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ISS Orbital Decay Dynamics

Introduction

Figure 1: The International Space Station orbiting above the Earth.

The International Space Station (ISS) currently maintains a nearly circular orbit with a minimum mean altitude of $370\text{ km}$ and a maximum of $460\text{ km}$, i.e. in the center of the thermosphere, at an inclination of $\theta=51.6^\circ$ (degrees) to Earth's equator. The trajectory of the spacecraft is similar to a spiral with a slowly changing distance from the station to the Earth's surface, and during one cycle of revolution the change in altitude is inconsiderable.

The ISS mass is $M_S=4.5\times10^5\:\text{kg}$ and its overall length is $L_S=109\:\text{m}$. Huge solar panels with a width of $W_S=73\text{ m}$ provide the ISS with electrical energy [NASA Official Report (2023)].

Including all batteries and other parts, the effective cross sectional area of the station is approximately $S\approx 2.5\times 10^3\, \mathrm{m^2}$ [European Space Agency, SDC6-23].

The ISS orbital decay is caused by one or more mechanisms which absorb energy from the orbital motion, the essential ones being:

  • atmospheric drag at orbital altitude is caused by frequent collisions of gas molecules with the satellite,
  • the Ampere force arising from the motion of the conductive apparatus in the Earth's magnetic field,
  • the interaction with the atomic oxygen ions.

... In May 2008, the altitude was 350 kilometers, the ISS lost 4.5 km and was re-boosted by the Progess-60 supply ship by 5.5 km. Again, the ISS continued to lose altitude by 5.5 km ..." [https://mod.jsc.nasa.gov]

Figure 2: The altitude of ISS $\text{(km)}$ over the years.

Figure 3: The ISS mean height $\text{(km)}$ in 2022-2023.

... The ISS loses up to 330 ft (100 m) of altitude each day ... " [NASA Control Data (2021)]. In 2023, the ISS flies at altitudes of 410 km with an orbital decay about 70 m every day ($\sim 2~km$ per month), and during magnetic storms the daily descent reaches 300 m. The ISS accomplishes the de-orbit maneuvers by using the propulsion capabilities of the ISS and its visiting vehicles [International Space Station Transition Report (2022)].

Figure 4: ISS model with the cross sections from different aspect angles ($\text{dm}^2$). The CROC provides $2481\text{m}^2$ cross section.

Denotations and Physical constants:

Universal gas constant$R$$=$$8.31\,\text{J}\cdot\text{K}^{-1}\cdot\text{mol}^{-1}$
Avogadro's number$N_A$$=$$6.022\cdot10^{23}\,\text{mol}^{-1}$
The molar mass of gas (for air)$\mu$$=$$0.029\,\text{kg}\cdot\text{mol}^{-1}$
Mass of the Earth$M_E$$=$$5.97\cdot10^{24}\,\text{kg}$
Radius of the Earth$R_E$$=$$6.38\cdot10^6\,\text{m}$
Universal Gravitational constant$G$$=$$6.67\cdot10^{-11}\,\text{m}^3\cdot\text{s}^{-2}\cdot\text{kg}^{-1}$
Density of air at Earth's surface$\rho_0$$=$$1.29\,\text{kg}\cdot\text{m}^{-3}$
Gravitational acceleration at Earth's surface$g_0$$=$$9.81\,\text{m}\cdot\text{s}^{-2}$
Average magnitude of Earth's magnetic field$B$$=$$5.0\cdot10^{-5}\,\text{T}$
The absolute charge of an electron$e$$=$$1.60\cdot10^{-19}\,\text{C}$

Part A: Modified barometric formula [2.0 points]

The pressure of atmospheric air, composed mainly of neutral $O_2$ and $N_2$ molecules, can be found by using the Clapeyron-Mendeleev law (the ideal gas law): $p \,V = \frac{M}{\mu} \, R \, T \,.$ Here $p,\:V,\:T,\:M$ and $\mu$ are the pressure, volume, temperature, mass and molar mass of a portion of air, and $R$ is the universal constant of an ideal gas.

There are two equations for computing air pressure as a function of height. The first equation is applicable to the standard model of the troposphere ($h<100\text{ km}$) in which the temperature is assumed to vary with altitude (the lapse rate).

The second equation belongs to the standard model of the thermosphere ($h>250\text{ km}$) in which the temperature is assumed not to change considerably with altitude and is applicable to ISS.

We may assume that all pressure is hydrostatic and isotropic (i.e., it acts with equal magnitude in all directions).

A1  0.50 Derive the general integral expression for the air pressure $p_{h}$ at ISS altitude $h$. This equation is called the general barometric formula.
Hint: the temperature and gravitational acceleration may depend on $h$.

Remark 1. The temperature of the Earth's thermosphere does not change considerably at altitudes of $300-600\text{ km}$ and reaches an average of $800-900\:\text{K}$ on the solar side [NASA data]. Therefore, one may set $T_h=T=\text{const}$ by investigating the ISS orbital flight. Since the spacecraft spends almost half of its flight time in the shadow side of the Earth where the temperature drops sharply, we may take the value of $T=425\:\text{K}$ as the average temperature at these altitudes.
This temperature is also in agreement with the air density value $\rho_h\sim10^{-12}\text{ kg}\cdot\text{m}^{-3}$ [MSISE-90 Model of Earth's Upper Atmosphere] at $h=400\text{ km}$.

Figure 5: The Earth’s thermosphere.

A2  0.30 Write down the air pressure (the standard barometric formula) $p^{sta}_{h}$, when the temperature and gravitational acceleration $g_h$ do not depend on $h$.
Calculate the parameter $h_0\equiv\frac{RT}{\mu g_0}$ for $T=425\:\text{K}$.

A3  0.60 Write down the air pressure (the improved barometric formula) $p^{imp}_{h}$ when the temperature is constant but the gravitational acceleration depends on $h$.
Hint: Use the leading-order correction only, with accuracy $O(z_h^2)$. This means that the flight altitude $h$ above the Earth's surface is significantly smaller than the Earth's radius: $z_h\equiv h/R_E\ll1$.

A4  0.40 Write down the ratio of the 'standard' and 'improved' versions of the barometric formula $p^{imp}_{h} / p^{sta}_{h}$. Evaluate it for $h=4.0\times10^5 m$. From here on, use the 'improved' version.

A5  0.20 Write down the air density $\rho_{h}$ and the concentration of neutral air molecules $n_{h}$ at height $h$ with accuracy $O(z_h^2)$.

Part B: Orbital deceleration and station descent rate [3.0 points]

Let us consider the problem of determining the rate of orbital decay of a satellite with mass $M_S$ that experiences a constant frictional force ${\vec F_{drag}}$ acting on it. We assume that the decrease in altitude $dh$ is much less than the flight altitude $h$ itself ($dh \ll h$).

B1  0.50 Write down the satellite velocity $v_h$ and revolution period $\tau_h$ for a stable orbit of altitude $h$.

B2  0.50 Write down the total energy $E_S$ of a satellite moving along a circular orbit with radius $R_E + h$.

B3  1.00 The total decelerating force exerted on a satellite of constant mass is given by some external braking force ${\vec F_{drag}}$. As a result, the ISS slows down and its altitude decreases by a height $dh$ in a small time interval $dt$. Write down the equation for the total energy balance of the ISS and surrounding system, given a value of${\ F_{drag}}$.

B4  0.50 Determine the rate of descent (de-orbiting speed) $u_h$ of the satellite. Hint: The de-orbiting speed depends on the friction force, the altitude of the satellite, and the mass of the satellite.

B5  0.50 Write down the descent distance $H_h$ that occurs during a revolution around the Earth (one-orbit descent distance), and the total time $T_h$ during which the satellite falls from altitude $h$ to the Earth's surface due to the frictional force.

Hint: Consider that $h_0 \ll h \ll R_E$.

Part C: Atmospheric drag [1.0 points]

The speed of the satellite $v$ is many times greater than the average speeds (hundreds of m/s) of the thermal motion of atmospheric molecules at a height $h\approx300-400\:\text{km}$, so we can assume that the molecules were at rest before colliding with the ISS. To roughly estimate the drag force, we assume that after the collision the molecules acquire the same speed as the satellite.

C1  0.50 Write down the air drag force $F_{air}$, the de-orbiting descending speed $u_h^{air}$ and the one-orbit descent distance $H_h^{air}$.

C2  0.50 Determine the total time $T_h^{air}$ for which the satellite will fall from the altitude $h$ to the Earth's surface due to air drag effect.
Hint: Consider that $h_0 \ll h \ll R_E$.

Part D: Drag by atomic oxygen ion [1.0 points]

In the thermosphere, under the influence of ultraviolet and X-ray solar radiation and cosmic radiation, air ionisation occurs ("polar lights''). Unlike $O_2$, $N_2$ does not undergo strong dissociation under the action of solar radiation, therefore, in general, there is much less atomic nitrogen $N$ in the Earth's upper atmosphere than atomic oxygen. At altitudes above $250\:\text{km}$, atomic oxygen $O$ dominates. Layers containing electrons and ions of oxygen atoms appear on the day side of the atmosphere. Here, the concentration of atomic oxygen ions reaches $n_{ion}\sim10^{12}\,\text{m}^{-3}$

D1  0.30 Write down the decelerating force $F_{ion}$, averaged during a 24-hour period, associated with the mechanical collisions of these particles. Note that there is a strong decrease in ionised layers at night, which you may assume are negligible. Let the density of ionized oxygen atoms be $\rho_{ion}$.

D2  0.70 Determine the speed of fall of the satellite $u_h^{ion}$ due to deceleration by ions of atomic oxygen and the one-orbit descent distance $H_h^{ion}$ caused by the ionisation effect. Hint: Consider that $h_0 \ll h \ll R_E$ .

Part E: Drag by the Earth's magnetic field [2.0 points]

We consider the influence on the motion of the satellite of the Earth's magnetic field, the value of which near the Earth's surface ranges between $(3.5-6.5)\cdot10^{-5}\:\text{T}$ with an average value of $B=5\cdot10^{-5}\:\text{T}$.

When a satellite moves at high speed in a magnetic field, an induced electric current (electromotive force (EMF) ) occurs in the current-conducting elements of the satellite's structure. This electromotive force causes a redistribution of electric charges
in these current-conducting elements. An electric field appears around the satellite, which affects the movement of electrically charged particles in the environment. Electrons are attracted to those parts of the satellite that have a positive potential (relative to the middle part of the satellite), and positively charged ions are attracted to those parts of the satellite that have a negative potential. Electrons and ions that hit the surface of the satellite structures are combined into neutral oxygen atoms, while the electrons 'travel' in the satellite's conductive structures, creating
an electric current. The satellite, moving in space, 'collects' electrons and ions from the surrounding space and collides with them.
For a rough estimate of the magnitude of the current that can
flow through the conductive structures of the satellite, we will assume that the collection occurs only from an area equal to the
cross-sectional area $S$ of the satellite, and that all ions and electrons participate in the creation of this current.

E1  0.60 Estimate the magnitude of the induced electric current $I_{ind}$.

E2  0.60

Determine an approximate expression for the induced 'braking' Ampere force $F_{ind}$ in the direction opposite to the direction of the satellite's motion.

Let $\phi$ be the angle between the Earth magnetic field $\vec B$ along the longitude lines and the velocity of the ISS, $\vec v_h$. To simplify, you may approximate the length of the satellite $L $ as the square root of the satellite area $S$. Additionally, instead of computing the average of $\sin(\phi)$, you may approximate it with $ \sin(\pi/2 - \theta)$. You may use a discrete number of sample points to compute an average value.

E3  0.80 Write down the de-orbiting descent speed $u_{ind}$ of the satellite due to Earth's magnetic field. Write down the one-orbit descent distance $H_h^{ind}$ caused by the magnetic drag effect.

Hint: Consider that $h \ll R_E$.

Part F: Numerical results and conclusion [1.0 points]

F1  0.40

Calculate and complete Table 1 in the Answer Sheet.

$h, \text{km}$$T_h^{air}, \text{days}$$u_{air}, \text{m/day}$$u_{ion}, \text{m/day}$$u_{ind}, \text{m/day}$$\sum, \text{m/day}$$u_{ISS}, \text{m/day}$
350      
375      
400      
410      

F2  0.40

Calculate and complete Table 2 in the Answer Sheet.

$h, \text{km}$$H_h^{air}, \text{m}$$H_h^{ion}, \text{m}$$H_h^{ind}, \text{m}$
350   
375   
400   
410   

F3  0.20

Rank the three orbital slowing processes in order of how strong an impact they have on ISS orbital altitudes higher than $380\:\text{km}$.

For the International Space Station orbiting at an altitude above $380\:\text{km}$, write down the most significant factors contributing to orbital decay.