3. External heat loads

36

3. External heat loads

This chapter and the following set boundary conditions for the spacecraft thermal analysis, building an “Environmental Model” for the thermal problem.

Figure 3.1 - Topics of Chapters 3 and 4 (yellow box)

Chapter 3 refers to a generic satellite³ in orbit around the Earth and consider it as a mass-point, in order to evaluate heat loads as a function of the selected orbit and perform a first-analysis thermal balance.

3.1 The space environment in Earth orbits

A satellite in a closed Earth orbit can be seen as a mass-point in thermal equilibrium with the surrounding environment, which is composed of, [19]:

 three main heat sources: Sun, Earth and planetary albedo;

 a heat sink, the deep space, whose Tds = 2.8 K is a standard value in thermal analysis.

3 For the sake of simplicity, one can address to a space vehicle orbiting around the Earth with the term “satellite” rather than the more general “spacecraft”. The generality of the models developed is not affected by this lexical choice!

37 More precisely, there are other two heating sources for this class of satellites. The former is the Free Molecular Heating (FMH), which is the result of bombardment on the space vehicle by molecules in the outer layers of the atmosphere and can become significant only for LEO satellites or at the end of the launch phase, after fairing jettisoning, [4]. The latter is the effect of charged-particle heating, which is not significant in the thermal design of satellites unless cryogenic temperature should be required, [7-19]. Therefore, they have been neglected in this work, as also done by [6].

Figure 3.2 - Schematics of the space environment for an Earth-orbiting satellite, with values of heat fluxes for the three main sources (adapted from [19])

The three external contributions in the figure above vary according to the relative position between the satellite and the sources throughout the mission, that is, in time.

The most convenient way to describe satellite’s position w.r.t. its heat sources is to introduce the Geocentric Reference Frame (GRF), [20], sometimes indicated as IJK, [21], whose origin is at the Earth’s centre and:

 X-axis (unit vector i) is in a direction from Earth to the Sun at the vernal equinox (∼21 March). The direction thus indicated is termed the “first point of Aries” and lies on the equatorial plane;

 Z-axis (unit vector k) is along the Earth’s spin axis, in the northerly direction;

38

 Y-axis (unit vector j) makes up a right-handed orthogonal set with the previous two.

For closed Earth orbits six classical orbital elements, [21] are required to univocally define satellite’s position at a fixed time in GRF are:

 a, [km], is the major semi-axis (defines the size of the orbit),

 e is the eccentricity (defines the shape of the orbit),

 i, [rad] is the inclination (defines the orientation of the orbit w.r.t. the equatorial plane),

 Ω, [rad], is the right ascension of the ascending node (defines the location of the nodal line w.r.t. the equatorial plane),

 ω, [rad], is the argument of perigee (defines where the perigee of the orbit is w.r.t. the Earth’s surface),

 ν, [rad], is the true anomaly (defines the satellite’s position w.r.t. the perigee) and is indicated as v0 at time t0 (= 0), the beginning of mission,

The orbital elements are shown in the figure below.

Figure 3.3 - The six classical orbital elements in GRF (adapted from [19])

39 For particular values of e and i, different orbital elements are defined, [22]:

 if i = 0, then Π = Ω + ω is the longitude of perigee, [rad],

 if e = 0, then u = ω + ν is the argument of latitude, [rad],

 if both i = 0 and e = 0, then l = Ω + ω + ν is the true longitude, [rad].

Contents and plots in the following paragraphs are the result of a Matlab^® script, external_power_inputs.m, which represents the “Environmental Model” block in the structure of the whole project. Below is a detail of this activity.

Figure 3.4 - Structure of external_power_inputs.m

The angular orbital elements are expressed in degrees as inputs in order to have a better mental visualization of the orbit, and then they are converted in radiants. The parameter R, [m], is the radius of a “reference sphere”, an isothermal sphere whose equilibrium temperature is a rough indicator of how “hot” or “cold” the local thermal environment is if α = ε = 1, [7]. Moreover, the value of R gives an order of magnitude of the size of the satellite and, consequently, of its external heat loads, while changing the optical properties of the sphere permits to discuss the effect of different coatings in the global thermal balance.

As outputs, the time history of the heat loads are obtained, according to the duration of the orbital simulation specified by N, the number of orbits.

Other important assumptions external_power_inputs.m is based on are:

 Keplerian orbits (no perturbations, so orbital parameters are constant),

 spherical Earth and Sun models,

40

 negligible value of Tds,

 uniform and constant thermo-optical properties.

The last assumption should be released if long missions are considered, since also the deterioration of the coatings could become significant and result in a variation of the optical properties in time. In the end, further reasonable simplifications are introduced in due time in the next paragraphs.

3.2 Sun heat loads

Sunlight is the greatest source of environmental heating incident on most Earth- orbiting satellites. Since Earth's orbit around the Sun is slightly elliptical, the intensity of sunlight reaching Earth varies approximately ± 3.5%, depending on Earth's distance from the Sun. At summer solstice, Earth is at aphelion and the intensity is at its minimum value of 1322 W/m²; at winter solstice, the intensity is at its maximum of 1414 W/m², [7]. From these data, the intensity of sunlight at a generic Earth’s distance r can be computed as:

𝐽

_𝑆𝑢𝑛

= 𝐽

_{𝑆𝑢𝑛,𝑚𝑒𝑎𝑛}^𝑟02

𝑟²

(3.1) where JSun,mean = 1367 W/m² is the so-called “solar constant” and r0 = 1 AU is the Earth-Sun mean distance through one year. Eq. (3.1) requires that r be expressed in AU. The value of the solar constant, though experimentally determined⁴, is consistent with the assumption of the Sun as a blackbody radiating at an equilibrium temperature of 5800 K, as depicted below.

4 Minimum, maximum and mean value of JSun are recommended by the World Radiation Center in Davos, Switzerland, [7].

41 Figure 3.5 - Comparison between measured solar spectral emittance and an

equivalent one for a blackbody at T = 5800 K

For a satellite considered as a reference sphere, the solar heat load is a function of time and is evaluated as:

𝑄

_𝑆𝑢𝑛

(𝑡) = 𝛼𝐽

_𝑆𝑢𝑛

(𝑡)𝐴

_𝑆𝑢𝑛

𝑓

_{𝑑𝑎𝑦−𝑛𝑖𝑔ℎ𝑡}

(𝑡)

(3.2) where

 ASun = π*R², [m²], is the projection of the satellite surface along the Sun‘s direction and is constant for this simple geometry,

 fday-night is a non-dimensional function of time that determines if the satellite is either in sunlight or in eclipse and is the subject of the rest of the paragraph.

In the previous equation, JSun is an implicit function of time through the satellite’s position. For the purposes of external_power_inputs.m, JSun = JSun,mean is a reasonable assumption and the Earth’s orbit along the ecliptic plane is considered circular.

The evaluation of fday-night can be done once established the eclipse conditions for the satellite throughout the mission. Neglecting the shadow projected by the Moon, which would add much more complexity to the problem, two models of increasing precision are proposed, [21]:

42

 cylindrical model, in which Sun is considered as point-source infinitely far away from the Earth;

 conical model, in which Sun is considered as a finite-size sphere at 1 AU from the Earth.

Figure 3.6 - a) Sun as a point-source in cylindrical eclipse model; b) Sun as a finite- size sphere in conical eclipse model

In both cases, the relative Earth-Sun position at any time is required. According to the hypothesis of circular Earth’s orbit, only the relative direction must be computed, indicated as rSun in GRF. The first-approximation algorithm described in [20], Ch. 5, Par. 5.3.2, is adopted:

𝒓_𝑺𝒖𝒏= cos(𝐿_𝑆𝑢𝑛)𝒊 + sin(𝐿_𝑆𝑢𝑛) cos(𝜖) 𝒋 + sin(𝐿_𝑆𝑢𝑛) sin(𝜖) 𝒌 (3.3) where:

 LSun is the Sun’s ecliptic longitude, [deg], measured East along the ecliptic plane from the vernal equinox (i axis),

 ϵ ≈ 23.5 deg is the angle between the ecliptic and the equatorial planes and is assumed constant,

43 and the longitude angle LSun is simply given by:

𝐿

_𝑆𝑢𝑛

≅ (𝐷 − 𝐷

₀

) ∙

^360°

365 (3.4) where D is the day number, assuming D = 0 on 1 January, 00:00 (e.g. 2 January, 12:00 gives D = 1.5), and D0 = 79.0 (this assumes that the Northern Hemisphere Spring Equinox occurs on 21 March, 00:00). Clearly, this equation is approximate and ignores complications such as leap years. Expressions that are more precise can be obtained from published solar ephemeris data.

Although Fig. 3.6 shows two different models of Sun, in order to evaluate QSun the assumption of “parallel rays” always stands, [6], since Earth-satellite distances are much smaller than 1 AU.

3.2.1 Cylindrical eclipse model

In the cylindrical eclipse model, the shadow projected by the Earth corresponds to a right cylinder of infinite height whose basis is the Earth’s maximum circle and the axis is aligned with the Earth-Sun direction.

Figure 3.7 - Cylindrical eclipse model (despite the image, the solar disk is a point- source)

The algorithm to compute at any time if the satellite is in eclipse or not is taken from the same reference used in computing rSun. At a generic time, the satellite’s direction in GRF is expressed as:

44 𝒓_𝑠𝑎𝑡= [cos(Ω) cos(𝜔 + 𝜈) − sin(Ω) cos(𝑖) sin(𝜔 + 𝜈)]𝒊 + [sin(Ω) cos(𝜔 + 𝜈) +

cos(Ω) cos(𝑖) sin(𝜔 + 𝜈)]𝒋 + [sin(𝑖) sin(𝜔 + 𝜈)] (3.5) while the distance of the satellite from the Earth’s centre is given by the trajectory equation in polar form, [21]:

𝑟

_𝑠𝑎𝑡

=

^{𝑎(1−𝑒2)}

1+𝑒 cos(𝜈) (3.6) A key-role in this model is played by the angle β, locally defined as:

𝛽 = sin

⁻¹

(

^𝑅𝐸

𝑟_𝑠𝑎𝑡

)

(3.7) where RE = 6378 km is the standard value for the radius in spherical Earth model.

Figure 3.8 - Geometry illustrating the role of β, [20]

The satellite is in eclipse if the following condition on β is satisfied:

𝛽 ≥ cos⁻¹(−𝒓_𝑆𝑢𝑛∙ 𝒓_𝑠𝑎𝑡) (3.8) Cycling through each orbit, the values of νentry and νexit respectively at eclipse entry and exit are obtained. Finally, the times tentry and texit, [s], are computed after solution of the Kepler’s equation. A Matlab^® function called cylindrical_eclipse.m has been created to perform this calculus.

As a result, fday-night has a square-wave shape whose values are 0 when the satellite is in eclipse and 1 when in sunlight.

45 Figure 3.9 – Square-wave shape of fday-night, cylindrical eclipse

The figure above has been replayed from [23] assuming a circular equatorial orbit of altitude h = 600 km and initial position aligned with the Earth-Sun direction, in order to satisfy the available data as much as possible. It represents a typical fday-night for this model, thus validating it.

3.2.2 Conical eclipse model

The main drawback of the previous model is that it does not take into account those points in space which are only partially shadowed. It is known that any Earth-orbiting satellite experiences a condition of “partial eclipse”, or penumbra, when passing from the eclipse region to sunlight and vice versa, [21]. Only exceptions are those satellites always illuminated (e.g. in a dawn-dusk orbit).

In penumbra region, fday-night must assume an intermediate value between 0 and 1 and the algorithm implemented for evaluating it is the one described in [21], Ch. 5, Par. 5.2.3.

46 Figure 3.10 - Conical eclipse model

The core of the model is the geometric description of two cones:

1. the penumbra cone, whose vertex lies along the segment connecting Earth and Sun’s centres;

2. the umbra cone, whose vertex lies along the Earth-Sun direction but is located

“behind” the Earth.

This means that the conical model better describes situations in which the satellite is far from the Earth, since penumbra region increases while umbra region becomes smaller and smaller. For very large distances, usually of interest for interplanetary missions, the umbra cone vertex could be overcome and spacecraft could enter an

“annular eclipse” region, [21], which has been neglected in this context.

Figure 3.11 - Geometry of the penumbra (vertex P) and umbra (vertex U) cones

47 The problem is to verify at any time if the satellite lies in one of the two cones or in the illuminated space. Each cone is identified by the distance d of its vertex from the Earth’s centre and its semi-aperture angle Φ. For penumbra cone:

Φ

₁

= sin

⁻¹

(

^{𝑅𝑆+𝑅𝐸}

‖𝒓_{𝑆𝑢𝑛,𝐸𝑎𝑟𝑡ℎ}‖

)

(3.9)

𝑑

₁

=

^𝑅𝐸

sin(Φ₁)

(3.10) while, for umbra cone:

Φ

₂

= sin

⁻¹

(

^{𝑅𝑆−𝑅𝐸}

‖𝒓_{𝑆𝑢𝑛,𝐸𝑎𝑟𝑡ℎ}‖

)

(3.11)

𝑑

₂

=

^𝑅𝐸

sin(Φ₂)

(3.12) where Rs ≈ 109.2*RE is the radius of the solar disk and rSun,Earth is the position-vector of the Sun in GRF, whose modulus is assumed constant.

Cycling through each orbit, four values are now obtained: νpen,entry and νpen,exit

respectively at penumbra entry and exit, and νu,entry and νu,exit respectively at umbra entry and exit. Finally, the times tpen,entry, tu,entry, tu,exit, tpen,exit are computed after solution of the Kepler’s equation. A Matlab^® function called conical_eclipse.m has been created to perform this calculus.

In the end, the criterion to evaluate fday-night in penumbra region is based on evaluating the portion of the solar disk, of area A, “seen” by the satellite. For this purpose, the following “shadow function” μSun in penumbra region is defined:

𝜇

_𝑆𝑢𝑛

= 1 −

^𝐴

𝜋𝑎_𝑆𝑢𝑛²

(3.13) where aSun is the semi-aperture angle of the satellite’s Field-Of-View (FOV) towards the Sun. Both expressions for A and aSun are not reported for the sake of brevity, but can be found in [21].

48 Figure 3.12 - Solar disk occluded by Earth and graphical definition of A and aSun,

[21]

3.2.3. Comparison between the models

In most orbits that have been simulated, the percentage of penumbra period w.r.t.

the orbital period P, [s], is so low (e.g. 0.3% for a GEO satellite) that such a complication of the model seems not to be worth. Nevertheless, one can find a class of orbits for which a satellite can experience penumbra periods of about 5%, without entering the umbra region. This is the case of a MEO (Medium Earth Orbit) having the following orbital elements:

 a = 9000 km

 e = 0.2

 i = 36 deg (things vary substantially for small variations of this value)

 Ω = 90 deg

 ω = 270 deg

 ν0 = 90 deg (initial position at the intersection with GRF Y-axis)

49 Figure 3.13 - Sketch of the orbit with orbital elements listed above

The satellite is a “black” sphere of radius R = 1 m and the simulation is carried out for N = 5 orbits, starting at D = 79 (21 March, 00:00). In this case, the difference between the two models can become significant, as depicted in the figures below.

Figure 3.14 - Solar power input on a MEO satellite, cylindrical eclipse model

50 Figure 3.15 - Solar power input on a MEO satellite, conical eclipse model

The first model clearly underestimates QSun, since it is zero in eclipse (about 2-3%

of the orbital period), while the second model shows that QSun is always large than zero and can reach relatively high local minima in penumbra region. Remembering that the Sun is the main source of heat in space and remains essentially the only one at larger altitudes (where both albedo and Earth contributions considerably decrease), a more detailed conical eclipse model should be adopted. Moreover, components with relatively low thermal inertia like the Optical Solar Reflectors (OSRs), which are employed as radiators in spacecraft, [22] could be very sensitive to rough heat load changes near eclipse region; therefore, a good estimate of QSun

is necessary in analyzing them.

3.3 Earth heat loads

Most of the sunlight incident on Earth’s surface is absorbed and eventually re- emitted as IR energy. The Earth’s emission spectrum is particularly influenced by the presence of a non-homogeneous atmosphere, whose constituents experience different thermo-optical phenomena. The most common model for Earth’s IR emission is to assume the planet as a blackbody radiating at an intermediate

51 temperature of 255 K, which is the mean value between the blackbody curves enveloping the measured spectrum.

Figure 3.16 - Comparison between measured Earth’s spectral emittance and possible approximating blackbody curves (adapted from [20])

The 288 K blackbody curve approximates the radiation from the Earth’s surface, while the 218 K blackbody curve approximates the radiation from the atmosphere in those spectral regions where the atmosphere is opaque, [20]. The intensity of Earth’s IR emittance is therefore a function of altitude h and is given by:

𝐽

_{𝐸𝑎𝑟𝑡ℎ}

= 𝐽

_{𝐸𝑎𝑟𝑡ℎ,𝑚𝑒𝑎𝑛}

∙ (

^𝑅𝐸

𝑅_𝐸+ℎ

)

²

(3.14) where JEarth,mean = 239 W/m² is the so-called “Earth’s constant”.

The concept of Earth-emitted IR can be confusing, since satellites are usually warmer than the effective Earth’s temperature, and the net heat transfer is from satellite to Earth, [7]. However, when computing satellite’s view factors toward space, a convenient practice is to ignore the presence of Earth and then adding its

52 IR emission as “incoming” in satellite’s thermal balance. In doing so, the Earth’s heat load to the satellite is expressed as:

𝑄

_{𝐸𝑎𝑟𝑡ℎ}

(𝑡) = 𝜀𝐽

_{𝐸𝑎𝑟𝑡ℎ}

(𝑡)𝐴

_𝑠𝑎𝑡

𝐹

_{𝐸𝑎𝑟𝑡ℎ,𝑠𝑎𝑡}

(𝑡)

(3.15) where:

 ε plays the role of the IR absorptivity of the satellite’s external surface, as a consequence of Kirchhoff’s law; this “improper” use of ε is also suggested by [19];

 Asat = 4π*R², [m²], is the external surface of the reference sphere;

 FEarth,sat is the view factor between Earth and satellite (≈0.5 for LEO satellites).

Both JEarth and FEarth,sat are implicitly functions of time by means of h: in fact, a rather precise expression for FEarth,sat can be obtained by analytical solution of the view factor between two spheres of radius R and RE. Since RE >> R, the Earth can be considered “flat” and the expression simplifies in:

𝐹

_{𝐸𝑎𝑟𝑡ℎ,𝑠𝑎𝑡}

=

¹

2

∙ {

1 − [1 −

¹

(^ℎ 𝑅𝐸+1)

2

]

1 2

}

(3.16)

As an example, the Earth heat load for the orbit introduced in the previous paragraph is plotted.

53 Figure 3.17 – Earth power input on a MEO satellite

Since the orbit is elliptic, QEarth shows a sine-like behavior in time. Intuitively, it is constant for circular orbits, while it becomes smaller at larger values of h.

The intensity of this energy emitted at any given time from a particular point on Earth can vary considerably depending on factors such as the local temperature of Earth's surface and the effect of local weather, like cloud coverage or moisture content.

Generally, highest values of Earth-emitted IR will occur in tropical and desert regions (as these are the regions of the globe receiving the maximum solar heating) and will decrease with latitude, [7]. To account for these second-order effects, experimental data are needed, as the one reported in Tables 2.1 through 2.4 of [7].

3.4 Albedo heat loads

For Earth, like any the other celestial body, the fraction of the solar radiation that is reflected from the surface and mitigated by the presence of atmosphere is known as albedo. By definition, albedo is the reflectivity of the Earth’s surface, whose value is highly dependent on local surface and atmospheric properties. For example, it varies from as high as 0.8 from clouds to as low as 0.05 over surface features such as water and forest, [20].

54 Table 3.1 - Common values of albedo coefficients for various surfaces, [6]

Because of greater snow and ice coverage, decreasing solar-elevation angle and increasing cloud coverage, albedo also tends to increase with latitude. These variations make selection of the best albedo value for a particular thermal analysis rather uncertain, [7].

From the standpoint of thermal analysis, an orbital average value can be used, since such changes occur rapidly in relation to the thermal inertia of most spacecraft. This value should be appropriately selected according to the orbit and, again, Tables 2.1 through 2.4 of [7] help the thermal analyst in classifying the best values.

A simpler way to express albedo heat load is adopted here:

𝑄

_𝑎𝑙𝑏

(𝑡) = 𝛼 ∙ 𝑎𝑙𝑏 ∙ 𝐽

_𝑆𝑢𝑛

(𝑡)𝐴

_𝑠𝑎𝑡

𝐹

_{𝐸𝑎𝑟𝑡ℎ,𝑠𝑎𝑡}

(𝑡)𝑓

_𝑎𝑙𝑏

(𝑡)

(3.17) where:

 alb = 0.33 is the 1-year-average albedo coefficient for Earth,

55

 falb is a non-dimensional function of time that accounts for the “geometrical”

part of the problem, that is, the relative position of Sun, Earth and satellite for a generic orbit.

The idea is to “recycle” the quantities Asat and FEarth,sat previously defined, [23], because albedo heat flux is numerically a fraction of JSun, but it has the direction of JEarth. This avoids the use of formulations that imply the introduction of a “visibility factor”, [6-19-20], which accounts for both the view factor and all the uncertainties listed before, and which is complex to translate in a computational code.

Also alb should be an implicit function of time: a possible way to account for its variation, which is also simple to automate, is to take into account the influence of latitude and longitude, as described in [24], Annex F, Par. F5. Here, alb is kept constant and two models of increasing precision are proposed for the evaluation of falb:

 sinusoidal model, only based on the variation along the orbit of the angle between Sun and satellite directions (in GRF);

 refined model, which takes into account the reflective contribution of the whole portion of Earth’s surface in the satellite’s horizon.

3.4.1 Sinusoidal albedo model

This model is based on the following observation: the albedo heat flux on a satellite decreases as the satellite moves along its orbit and away from the subsolar point (the point on Earth where the Sun is at the zenith, i.e., directly overhead), even if the albedo constant remains the same. As the spacecraft moves away from the subsolar point, it is over regions of Earth's surface where the local incident solar energy per square meter is decreasing with the cosine of the angle from the subsolar point. The albedo heat load on the satellite will therefore approach zero value near the terminator (the dividing line between the illuminated hemi-sphere and the shadowed one), even if alb >0, [7].

56 The angle from the subsolar point, θs, is therefore a key-parameter in this model and is defined as the angle between the Sun direction and the direction of the local vertical from the Earth’s surface.

Figure 3.18 - Definition of θs angle

Again, under the assumption of “parallel rays”, since in GRF the local vertical coincides with the satellite’s position-vector, one obtains:

𝜃_𝑠 = cos⁻¹(𝒓_𝑺𝒖𝒏∙ 𝒓_𝒔𝒂𝒕) (3.18) Beyond the terminator, sunrays are not able to reach the Earth’s surface and no reflection can occur, so it is:

𝑓_𝑎𝑙𝑏 = {cos(𝜃_𝑠) (𝑠𝑎𝑡𝑒𝑙𝑙𝑖𝑡𝑒 𝑖𝑛 𝑖𝑙𝑙𝑢𝑚𝑖𝑛𝑎𝑡𝑒𝑑 ℎ𝑒𝑚𝑖 − 𝑠𝑝𝑎𝑐𝑒)

0 (𝑠𝑎𝑡𝑒𝑙𝑙𝑖𝑡𝑒 𝑖𝑛 𝑠ℎ𝑎𝑑𝑜𝑤𝑒𝑑 ℎ𝑒𝑚𝑖 − 𝑠𝑝𝑎𝑐𝑒) (3.19) A Matlab^® function called sinusoidal_albedo.m has been created to integrate this model in the main script. The procedure above can be found in [4] and [25]. It is also adopted in [23], where falb is defined as an explicit function of time. This turns out to be convenient for circular orbits, while an error that increases with increasing e occurs because the time the satellites spends in the illuminated hemi-space may not be equal to the one spent in the shadowed hemi-space.

57 Figure 3.19 – Sinusoidal shape of falb

Despite that, the figure above replays falb as reported in [23], assuming a circular equatorial orbit of altitude h = 600 km and initial position aligned with the Earth-Sun direction, in order to satisfy the available data as much as possible. This can be seen as a particular case of falb for circular orbits, thus validating the model.

It is worth to notice that the sinusoidal behavior is w.r.t θs and not time; therefore, falb

usually shows a “tilted” sine-like plot in time when computed for elliptic orbits. This is the case of an orbit having the following orbital elements:

 a = 9000 km

 e = 0.1

 i = 60 deg

 Ω = 30 deg

 ω = 90 deg

 ν0 = 0 deg (initial position in GRF 2^nd octant)

58 Figure 3.20 - Sketch of the orbit with orbital elements listed above

The satellite is again a “black” sphere of radius R = 1 m and the simulation is carried out for N = 5 orbits, starting at D = 79 (21^st March, 00:00).

Figure 3.21 - Albedo power input on a MEO satellite, sinusoidal albedo model

59 3.4.2 Refined albedo model

The previous model is based on the approximation that a satellite receives reflected solar radiation from only that portion of the Earth directly beneath it, ideally, a point at its nadir. The main drawback of this assumption is that Qalb is zero at the terminator, and would remain zero at any time if the satellite were flying along a dawn-dusk orbit, whose orbital plane is almost exactly perpendicular to rSun. This is situation is unacceptable since dawn-dusk and, more generally, polar satellites receive albedo heat loads when they are flying over the poles and even beyond.

The refined model assumes the Earth to be a diffusive reflecting sphere or, equivalently, a Lambertian surface, [20-26]. Therefore, the satellite receives the reflective contribution from the whole Earth’s surface it can “see”, which is called satellite’s horizon.

Figure 3.22 - Satellite's horizon

It can happen that part of the horizon is in the shadowed hemi-space, thus giving no contribution to the whole albedo heat load. The algorithm, implemented in a Matlab^® function called refined_albedo.m, is described in [26], which refers to θs as the

“solar zenith angle”. In the figure above, θm is the central semi-angle of the satellite’s horizon (a spherical cap) and is given at a certain time by:

𝜃

_𝑚

= cos

⁻¹

(

^𝑅𝐸

𝑟_𝑠𝑎𝑡

)

(3.20)

60 The whole geometry involved in the problem is highlighted below.

Figure 3.23 - Geometry for refined albedo model, [26]

The unit of measure for distances is [DU], the canonical unit also called “Earth-mean- radius” (1 DU = RE), while angles can be expressed in [deg] or [rad]. With reference to the figure above (and only for this section):

 ρ is the distance of the satellite from the Earth’s centre,

 θ is the co-latitudinal angle defining the position of the surface element dS, [DU²], w.r.t. rsat,

61

 ϕ is the azimuthal angle of integration defining dS w.r.t. the plane formed by rsat and rSun,

 ξ is the angle between ρ and the normal to dS,

 λ is the angle between rSun and the normal to dS The most general form of falb is:

𝑓

_𝑎𝑙𝑏

=

¹

𝜋

∫

cos 𝜆 cos 𝜉 𝜌²

𝑑𝑆

𝑆

(3.21) In spherical coordinates, the surface element is expressed as:

𝑑𝑆 = sin 𝜃 𝑑𝜃 𝑑𝜙 (3.22) in addition, the following relationship from spherical trigonometry is exploited, [22]:

cos 𝜆 = cos 𝜃 cos 𝜃_𝑠+ sin 𝜃 sin 𝜃_𝑠cos 𝜙 (3.23) After introducing geometric relationships among the quantities above, which are here omitted for the sake of brevity, Eq. (3.21) can be rewritten as:

𝑓

_𝑎𝑙𝑏

=

²

𝜋

∫ ∫

^{(𝑟𝑠𝑎𝑡}cos 𝜃−1)(cos 𝜃 cos 𝜃_𝑠+sin 𝜃 sin 𝜃_𝑠cos 𝜙) (𝑟_𝑠𝑎𝑡²+1−2𝑟_𝑠𝑎𝑡cos 𝜃)³²

sin 𝜃 𝑑𝜃

𝜙_𝑚

0

𝑑𝜙

𝜃_𝑚

0

(3.24)

where ϕm can be evaluated through the relation:

cos 𝜙

_𝑚

= −

cos 𝜃 cos 𝜃_𝑠

sin 𝜃 sin 𝜃_𝑠

(3.25) Cycling through each orbit, the double integral in Eq. (3.24) has different extremes of integration, according to the relative position between Earth and satellite. In particular, the integration area is given by the intersection between the illuminated hemi-sphere and the satellite’s horizon.

It is interesting to notice that the condition for which the integration area reduces to zero, and so is falb, is when Eq. (3.8) is satisfied; in other words, a cylindrical eclipse well describes conditions for absence of albedo heat loads. Therefore, the “albedo period” Palb (the period during which Qalb > 0) plus the “cylindrical eclipse period” Pecl

returns P.

62 Details about how to handle Eq. (3.24) in various cases can be found in [26], together with a graph that shows the variation of albedo heat flux as a function of altitude.

This graph is replayed below by means of a Matlab^® script called Cunningham_fig_2.m in order to assess the correctness of the model implemented. Only for this plot, alb = 0.34 and JSun = 1353 W/m². SZA coincides with θs.

Figure 3.24 - Albedo heat flux variation with altitude

3.4.3 Comparison between the models

The “philosophy” behind the refined model lies in the “π” at the denominator of Eq.

(3.24): in fact, this model takes only the directional contribution of the albedo radiation from any element dS (see Eq. (2.15)). On the other hand, the sinusoidal

63 model takes the whole contribution of the albedo radiation at the satellite’s nadir on the Earth’s surface and project it along rsat.

To show what the limit of the sinusoidal model is, consider two concentric noon- midnight orbits, the former (green circle) at h = 600 km and the latter (red circle) at h = 3000 km.

Figure 3.25 - Concentric noon-midnight orbits

From Eq. (3.19), falb in the sinusoidal model is expected to have the same pattern for both orbits (apart from the change of P value), while falb in the refined model should change because the satellite’s horizon changes at different altitudes. In other words, if the sinusoidal model is adopted, the variation of albedo heat loads with altitude does not depend on falb in Eq. (3.17); instead, making use of the refined model, a dependence on falb also exists. The two following figures confirm this trend.

64 Figure 3.26 – Albedo models comparison, noon-midnight orbit at h = 600 km

Figure 3.27 – Albedo models comparison, noon-midnight orbit at h = 3000 km

To conclude, the sinusoidal model slightly underestimates the albedo loads at lower altitudes, while the overestimation error increase at higher altitudes.

The improvement in adopting the refined model is plotted below, with reference to the orbit introduced for the sinusoidal model (simulation with N = 1).

65 Figure 3.28 – Albedo power input comparison for orbit in Figure 3.20

The results are rather different and, in this case:

 max(Qalb) ≈ 741 W for sinusoidal model;

 max(Qalb) ≈ 421 W for refined model;

 overestimation error on max(Qalb) ≈ 76%.

As expected, albedo heat loads are more realistic when the satellite crosses the terminator. This is definitely confirmed if a dawn-dusk orbit is considered.

Figure 3.29 - Albedo power input comparison for a dawn-dusk orbit, h = 600 km

66 In the end, the refined model could be further improved, for example, introducing an expression of alb as a function of θs that “enters” the integral in Eq. (3.24). The subject of albedo coefficient “corrected” along the orbit is treated in [27], Par. 2.2.

3.5 Thermal balance of a satellite

The energy conservation for a generic system, written in terms of powers, simply requires that, at any instant, heat entering is equal to heat exiting:

𝑄

_𝑖𝑛

= 𝑄

_𝑜𝑢𝑡

(3.26) Up to now, the external power inputs have been evaluated and, according to Fig.

3.2, the satellite must also withstand the internal heat generated by its subsystems (electronics, batteries et al.). In the following, the term Qint is introduced to account for this contribution.

The only way the satellite can dispose heat in excess is by radiation towards deep space. In this sense, the satellite can be treated as a grey-body emitting at its equilibrium temperature Tsat through its surface Asat. Therefore:

𝑄

_𝑜𝑢𝑡

= 𝑄

_{𝑒𝑚𝑖𝑡𝑡𝑒𝑑}

= 𝜀𝐴

_𝑠𝑎𝑡

𝜎𝑇

_𝑠𝑎𝑡⁴

(3.27) Moreover, since the satellite has its own mass M, [kg], it is able to store energy in time thanks to its “thermal inertia”, which can be defined as a measure of the capability of a system to oppose resistance to variations of its external conditions. In the end, Eq. (3.26) rewrites as follows:

𝑀𝑐

_𝑝^{𝑑𝑇𝑠𝑎𝑡}

𝑑𝑡

= 𝑄

_𝑆𝑢𝑛

+ 𝑄

_{𝐸𝑎𝑟𝑡ℎ}

+ 𝑄

_𝑎𝑙𝑏

+ 𝑄

_𝑖𝑛𝑡

− 𝑄

_{𝑒𝑚𝑖𝑡𝑡𝑒𝑑}

(3.28) Eq. (3.28) is often called the satellite’s “one-node” equation, [19-27]. As discussed in Chapter 6, each node of a thermal network is by definition isothermal (i.e., the volume of material represented by the node is all at the same temperature), so Tsat

can be referred to as the satellite’s “isothermal temperature”, in the sense that at any time the satellite is at a uniform temperature.

67 3.5.1 Equilibrium temperature

When the contribution of thermal inertia is neglected, the satellite is said to be at

“equilibrium condition”; In other words, it instantly responds even to sudden variations of external conditions. At equilibrium, Eq. (3.28) simplifies in:

𝑄

_𝑆𝑢𝑛

+ 𝑄

_{𝐸𝑎𝑟𝑡ℎ}

+ 𝑄

_𝑎𝑙𝑏

+ 𝑄

_𝑖𝑛𝑡

= 𝑄

_{𝑒𝑚𝑖𝑡𝑡𝑒𝑑}

(3.29) At any instant of the satellite’s motion, Eq. (3.29) is an algebraic equation with Tsat

as only unknown. By means of Eqs. (3.2) - (3.15) - (3.17) - (3.27) and after algebraic manipulations, a closed-form solution for Tsat is obtained:

𝑇_𝑠𝑎𝑡 = √1 𝜎[ 𝑄_𝑖𝑛𝑡

𝜀𝐴_𝑠𝑎𝑡 +𝛼

𝜀𝐽_𝑆𝑢𝑛(𝑓_{𝑑𝑎𝑦−𝑛𝑖𝑔ℎ𝑡}

4 + 𝑎𝑙𝑏 ∙ 𝐹_{𝐸𝑎𝑟𝑡ℎ,𝑠𝑎𝑡}𝑓_𝑎𝑙𝑏) + 𝐽_{𝐸𝑎𝑟𝑡ℎ}𝐹_{𝐸𝑎𝑟𝑡ℎ,𝑠𝑎𝑡}]

4

(3.30) Assuming Qint = 0, the dependence on Asat vanishes. This confirms the role of the reference sphere introduced before as an indicator of the external environment, [7]

rather than a geometric parameter of the satellite.

Table 3.2 – Equilibrium temperatures for a satellite in LEO, [20]

The equilibrium solution is important for at least two reasons:

1. At initial time, it can be used as initial condition for the differential problem in time, Eq. (3.28). In this case, the transient solution should be more realistic than the ones obtained, for example, by setting room-temperature as initial condition. Despite that, it has been proven that there exists a limit-cycle to which the numerical solution of Eq. (3.28) converges independently of initial conditions [28-29];

68 2. Maximum and minimum values given by the equilibrium solution along the duration of the mission allow confining the differential solution inside a range of values that, due to thermal inertia, are never trespassed;

3. Eq. (3.30) strongly underlines the role of the ratio α/ε, which is a fundamental parameter in sizing satellite’s passive thermal control. By only adjusting α/ε (i.e., selecting the appropriate coating), Tsat can change even by hundreds degrees, [19].

As an example, consider a LEO (Low Earth Orbit) satellite covered with a black paint with optical properties defined in Tab. (3.2). In order to verify the results, assume dawn-dusk and noon-midnight orbits whose h = 240 km, so that P is the same.

Figure 3.30 - Dawn-dusk and noon-midnight orbits, h = 240 km

Combining the previous models of eclipse and albedo, two simulations for each orbit have been done and the following values of mean temperature Tsat,0 have been obtained.

Table 3.3 – Values of Tsat,0 for the selected orbits

69 Taking into account that Tab. (3.2) gives no reference about the orbit, these values are accurate enough. Below are the temperature plots.

Figure 3.31 - Noon-midnight orbit comparison

Figure 3.32 - Dawn-dusk orbit comparison

3.5.2 Effect of thermal inertia

The effect of thermal inertia on the temperature history of a satellite is associated to its “thermal time constant”, indicated as τ, [27], which is proportional to the satellite’s

“thermal mass” CM = M*cp, [J/K]:

70

𝜏 =

^𝑀𝑐𝑝

4𝐴_𝑠𝑎𝑡𝜀𝜎𝑇_{𝑠𝑎𝑡,0}³

=

^{𝑀𝑐𝑝𝑇𝑠𝑎𝑡,0}

4𝑄_𝑖𝑛,0

(3.31) where Tsat,0 and Qin,0 are the orbital averages of equilibrium temperature and total heat load, respectively. After computing Qin,0, the other value is given by:

𝑇

_{𝑠𝑎𝑡,0}

= √

_𝜀𝜎𝐴^{𝑄𝑖𝑛,0}

𝑠𝑎𝑡

4

(3.32)

The expression for τ is obtained after linearization of Eq. (3.28). A preliminary estimate of this parameter allows the thermal analyst to judge, once the equilibrium solution is known, how smooth the real temperature plot at regime will be.

As an example, consider a satellite made of polished aluminum (α = 0.15, ε = 0.05, cp = 961.2 J/kg*K) in a MEO with the following orbital elements:

 a = 8500 km

 e = 0.1

 i = 30 deg

 Ω = 0 deg

 ω = 90 deg

 ν0 = 270 deg (initial position along the GRF X-axis)

Figure 3.33 - Sketch of the orbit with orbital parameter as above

71 Three different values of τ have been selected in order to see the change in the smoothing of temperature plot. Numerical solution of Eq. (3.28) has been obtained by means of Matlab^® ode45 command.

Figure 3.34 - Temperature profiles for different τ, solution at regime

Apart from the values of Tsat, plots in the previous figure are qualitatively similar to the ones in [27], Fig. 2.1, for the same values of τ. Considerable smoothing of the time variation of temperature is obtained as the value of time constant is increased.

However, range between extreme values of Tsat (maximum-to-minimum) is smaller for large values of τ than for small. There is also less orbit-to-orbit variability in temperature extremes for large values of τ than for small values.

It is intuitive that the case τ = 0 recalls the equilibrium solution, thus confirming the role of the equilibrium solution in “englobing” all the other possible solution for a satellite having non-zero thermal inertia.