Recently, there is significant interest in developing a new class of standardized pico-satellites called cubesats. Cubesats have been the best choice for universities and research institutes toward starting space technology developments. The mass of the cubesat is restricted to be 1 kg and its size is confined to be .1 m on all sides. This small size is considered as the main advantage of cubesats which enable them to be launched as an auxiliary payload and result in less cost and test requirements. The surface of the cubesat is covered by six aluminum plates, each of which has a solar chip attached to it. The majority of these satellites are placed in a low Earth sun synchronous orbit (the inclination is close to 98°) [1] and [11].

Many factors affect the satellite orbital motion. The Earth gravitational field has the master role in perturbing the satellite dynamical motion. However, the non-gravitational factors (e.g. solar radiation pressure, air drag, luni-solar attraction… etc) play a significant role in perturbing the orbital motion. Several authors have performed a suite of studies and models to illustrate those effects on various types of satellites and orbits. Over the last few years, a vast knowledge of natural radiation pressure effects on satellite dynamics and rotation has been achieved. The main contribution of the natural radiation pressure is due to the direct solar radiation so a variety of models were constructed to estimate their force [2], [3], [9], [11], and [19]. The second main contribution of radiation forces is due to the Earth reflected radiation known as the albedo. It is an extremely complex phenomenon which shows relevant spatial and temporal variations. Albedo depends upon the reflectivity of the illuminated surface of the Earth that is visible to the spacecraft, the solar angle, and the position of the spacecraft in space. Moreover, it depends on seasonal variations and geographical longitude and latitude of the Earth surface that is illuminated by the Sun and seen by the satellite [4], [6] and [17].

Based on a new numerical modeling of the albedo in which the irradiance is calculated depending on the geographical latitude, longitude and altitude of the satellite, the main issue of this work is to investigate the orbital perturbations of the cubesats lying on LEO due to Earth albedo.

The albedo irradiance, which reaches the satellite surface, is determined using a numerical model. This model is based on partitioning the Earth surface into a number of cells forming a grid. Then the incident solar irradiance on each cell is used to calculate the total radiant flux. The total albedo irradiance, S, at the satellite surface is given by [4] and [6]:

where V_Sun ？ V_Sat is the set of grid points that are illuminated by the Sun and visible by the satellite, σ(φ_g, θ_g) is the reflectivity of the grid points of latitude φ_g and longitude θ_g, E_AM0 = 1367 W/m² is the incident solar irradiance

is the unit vector normal to the grid cell and ρ_sun are ρ_set unit vectors directed from the grid center to the Sun and satellite , respectively. The cell area A_C(φ_g) is found using the surface revolutions as in [4] and [6]:

where Δθ_g = 1.25°, Δφ_g =1° and r_E is the Earth mean radius.

It was obvious that the albedo irradiance had a large dependency on the geographical latitude of the grid points. The maximum albedo is observed over the poles and decreased by moving away from them and towards the shadow side of the Earth. Moreover, there is a significant dependency on the longitude and the solar angle, where albedo has maximum values for low solar angles [6], [16] and [17]. However, as previously mentioned, the longitude dependency is disregarded in this work.

These equations represent the albedo contribution of a single Earth cell. However, the sunlit area visible to the satellite is decomposed into a definite number of cells. So, the total albedo irradiance reaching to the satellite can be obtained by summing up the contribution of each cell [6] and [17]:

where k is defined as the number of illuminated Earth cells visible from the satellite.

The total radiant force exerted on a flat non-perfectly reflecting surface is given by [13]:

where,

where

is a unit vector directed the force direction, c is the speed of light and S is the radiation irradiance at the satellite surface, η is the radiation incident angle to the satellite surface normal, β is the satellite surface specularity, ρ' is the satellite surface reflectivity, B_f and B_b are the non-Lambartian coefficient of the front and back surfaces of the spacecraft, respectively, α' is the spacecraft absorption coefficient, and ε_f and ε_b are the front and back satellite surface emissivity, respectively.

For non-perfectly reflecting surfaces, the force vector,

, is not directed normal to the surface. But, it inclines by an angle ？, known as the cone angle, to the incident direction as depicted in figure 1. So, the force components in the incident direction will be:

The geocentric equatorial system with e_x directed parallel to the Earth equatorial plane, e_y directed in the plane that contains the meridian of the sub-satellite point and e_z directed normal to the equatorial plane is used. As shown is Fig. 1, the vector directed through incident radiation,

, is given by:

where

is the satellite position vector and

is the Earth radius vector. For oblate Earth, the Earth radius vector is given by [7]:

with

where a_e is the Earth’s Equatorial radius, f’_e is the Earth’s flattening, φ_g is the geodetic latitude of the grid point, h is the height above sea level and θ is the sidereal time.

The satellite position vector,

, in the geocentric coordinate system, is given by [7]:

with

where Ω is the longitude of the ascending node, ω is the argument of perigee, v is the true anomaly, i the inclination, a is the semi-major axis and e is the eccentricity. Using the geocentric coordinate system, the incident radiation vector is:

with

The acceleration experienced by a satellite of mass M and cross sectional area A is:

The radial components of the disturbing accelerations,

is given by:

The tangential component of disturbing acceleration is given by:

where

is a unit vector in the direction of the angular momentum vector. It can be expressed as [7]:

The perpendicular component of disturbing acceleration is given by:

It is worth noting that the spacecraft is decomposed into a finite number of small elementary surfaces. Consequently, the total force acting on the cubesat is considered to be the sum of all forces acting on each elementary surface [3], [10] and [14]. At a given time, the total radiant acceleration affecting the spacecraft is

where k is the number of satellite surfaces and

The Cubesat is a cube shaped picosatellite constrained to CalPoly’s specifications. The structure of these cubesats may be consisted of one cube “1U”, two cubes “2U” or three cubes “3U” as specified in table 1 [5] and [12].

The cubesat’s thermo-optical properties are also constrained to CalPoly's specifications as tabulated in table2 [5] and [12]:

Based on Eqs. 15, 17 and 18 and make use of the thermooptical properties and area to mass ratio of cubesats, the acceleration components experienced by a cubesat are given by the following equations:

On account of the different area to mass ration of each cubesat, the constant C1=2.9×10^-9 is used for 1U cubesat, where C1=2.4×10^-9 is used for 2U cubesat and C1=1.8×10^-9 is used for 3U cubesat in case the radiation falls normal to the satellite surface (i.e. η=0).

For orbits of e ≠ 0 and i ≠0, the effect of radiation pressure on the satellite’s orbital elements; the semi-major axis a, the eccentricity e, the mean anomaly M, the inclination i, the longitude of the node Ω and the argument of the perigee ω can be evaluated using the general perturbation equations of Gauss [18].

The variation of M contains both mean motion of the osculating ellipse and the effect of the perturbations. So, to find an element that changes slowly with a time derivative going to zero, a new variable, ？, will be defined as [18]:

where M(t) is the osculating mean anomaly at time t and

is the mean motion. The perturbation equations are obtained as functions of the acceleration components

as the following [18]:

where E is the eccentric anomaly.

Having implemented the Earth’s reflectivity data measured by NASA’s Earth Probe satellite, which is part of the TOMS project (Total Ozone Mapping Spectrometer) [15], the Earth’s reflectivity as a function of latitudes is illustrated in fig. 2.

As illustrated in Fig. 2, the albedo has a maximum value of ~ 88 % over the latitude of 88.75 South (close to the South Pole). However, it has a minimum value of ~ 13.8 % over the latitude of 1.25 South (close to the equator).

In the present work, the albedo force is calculated for some LEO cubesats of different configuration and perigee heights. The results are tabulated in the following table:

As illustrated in table 1, the maximum albedo force is in the order of ~ 10^-7 N and the minimum value is in the order of ~ 10^-10 N which is experienced by a cubesat of ~ 823 perigee

height.

The dependency of the albedo force on the orbital eccentricity is studied. Some cubesats are chosen approximately for the same inclination and different eccentricities and arranged in ascending order according

to their eccentricities. The results showed that albedo force depends directly on the orbital eccentricity as seen in table 2.

The mean orbital perturbations of the orbital elements are studied and illustrated in table 3.

Based on the tabulated results in table 3, the maximum perturbations of the semi-major axis, eccentricity and argument of perigee are in the order of ~10^-4, 10^-10 and 10^-9, respectively. However, the minimum values are in the order of ~10^-6, 10^-12 and 10^-11, respectively.

The perturbations of the semi-major axis, eccentricity and argument of perigee of the satellite XaTcobeo are illustrated in details over one revolution in the following figs 3-5:

As illustrated previous figures, the maximum albedo perturbations for the semi-major axis, eccentricity and argument of perigee were experienced by the satellite when

it passes close to the Earths poles where the Earth’s reflected radiation has its maximum percentage. However, the minimum perturbations occurred when the satellite passes close to the Earths equator.

A simple analytical model of the terrestrial albedo force was constructed w.r.t. the geocentric equatorial coordinate system in which the Earth was considered as an oblate spheroid.

The current numerical test has validated that the albedo force has significant contribution on the satellite dynamics and also a great dependency on the geodetic latitudes which depends on the satellite field of view. The maximum albedo force is in the order of ~ 10^-7 N. Nevertheless, the minimum value is in the order of ~ 10^-10 N.

The results show that the maximum perturbations of the semi-major axis, eccentricity and argument of perigee are in the order of 10^-4, 10^-10 and 10^-9, respectively. However, the minimum values are in the order of 10^-6, 10^-12 and 10^-11, respectively. Therefore, we can conclude that the effects of the albedo have a considerable disturbance on the orbit and it can affect the cubesats life time.

In our future work, the Sun’s apparent path (the ecliptic) in the theoretical modelling and application will be considered. Moreover, the dependency of the albedo intensity on the Earth’s longitude is to be investigated.