Accurately Predicting The Student Nitric Oxide Explorer’s Orbital Lifetime

Submitted by Aaron Cannon

To Professor Nerem

For ASEN 5050 Space Flight Dynamics

Date: December 14, 2000

Abstract

The SNOE’s orbit is decaying at an increased rate. Atmospheric drag has the greatest long term affect on the amount of orbital decay per time. The amount of atmospheric drag that SNOE experiences is dependent on the density of the atmosphere. To correctly predict SNOE’s future orbital decay and thus lifetime, a good model for future upper atmospheric density is needed. Models do exist for future upper atmospheric density but these models are not perfect and the errors from these models increase as the time that the orbital decay is propagated for increases.

Introduction

On February 27, 1998 the University of Colorado launched the Student Nitric Oxide Explorer (SNOE) spacecraft from Vandenburg Air Force base in California. SNOE was built at the University of Colorado at the Laboratory for Atmospheric and Space Physics (LASP) and is still operated there today. SNOE is a hexagonal structure, 36” high and 39” across its widest dimension and has a mass of about 115 kg¹. SNOE was launched into a 556 km near circular sun synchronous orbit with an average local time of 10:30 AM/PM. SNOE is a spinning satellite at 5 RPM with the spin axis normal to the orbit plane.² The scientific goals of SNOE is to measure the nitric oxide density in the terrestrial lower thermosphere (100-200 km altitude) and to analyze the energy inputs to that region from the sun and magnetosphere that create it and cause its abundance to vary dramatically. SNOE seeks to meet these science goal using 3 instruments: an ultraviolet spectrometer to measure nitric oxide altitude profiles an auroral photometer to measure auroral emissions (which has now been shut off) beneath the spacecraft and a five-channel solar soft X-ray photometer.¹

The Problem

The problem is simply this: SNOE’s orbit is decaying. How long does SNOE have before it will enter the atmosphere and burn up? SNOE was originally proposed to be a two year mission therefore most of the calculations for the orbit decay only went for two years. SNOE will celebrate its three-year anniversary in February of 2001. This was made possible by increases in funding and the success of the science data that SNOE is gathering. No significant studies have been done on how long SNOE will remain in orbit with the new funding available. Figure 1 shows SNOE’s orbit decay from launch to present. The data used in figure 1 was taken from Norad TLE’s collected and used weekly by the SNOE ADCS mission operation team. Data points are approximately fifteen days apart.

Figure 1

Figure 1 shows that SNOE’s average mean motion is increasing. It can be inferred from this graph that because SNOE has more orbits per day that it must be getting closer to the Earth’s surface and since SNOE is in a nearly circular orbit the average mean motion can be converted to the average semi-major axis. The equations for this are:

n(rad/s) = n(rev/day) * 2 * pi/86400 (1)

a(km) = (mu/n²)^1/3 (2)

where:

n = average mean motion

a = average semi-major axis

mu = Earth gravitational parameter

Figure 2 shows the decay of SNOE’s semi-major axis from launch to present using the same data as figure 1 but with the average mean motion converted into average semi-major axis.

Figure 2

Importance of Satellite Lifetime Estimation

The importance of satellite lifetimes is important to the field of astrodynamics because nearly all spacecraft must estimate how long they will be in orbit. For low Earth orbiters (LEO) gathering science data like SNOE this is especially important because their lifetime must allow the amount of science data to meet its scientific goals. This field is also important to larger satellites because they may not completely burn up during reentry. Detailed analysis must be done for these larger satellites to try and avoid land impacts that can cause death and destruction. This paper does not go into the detail needed to describe Earth impacts but it is a start.

Reasons for SNOE’s Orbital Decay

This study will focus on the effects of atmospheric drag on the orbit of SNOE. Although no new in depth studies have been done for the SNOE project William Tobiska did a detailed study in 1972 for the Solar Mesosphere Explorer (SME). The SME project is a similar to SNOE it that it had a similar initial orbit: polar orbit with an inclination of 97.5 degrees, circular orbit of 540 km, and sun synchronous orbit. Therefore many of the equations and assumptions in this paper rely heavily on Tobiska’s document.³

Atmospheric drag is not the only perturbing force on SNOE’s orbit, but because SNOE has a low Earth, circular, nearly sun synchronous orbit most of the other perturbing forces are periodic or minor compared to atmospheric drag. In other words atmospheric drag is the major force that affects the lifetime of SNOE.

Description of Solution

An accurate prediction for the long-term effects of atmospheric drag on a satellite in SNOE’s orbit is very difficult. The reason for this is that there are several variables that are time dependent and cannot be well predicted. Luckily, spacecraft lifetimes are usually measured in years so there is a little room for discrepancies. As I go through the following calculations I will try and point out where the discrepancies can occur and how those discrepancies would affect the results.

Describing the Spacecraft and its Environment

The first step in estimating the lifetime of SNOE is to describe the spacecraft and the environment it is in so that we can examine their interactions. The initial equation used for this is:

Fd = ½ * Cd * A * r * v² (3)

Where:

Fd is the aerodynamic drag force

Cd is the drag coefficient

A is the effective area

r is the atmospheric density

v is the velocity

using the expressions for work and rate change of work:

W = ∫²₁Fds (4)

Wdot = Fsdot = Fv (5)

Substituting equation (3) and letting the force in the rate change of the work equation be the drag force by the atmosphere upon the spacecraft:

Wdot = ½ * Cd * A * r * v³ (6)

The total energy in the system is:

E = T + V (7)

And:

e = -mu/2r

for this and all of the future equations and calculations the orbit is assumed to be circular so r = a

Taking the time derivative of this equation:

de/dt = (mu * (dr/dt))/2r² (8)

So the rate change of the total energy is:

dE/dt = (mu * (dr/dt))/2r² (9)

From Mr. Kepler we know that the orbit period, P, is:

P = (2pi*r^3/2)/mu^1/2 (10)

The rate change of the period is the first time derivative:

dP/dt = (3/2) * (2pi/ mu^1/2) * r^1/2 * (dr/dt) (11)

simplifying the equation and substituting in the equation for P:

(dP/dt)/P = (3/2) * ((dr/dt)/r)

= (3/2) (A/m) * Cd * r * v³ * (r/mu) (12)

equating dW/dt and dE/dt gives

dr/dt = (Cd * A * r *v³ *2r²)/(2mu * m) (13)

eliminating terms:

(dP/dt)/P = (3/2) * (A/m) * Cd * r (14)

the rate change of the period can be written:

dP/dt = 3pi * r * (A/m) Cd * r (15)

this equation can be solved for atmospheric density:

r = (dP/dt)/(3pi * r *(A/m) * Cd) (16)

or for the radius:

r = (dP/dt)/(3pi * r *(A/m) * Cd) (17)

A related derivation of these equations can be found in Tobiska.³ Equations (15), (16) and (17) are the results that we needed. They can be integrated numerically with a computer by taking a constant time interval dt and stepping through the iteration to find the density or radius at each interval. They can also be solved analytically for any particular time interval. The problem with this solution is that we need to predict the radius/semi-major axis into the future, so if someone had a window in to the future we could get a value for the density of the atmosphere at all the future times. A perfect solution for the lifetime of SNOE would then be available. This discrepancy in the future density of the atmosphere is the major error in the calculation for the lifetime of SNOE and an explanation of this is below in the Atmospheric Models section. Other discrepancies are that the exact values of Cd and A are not known. The value for Cd for SNOE is estimated to be approximately 2. The value for A for SNOE is estimated to be approximately 1 m², although A does vary with time it is small because science observations require SNOE to with a relatively constant orientation to the velocity vector. These numbers came from conversations with a current LASP employee that worked on the SNOE project before launch.

Atmospheric Models

For a complete understanding of atmospheric models one must investigate the physics of the upper atmosphere. Whole books have been published concerning this topic and there is not room in this paper to go into all of the equations and calculations needed for a complete investigation. Instead this paper relies on the work of others and only highlights the effects that they have on the calculation of the lifetime of SNOE. The density of the atmosphere at SNOE’s altitude is affected by three basic variables: the molecular composition of the atmosphere, the incident solar flux, and the geomagnetic interactions. The sun goes through an 11-year cycle of activity. As solar activity increases so do the incident solar flux and geomagnetic interactions. The increases in these parameters cause the upper atmosphere to heat up. Solar flux causes nearly instantaneous heating while geomagnetic activity causes delayed heating. This heating causes increased particle collisions, which in turn causes increased density in the upper atmosphere. From the equations above we see that increased atmospheric density means increased atmospheric drag and a shorter lifetime for SNOE. When SNOE was launched in February of 1998 solar activity was expected to increase almost immediately. The most accepted way to measure the solar activity is to measure radiation with a wavelength of 10.7 cm (F10.7). The greater the solar activity the more F10.7 will be measured on Earth.⁴

Figure 3 was taken from Vallado⁴ and shows three Schatten predictions for F10.7. By looking at the difference between the three predictions we can see the difficulty in predicting the future values of F10.7.

Figure 3

Table 1 is a table of density by altitude taken from Vallado.⁴ This table shows the differences in the maximum and minimum values for density of the atmosphere. These differences are accounted for by the three parameters mentioned above. What we are concerned with here is the amount of difference shown here. For example at an altitude of 500 km, which is approximately, SNOE’s current orbit the minimum is 3.916E-13 and the maximum is 2.042E-12. This is more that a factor of five. In equation (15), r and the rate change of the period are linearly dependent. So an increase by a factor of five in the density would cause a decrease in the rate change of the period by a factor of five. Most often the predictions are not off by nearly this much but these errors grow as the time interval increases. This is because error at the beginning of the time interval affects the rest of the integration. Let’s say that the actual value for the atmospheric density is larger than the predicted value. This would cause SNOE’s orbit to decay more than expected. So even if the predict for the next time frame was correct SNOE would be at a lower radius and thus have a higher atmospheric drag than expected. See how these errors can add up quickly.

Future Work

There are several different models for the atmospheric density and the predicted solar flux. Luckily the integration needed to calculate the orbital decay is not complicated. I propose that several of these models could be analyzed to determine a possible range of values for the lifetime of SNOE. Since we do have orbital data from SNOE, atmospheric data, and solar flux in the past it could be used find more accurate values for Cd and A. This data from the past could also be used to validate some of the models, but one must be careful “past performance does not always guarantee future result.” One could also do some research to see if someone else has already done some of the work for them. For example one could check to see if Satellite Tool Kit (STK) could predict SNOE’s orbital decay. To check if STK could predict the orbital decay correctly, the past orbital decay of SNOE could be compared to the calculations done by STK.

Summary and Conclusions

SNOE’s orbital decay can be modeled analytically. Problems arise when orbital decay is propagated for long periods of time because of errors in future model of the upper atmosphere. I believe that a fairly accurate estimate of SNOE’s lifetime is possible because, although modeling the upper atmosphere is imperfect, lots of research has been put into this subject and the models are getting better all the time as more data is taken.

References

1. http://lasp.colorado.edu/snoe/overview.html

2. Bailey, S., Measurements of the Solar Soft X-ray irradiance from the Student Nitric Oxide Explorer, Laboratory for Atmospheric and Space Physics, University of Colorado, Boulder, Colorado, 1999.

3. Tobiska, W., Predictive Model of the Orbit Decay of the Solar Mesosphere Explorer, University of Colorado, Boulder, Colorado, 1985.

4. Vallado, D., Fundamentals of Astordynamics and Applications, The McGraw-Hill Companies, Inc. College Custom Series, New York, 1997.

Appendix

Calculations and data from excel for the past orbital decay plots.

Dt(day)	doy	Mean Motion (deg)	nbar	abar (km)	Altitude (km)	P (sec)	dP	dP/dt
16	59	15.03366749	0.001093279	6934.660661	556.5244	5747.101	0.161101	1.16537E-07
17	75	15.03408892	0.00109331	6934.531067	556.3948	5746.94	0.227004	1.54551E-07
13	92	15.03468279	0.001093353	6934.348457	556.2122	5746.713	0.127375	1.13404E-07
15	105	15.03501604	0.001093377	6934.24599	556.1097	5746.585	0.243953	1.88235E-07
16	120	15.03565433	0.001093424	6934.049741	555.9134	5746.341	0.176428	1.27625E-07
14	136	15.03611598	0.001093457	6933.907811	555.7715	5746.165	0.171775	1.42009E-07
16	150	15.03656548	0.00109349	6933.769623	555.6333	5745.993	0.168765	1.22081E-07
15	166	15.03700713	0.001093522	6933.633855	555.4976	5745.824	0.096593	7.45313E-08
15	181	15.03725992	0.00109354	6933.556148	555.4198	5745.728	0.233384	1.80081E-07
14	196	15.03787074	0.001093585	6933.368392	555.2321	5745.494	0.064825	5.35919E-08
15	210	15.03804041	0.001093597	6933.31624	555.1799	5745.429	0.285744	2.20481E-07
15	225	15.03878835	0.001093652	6933.086357	554.9501	5745.144	0.197773	1.52603E-07
14	240	15.03930607	0.001093689	6932.927244	554.7909	5744.946	0.373044	3.08403E-07
16	254	15.0402827	0.00109376	6932.627118	554.4908	5744.573	0.359266	2.59886E-07
15	270	15.04122338	0.001093829	6932.33807	554.2018	5744.214	0.394397	3.04319E-07
15	285	15.04225618	0.001093904	6932.02075	553.8845	5743.819	0.410542	3.16777E-07
15	300	15.04333141	0.001093982	6931.690433	553.5541	5743.409	0.465671	3.59314E-07
15	315	15.04455121	0.001094071	6931.315751	553.1795	5742.943	0.293619	2.26558E-07
15	330	15.04532043	0.001094127	6931.079498	552.9432	5742.649	0.543651	4.19484E-07
15	345	15.04674489	0.00109423	6930.642052	552.5058	5742.106	0.329043	2.53891E-07
15	360	15.04760717	0.001094293	6930.377283	552.241	5741.777	0.568501	4.38658E-07
15	375	15.0490972	0.001094401	6929.919819	551.7835	5741.208	0.398036	3.07126E-07
15	390	15.05014062	0.001094477	6929.599516	551.4632	5740.81	0.307097	2.36957E-07
15	405	15.05094575	0.001094536	6929.352387	551.2161	5740.503	0.521085	4.02072E-07
15	420	15.0523121	0.001094635	6928.933047	550.7967	5739.982	0.340535	2.62759E-07
15	435	15.05320516	0.0010947	6928.658996	550.5227	5739.641	0.527962	4.07378E-07
14	450	15.05458996	0.001094801	6928.2341	550.0978	5739.113	0.331494	2.74052E-07
17	464	15.05545957	0.001094864	6927.967312	549.831	5738.782	0.469342	3.19541E-07
14	481	15.05669097	0.001094953	6927.589575	549.4533	5738.313	0.314184	2.59742E-07
15	495	15.0575154	0.001095013	6927.336706	549.2004	5737.998	0.456389	3.52152E-07
15	510	15.05871314	0.001095101	6926.969377	548.8331	5737.542	0.434073	3.34933E-07
16	525	15.05985249	0.001095183	6926.62	548.4837	5737.108	0.370743	2.68188E-07
14	541	15.06082575	0.001095254	6926.321589	548.1853	5736.737	0.603236	4.98707E-07
16	555	15.06240961	0.001095369	6925.836031	547.6997	5736.134	0.404438	2.92562E-07
15	571	15.06347169	0.001095447	6925.51048	547.3742	5735.73	0.602906	4.65205E-07
14	586	15.06505524	0.001095562	6925.025159	546.8889	5735.127	0.360023	2.97638E-07
15	600	15.06600101	0.001095631	6924.735343	546.599	5734.767	0.709727	5.47629E-07
15	615	15.06786579	0.001095766	6924.164001	546.0277	5734.057	0.341457	2.6347E-07
15	630	15.06876312	0.001095831	6923.889113	545.7528	5733.715	0.515148	3.97491E-07
15	645	15.0701171	0.00109593	6923.474387	545.3381	5733.2	0.821013	6.33498E-07
15	660	15.0722755	0.001096087	6922.813395	544.6771	5732.379	0.790882	6.10248E-07
15	675	15.07435527	0.001096238	6922.176632	544.0403	5731.588	1.248272	9.63173E-07
15	690	15.077639	0.001096477	6921.171551	543.0353	5730.34	1.070283	8.25836E-07
15	705	15.08045565	0.001096682	6920.309723	542.1734	5729.27	0.95704	7.38457E-07
11