Section 13. Orbit Perturbation. Orbit Perturbation. Atmospheric Drag. Orbit Lifetime

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1 Section 13 Orbit Perturbation Orbit Perturbation A satellite s orbit around the Earth is affected by o Asphericity of the Earth s gravitational potential : Most significant o Atmospheric drag : Orbital decay due to energy loss (apogee lowering : circular) o Solar radiation pressure : Generates force away from the Sun o Lunisolar gravitational effect : N body problem 1 Atmospheric rag Orbit Lifetime Lifetime estimates the amount of time a low Earth orbiting satellite can be expected to remain in orbit before the drag of the atmosphere causes reentry. While the computational algorithms are similar to those implemented in the Long-term Orbit Predictor, there are some important differences. The effect of drag on an eccentric, low-earth orbit. As a satellite passes through the upper atmosphere at perigee, drag acts to gradually slow it down, circularizing the orbit until it eventually decays. This drag force can effect at altitude below 800 km. First, a much more accurate atmospheric model is implemented to compute the drag effects. Second, the gravitational model for the Earth Jn (J-J10), however, is significantly simplified since the inclusion of the higher order terms doesn t impact orbit decay estimates. 3 4

2 Orbit Lifetime Estimation Perturbations from Atmospheric rag a = -(1/) ρ(c A/m)V a = acceleration due to drag on a satellite ρ = atmospheric density C = the coefficient of drag. A = satellite cross-sectional area m = satellite mass V = satellite velocity with respect to the atmosphere 5 6 Perturbations from Atmospheric rag For circular orbits, the previous equation can be used to derive the much simpler expression : arev = π( CA/ m) ρa P r ev = -6 π ( CA/ m) ρa / V Vrev = π( CA/ m) ρav e = 0 rev arev = semi major axis (km/rev) P r ev = period(sec ond / rev) Vrev = velocity( km / s)/ rev e = eccentricity rev 7 ata = # of revolution / Ballistic Coefficient (Approximate) Orbit Lifetime Estimation (Example) 8

3 Orbit Lifetime Estimation (STK) Earth s Oblateness J Naut. Mile bulge = come out Columbus was wrong! The Earth isn t really round. We call this squashed shape oblateness. This bulge can be modeled by complex mathematics and is frequently referred to as the J effect. J is a constant describing the size of the bulge in the mathematical formulas used to model the oblate Earth. S/L 1000 kg/circular orbit/altitude 300 km/lifetime = 1 days Reentry after 198 orbits 9 The Earth s oblateness, shown here as a bulge at the equator (highly exaggerated to demonstrate the concept) causes a twisting force on satellite orbits that change various orbital elements over time. 10 Why J? (MAX J10) This term arises from the mathematical short-hand used to describe Earth s gravitational field. (Gravitational acceleration at any point on Earth is commonly expressed as a geopotential function expressed in terms of Legendre polynomials and dimensionless coefficients Jn. J, J3 and J4 are the zonal coefficients that depend on latitude. Of these, J is by far the most important; it is roughly 1000 times greater than either J3 or J4. However, for more precise modeling of the Earth s oblateness, all three of these must be taken into account. In addition, other, higher order terms can be included in the model. These terms serve to slice the Earth into wedges that depend on longitude (sectoral terms) and slice it again into regions of longitude and latitude (tesseral terms). Wedges = Any shape that is triangular in cross section Zonal = Relating to or of the nature of a zone 11 How J Affects the Right Ascension of the Ascending Node. Ω The nodal regression rate caused by the Earth s equatorial bulge. Positive numbers represent eastward movement; negative numbers represent westward movement. The less inclined an orbit is to the equator, the greater the effect of the bulge. The higher the orbit, the smaller the effect. 1

4 How J Affects the Argument of Perigee. ω Perturbations because of a Nonspherical Earth The geopotential Function is : Satellite angular momentum vector change!!!! Effects of J (Sufficient for mathematical analysis) J J J 3 4 = = = The perigee rotation rate caused by the Earth s equatorial bulge depends on inclination and altitude at apogee J simple equations Sun-synchronous Orbit Ω day J J 14 7 / a (cos i)(1 e ) deg/day ω a i e day 14 7 / (4 5sin )(1 ) deg/day a = semi-major axis (km) e = eccentricity i = inclination (deg) n = the mean motion (deg/day) R = Earth's equatorial radius e Sun-synchronous orbits take advantage of the rate of change in right ascension of the ascending node caused by the Earth s oblateness. By carefully selecting the proper inclination and altitude, we can match the rotation of W with the movement of the Earth around the Sun. In this way, the same angle between the orbit plane and the Sun can be maintained without using rocket engines to change orbit. Such orbits are very useful for remote sensing missions that want to maintain the same Sun angle on targets on the Earth s surface

5 Sun-synchronous Orbit Molniya Orbit Molniya orbit and ground tracks. Molniya orbits take advantage of the fact that w, due to Earth s oblateness, is zero at an inclination of Thus, apogee stays over the Northern Hemisphere, covering high latitudes for 11 hours of the 1-hour orbit period Molniya Orbit Other Perturbations Other perturbing forces can affect a satellite s orbit and its orientation within that orbit. These forces are usually much smaller than the J (oblate Earth) and drag forces but, depending on the required accuracy, satellite planners may need to anticipate their effects. These forces include: Solar radiation pressure, which can cause long-term orbit perturbations and unwanted satellite rotation. Third-body gravitational effects (Moon, Sun, planets, etc.), which can perturb orbits at high altitudes and on interplanetary trajectories. Unexpected thrusting caused by either out-gassing or malfunctioning thrusters, which can perturb the orbit and cause satellite rotation. 19 0

6 Solar radiation pressure, Solar radiation pressure causes periodic variations in all of the COEs. Its effect is strongest for satellite with low ballistic coefficients, which is light vehicles with large frontal areas such as Echo. The magnitude of the acceleration is : ar / A m A is the cross-sectional area of the satellite exposed to the Sun (m ) m is the mass of satellite (kg) Third-body gravitational effects Lunisolar effects on for nearly circular orbits (deg/day) i = orbit inclination (deg) n = number of orbit revolutions per day S/L Satellite 1 How to design Orbit? Orbit Propagation 3 4

7 Propagators NORA Propagators 5 6 STK Propagators The two-body propagator or Keplerian motion propagator it uses the same basic technique outlined in the two-body equation of motion development. This technique assumes the Earth is a perfect sphere and the only force acting on a satellite is gravity. This propagator doesn t account for any perturbations. TLE (Two Line Element) The J propagator it accounts for the 1st order effects of J Earth oblateness. This effect causes secular changes to the orbital elements over time. The J4 propagator it accounts for 1st and nd order J effects as well as 1st order J4 effects. J3, which causes long-term periodic effects, is not modeled. Because the nd order J and 1st order J4 effects are very small, you ll see very little differences between the J and J4 propagators for most orbits considered. MSGP-4 stands for Merged Simplified General Perturbations-4. It is one of the most widely used propagators in the industry. This technique uses the generalized approach to model orbit perturbations, including both secular and periodic variations such as Earth oblateness, solar and lunar gravitational effects and drag. 7 8

8 Predicting Orbits in the Real World Now, we know how the first five elements change with time, so let s update them by multiplying the rate of change by the time interval and adding this to the value of the orbital elements (COEs) future km/day future future 1/day future a = a + a (t -t ) km e = e + e (t -t ) i = i deg future future deg/day future ω = ω + ω (t -t ) deg Ω = Ω + Ω deg/day (tfuture -t ) deg future? future,? = and future value of?? = time rate of change of? 9 Example (Orbit Perturbation) A Remote-sensing satellite has the following orbital elements a = 7303 km e = i = 50 degree ω = 45 degree Ω = 0 degree U.S. Space Command has told you the semi-major axis is decreasing by km/day. Estimate your satellite lifetime in case your orbit correction thrusters stop operating. Assume your satellite will reentry almost immediately if your semi-major axis drops below 6500 km. Question 1 : How long will it take for satellite will reentry to earth? (Atmospheric rag) Question and 3 : Find Nodal Regression (eg/day) and Find Perigee Rotation (eg/day) Question 4 : Because of the Earth s equatorial bulge will move the ascending node naturally from 90 degree () to 30 degree (future). How long it will happen? 30 Example (Orbit Perturbation) 1- Find decay time - Find Nodal Regression (eg/day) a - a decay decay time = decay rate 14-7/ - Ω a (cosi)(1-e ) 3- Find Perigee Rotation (eg/day) 14-7/ - ω a (4-5sin i)(1-e ) 4- Find wait time until argument of ascending node reach 30 degree Ω wait time = = Ω Ω -Ω inital future Ω decay time = days Ω= -5 deg/day ω =? deg/day wait time = 1 days 31 Example (Orbit Perturbation) Satellite has an inclination 13 degree and altitude is 650 km Calculate 1. The drift of argument of the perigee. The drift of the right ascension of ascending node (The nodal regression) 3. From obtained value of the nodal regression, find when satellite will return over the same point of ground track. d ω = degree/day d Ω = degree/day 360 Repeat = 5 days dω / 3

9 Example (Orbit Perturbation) In case of third-body perturbation, the satellite has this effect because of the gravitational force of the Sun and the Moon. Thus find out rift of argument of perigee due to Moon and Sun rift of Right ascension of ascending node due to Moon and Sun d ω ( moon) = degree/day d ω ( Sun) = degree/day d Ω ( Moon) = degree/day d Ω ( Sun) = degree/day Example (Orbit Perturbation) Orbit ecay rate arev = π( CA/ m) ρa P r ev = -6 π ( CA/ m) ρa / V Vrev = π( CA/ m) ρav e = 0 rev Calculate 1. Satellite Period (P = min). Mean orbit decay rate 3. Maximum orbit decay rate ρ650 = atmospheric density at 650 km altitude ρ650(mean) = kg/m ρ (max) = kg/m 650 a = =708 km C =.0 ragarea = 1 m Satellite Mass = 00 kg BC = 100 kg/m a rev(mean)= m/rev = km/year a rev (max) =1.48 m/rev = 7.99 km/year C =.0 ragarea = 1 m Satellite Mass = 00 kg m BC= = 100 kg / m C A No #Rev read = = 800 BC No #Rev = = rev Lifttime = No #Rev Period = sec = min = 549 days 15years 35