASA
3108:. NASA
2973:(AMS).
2923:
2411:where
1788:
1666:
1643:0.0306
1507:where
1272:with:
874:Uranus
860:Saturn
806:ρ
796:Note:
354:where
305:Thus:
284:ρ
69:radial
51:, and
3334:: 1.
3209:. ESA
3067:(PDF)
3060:(PDF)
2802:, is
2012:pinch
1805:with
1038:6.674
886:Pluto
866:Titan
842:Earth
836:Venus
730:4.808
722:1.381
698:4.808
690:1.660
210:1.381
49:earth
3357:2022
3302:NIST
3259:2013
3237:2013
3215:2013
3188:2013
3162:2013
3136:2013
3114:2013
3092:2013
3085:NASA
3011:NIST
1884:the
1862:and
1858:the
1832:the
848:Mars
740:) =
667:1.01
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3392:doi
3380:299
3344:doi
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2123:cut
2064:exp
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696:=
688:×
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677:Da
360:dz
302:.
300:dz
296:dP
292:dz
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