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JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 111, A04301, doi:10.1029/2005JA011399, 2006
Thermospheric densities derived from spacecraft orbits: Application to the Starshine satellites J. L. Lean,1 J. M. Picone,1 J. T. Emmert,2 and G. Moore3 Received 1 September 2005; revised 15 November 2005; accepted 30 November 2005; published 6 April 2006.
[1] The amplitude and variability of the total mass density of the Earth’s atmosphere at
altitudes between 200 and 475 km are analyzed for the period from June 1999 to January 2003, around the maximum of solar activity in cycle 23. The densities are derived with uncertainties of order ±6% from analysis of the approximately circular orbits of three Starshine spacecraft of known ballistic coefficients. Local densities averaged over all three Starshine missions are 4% lower than the corresponding NRLMSISE-00 model values, which may reflect a secular decrease from thermospheric cooling. Differences of order ±15% occur on timescales of days to weeks, and larger differences of as much as 30% can persist on timescales of several months. Some differences are traceable to the NRLMSIS model’s use of the 10.7 cm proxy of EUV radiation, since they are reduced when NRLMSIS is evaluated using the Mg II index, which is a better proxy of variations in EUV irradiance that heats the thermosphere. Comparisons of 27-day density cycles extracted by complex demodulation indicate that NRLMSIS underestimates upper atmospheric density increases associated with solar rotational modulation of EUV radiation by as much as a factor of two. Citation: Lean, J. L., J. M. Picone, J. T. Emmert, and G. Moore (2006), Thermospheric densities derived from spacecraft orbits: Application to the Starshine satellites, J. Geophys. Res., 111, A04301, doi:10.1029/2005JA011399.
1. Introduction [2] Solar control of thermospheric temperature and density is well established theoretically [e.g., Roble and Emery, 1983]. Incoming solar extreme ultraviolet (EUV) radiation is the primary heating source of the atmosphere above about 150 km. This energy, absorbed by atmospheric gases (primarily N2, O2, and O), maintains the global mean temperature which establishes the composition and total mass density structure [Roble et al., 1987]. Temperature and density fluctuations are therefore expected to track variations in EUV radiation imposed by the Sun’s rotation on its axis (every 27 days) and the evolution of solar activity during the 11-year cycle. From activity minimum to maximum, the energy flux of solar radiation at wavelengths from 5 to 105 nm increases by about a factor of two, from about 2 to 4 mWm2 [Lean et al., 2003]. Global exospheric temperature increases correspondingly, from 750 K to 1150 K according to the NRLMSISE-00 empirical model [Picone et al., 2002], as shown in Figure 1, with attendant mass density profile changes. [3] The thermosphere measurably impedes the motions of objects orbiting the Earth at altitudes below about 1000 km, causing orbits to decay in altitude and eccentricity and their 1 E. O. Hulburt Center for Space Research, Naval Research Laboratory, Washington, D. C., USA. 2 School of Computational Sciences, George Mason University, Fairfax, Virginia, USA. 3 Project Starshine, Monument, Colorado, USA.
Copyright 2006 by the American Geophysical Union. 0148-0227/06/2005JA011399
periods to decrease. Since atmospheric drag is proportional to total mass density [see, e.g., Picone et al., 2005], these effects accelerate as solar activity approaches a maximum, at rates that depend on altitude. For example, the lifetime of a typical spacecraft in a circular orbit at 500 km is 30 years for solar minimum conditions compared with 3 years at solar maximum conditions [Walterscheid, 1989]. At 400 km, the solar minimum and maximum lifetimes are 4 years and 7 months, respectively. Knowledge of thermospheric density fluctuations is therefore needed for practical applications that include determination of spacecraft orbit lifetimes, onboard resources, and reentry (e.g., the Solar Maximum Mission [Covault, 1989]) and collision avoidance for manned platforms. This knowledge is needed also to ensure precise position predictions of the Space Object Catalog’s 9500 objects [Wagner, 2004], many of which are in lowEarth orbit. Special Perturbations orbit determination, to which the US Strategic Command plans to transition for routine space object catalog maintenance, requires an accurate specification of upper atmospheric density as a function of solar activity, geographical position and altitude, and local time. [4] Density specification tools such as the Naval Research Laboratory Mass Spectrometer and Incoherent Scatter (NRLMSIS) model [Picone et al., 2002], the earlier MSIS models [e.g., Hedin, 1991], and the Jacchia models [e.g., Jacchia, 1970] input the solar 10.7 cm radio flux (F10.7) to account for variations in solar EUV radiation, and the Ap index as a proxy for geomagnetic disturbances. The actual upper atmospheric responses to these influences are not well known, in part because there are few direct measurements of thermospheric neutral densities available
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Figure 1. Shown are thermospheric temperature and mass densities during solar cycle minimum (F10.7 = 70, F10.7A = 70, Ap = 11) and maximum (F10.7 = 200, F10.7A = 174, Ap = 11) conditions, estimated from the NRLMSISE-00 empirical density specification model [Picone et al., 2002].
for model validation, even in recent times. Deficiencies in F10.7 as a proxy for variations in solar EUV radiation also contribute. Day-to-day fluctuations in F10.7 primarily reflect changes in radiation emitted from the corona, the Sun’s hot outer layer. However, the strongest sources of upper atmospheric heating fluctuations, such as the He II 30.4 nm line, are emitted from the cooler and lower chromosphere, for which a better proxy is the Mg index [Viereck et al., 2001, 2004]. Thermospheric N2 densities measured by Atmospheric Explorer-E, for example, correlate differently with different solar indices and emission bands over short and long timescales [Hedin, 1984]. Densities inferred from atmospheric drag on the Long Duration Exposure Facility spacecraft and measured by the Dynamics Explorer-2 mass spectrometer appear to track Mg better than F10.7 [Bass et al., 1996; Thuillier and Bruinsma, 2001]. [5] Since the upper atmosphere produces drag on Earthorbiting objects, the evolution of spacecraft orbits contains information about the atmospheric density. Spacecraft drag effects composed the primary database for early empirical models of upper atmosphere density [Jacchia, 1970]. Recently, the MSIS model, formulated primarily from incoherent scatter radar temperatures and mass spectrometer composition data, has been revised to include total mass densities from satellite accelerometers and orbit determinations, and designated NRLMSISE-00 [Picone et al., 2002]. The numerous objects now in low-Earth orbit (LEO) afford a global long-term database for characterizing upper atmosphere climatology. The recent development of a fast and accurate method for retrieving total mass densities from routinely compiled, easily accessible orbital elements of LEO objects [Picone et al., 2005] enables the use of this vast and growing archive of global data for analysis of thermospheric evolution, secular change [e.g., Emmert et al., 2004], and responses to solar and geomagnetic forcing. [6] We employ this new methodology to derive total mass densities from the orbital elements of the three Starshine spacecraft [Maley et al., 1999]. These spacecraft are especially suitable for estimating average upper atmospheric densities. The orbits are nearly circular, so their altitudes
and velocities are approximately constant within each orbit, and the spacecraft are mirrored spheres for which the ballistic coefficients are known and are essentially independent of orientation with respect to the direction of motion. The densities along the Starshine orbits characterize the thermosphere from 1999 to 2003, a period of high overall solar activity. Derived independently of upper atmosphere specification models, the densities are compared with the NRLMSIS model estimates as part of that model’s validation.
2. Atmospheric Density Derivation From Orbital Elements [7] Orbital parameters of Earth-orbiting objects are archived in the form of Two-Line Element sets (TLEs). The North American Aerospace Defense Command (NORAD) tracks the 9500 objects that currently reside in the Space Object Catalog and derives from General Perturbations theory an analytical description of each orbit parameterized by six Kozai mean elements [Kozai, 1959]: inclination, right ascension of the ascending node, argument of perigee, eccentricity, mean motion, and mean anomaly, all tagged by a specific time identified as epoch. (Unless specified otherwise, ‘‘elements’’ refers to the mean elements rather than osculating elements, which represent the complete time-dependent solution, as described by Picone et al. [2005]). Values of these six parameters at successive epochs, together with the object’s identification and additional elements related to drag (1=2 dn/dt, the mean motion time derivative, and B*, the scaled inverse ballistic coefficient) comprise the TLEs. Typically, the TLEs are updated and archived with sufficient time resolution to characterize the evolution of the orbit. For the Starshine spacecraft, there are from one to four TLEs per day. [8] Building on King-Hele’s [1987] theoretical analysis of the trajectories of Earth-orbiting objects, Picone et al. [2005] derived specific relationships between density and the Kozai mean elements. The drag of the Earth’s atmosphere, determined by its density, causes the mean motion, n, of a spacecraft orbiting the Earth with velocity v, to
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increase with time, t. The total mass density, r, of the upper atmosphere along the spacecraft track relates to the change in the mean motion, dn, by dn ¼ 3=2 nðt Þ1=3 m2=3 rðt ÞBvðt Þ3 F ðt Þdt;
ð1Þ
where m = GM is the gravitational parameter defined by G, the gravitational constant, and M, the mass of the Earth. The inverse ballistic coefficient, B = CDA/m is obtained from the drag coefficient, CD, the cross-sectional area, A, perpendicular to the object’s velocity in the frame of the thermospheric wind, and the mass, m, of the spacecraft. [9] The quantity F in equation (1) accounts for the effect on the spacecraft of winds in the Earth’s upper atmosphere. Assuming a corotating atmosphere, F depends approximately on the orbital inclination, i, spacecraft velocity, v, in the Earth’s inertial frame, and distance, r, from the center of the Earth according to F ðt Þ ffi
2 rðt Þw 1 cosðiÞ ; vðt Þ
ð2Þ
where w is the Earth’s angular velocity [King-Hele, 1987]. [10] During any single orbit the osculating (instantaneous) mean motion can vary significantly because of gravitational perturbations related to the asymmetry of the Earth’s mass distribution. The effect of atmospheric drag (a nonconservative force that produces a monotonic effect) becomes evident primarily on longer timescales, over multiple orbits, and relates to the mean motion through equation (1). This means that in general, the determination of thermospheric density, r, requires the integration of equation (1) over sufficient time Dt = t2 t1 to accumulate a measurable change due to drag. Using equation (1) and defining an effective (average) density, rA, over the time interval t2 t1 as Z rA ¼
t2
rðtÞvðt Þ3 F ðt Þdt
t1
Z
t2
ð3Þ vðt Þ3 F ðt Þdt
t1
the change in the Kozai mean motion, Dn = n2 n1 from t1 to t2 is n2 n1 ffi
1=3 =2 nA m2=3 BrA
3
Z
[11] The use of equation (5) to derive the effective atmospheric density therefore requires that the orbit be propagated over the time interval between R successive TLE epochs. This is needed to evaluate v(t)3F(t)dt and is accomplished with the General Perturbations propagator that was used to compute the original TLEs by fitting observations of the object’s location (Picone et al. [2005] provide details). The orbit propagator, Simplified General Perturbations 4 (SGP4) [Hoots and Roehrich, 1988], estimates the position, x, y, z, and velocity, dx/dt, dy/dt, and dz/dt, of the object at a specified time after epoch. In geocentric inertial coordinates, the object’s height above pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 the Earth’s center is then ffi r = x þ y þ z and its speed qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 is v = dx=dt þ dy=dt þ dz=dt, which enable estimates of F(t) at all points along the orbit between successive epochs using equation (2) and hence the computation of densities using equation (5). [12] For near-circular orbits, an appropriate density value can also be derived directly from the TLEs at each epoch without propagating the orbit forward in time. This is possible because the velocity and distance from the Earth’s center are approximately constant during an orbit. Using the subscript K to denote quantities pertaining to the Kozai mean element at the epoch, then Zt2
vðtÞ3 F ðt Þdt ffi FK v3K ðt2 t1 Þ
3
vðt Þ F ðt Þdt
where vK is the velocity obtained from SGP4 at epoch and FK is the corresponding wind factor. Noting that dn/dt ffi (n2 n1)/(t2 t1) for proximate epochs, an estimate of the density, rK, at each TLE epoch is obtained from equation (5) as rK ffi 2=3
dnK m2=3 1=3 dt Bn v3 FK K K
rK ¼
ð4Þ
m n2K
1=3 ð8Þ
with speed vK ¼
1=2 m rK
ð9Þ
2
rA ¼
=3 ðn2 n1 Þm2=3 Zt2 1=3 BnA vðt Þ3 F ðt Þdt
ð7Þ
where nK and dnK/dt are the Kozai mean motion and the time derivative of the mean motion, respectively. In this case FK is given by equation (2) with i = iK, r = rK and v = vK. Furthermore, for a circular orbit rK = aK, the semimajor axis, and relates directly to the mean motion, nK (= 2p/T, for period, T), according to
t1
where nA may be designated the mean motion at tA = t + Dt/2, the center of the time interval so that nA = (n1 + n2)/2. The observed density tagged by tA is then
ð6Þ
t1
t2
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ð5Þ
t1
where F(t) is given by equation (2) with i = iA, the mean orbit inclination over the time interval.
3. Starshine Satellites and Orbits [ 13 ] The small spherical Starshine spacecraft were launched into approximately circular, low-Earth orbits with moderate inclinations, near the maximum of solar activity in cycle 23. A primary goal of Project Starshine is the education of students about upper atmospheric drag on
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Table 1. Characteristics of the Three Starshine Spacecraft and Their Orbits Starshine Spacecraft (NORAD Number)
Launch Date
Reentry Date
Launch Height, km
1 (25769) 2 (26996) 3 (26929)
1999 05 – 27 2001 12 – 05 2001 09 – 29
2000 02 – 18 2002 05 – 01 2003 01 – 21
380 360 470
Eccentricity
Inclination, deg
Mass, kg
Diameter, cm
CD
Inverse Ballistic Coefficient, cm2 gm1
