Sustainable management of space activity in low Earth orbit

ABSTRACT

This paper extends the analysis initiated by Rouillon [2020. “A Physico-economic Model of Low Earth Orbit Management.” Environmental and Resource Economics 77 (4): 695–723. https://doi.org/10.1007/s10640-020-00515-z] of the externality caused by space debris. Satellite operators make choices about the design and launch of satellites, while in-orbit servicing firms supply efforts to remove space debris. Focusing on the long-term orbital state, we compare two management regimes. The open access equilibrium occurs when the orbit is a common resource. The optimal policy maximizes the net present value generated periodically by the space industry. We investigate economic instruments capable of effectively regulating space activity. We show that the combination of an ad valorem tax, a launch tax, and a market for removal effort certificates can provide the right incentives. A numerical application using a realistic calibration illustrates our results.

KEYWORDS:

• Space economics
• orbital debris
• sustainability

JEL CLASSIFICATION:

• L1
• L9
• Q2

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Mark and Kamath (2019) produced a review of space object removal methods.

2 Low Earth orbit is at altitudes between 200 and 2000 km. The reason we focus on this region is that it is already very crowded with debris and highly coveted by the space industry. We also ignored medium and geostationary orbits because the physical and economic heterogeneity between orbital regions (i.e. decay rate, launch cost, satellite design) would affect the quality of our results.

3 Satellites differ in many respects (mass, surface, power, etc.). To account for this diversity, we assume that satellites can be produced at different scales (from large to nano satellites), making the launch rate a continuous variable.

4 The operational lifetime $min\{T_{\lambda },T_{\mu }\}$ is distributed according to $1-e^{-(\lambda +\mu )t}$.

5 Equivalently, the operator chooses the intrinsic lifetime, equal to $1/\lambda$.

6 The duration of the natural fallout of a satellite varies from decades to centuries depending on the altitude.

7 In our physical model (see Appendix A1), the removal activity causes inactive objects to decay at rate r. Thus, a cleaning effort r is equivalent to removing a fraction $1-e^{-r}$ of inactive objects currently in orbit in one year. Alternatively, this is equivalent to removing inactive objects in an average of 1/r years after their mission ends.

8 This assumption is justified in Appendix A1.

9 Rouillon (2020) found 67 operators owning the 589 commercial satellites in low Earth orbit in 2018. Updated today (Union of Concerned Scientists, 2022), we count 162 operators and 3797 commercial satellites. Additional data describing recent developments in the satellite services market are available in a supplementary material in an appendix.

10 Cost $c\left(\lambda \right)$ is the counterpart of the remuneration of subcontractors for the manufacture and placing in orbit of the satellite. It therefore includes the cost of design, construction, testing, launch and insurance.

11 From footnote 7, a cleaning effort r = 1 means removing $1-1/e\simeq 63$ % of currently end-of-life objects in one year. By definition, parameter d gives the annual cost of the inputs required to achieve this. See Mark and Kamath (2019) for a survey of methods currently in development.

12 Up to Section 8, the market for satellite services is assumed perfectly competitive. Several arguments can justify this, at least as an approximation. First, competition in this market goes beyond the space sector, as most of these are also available by terrestrial means. Second, since the lifetime of a satellite ranges from 1 to 25 years (Union of Concerned Scientists, 2022), the addition or removal of satellites in a given year has a negligible effect compared to the size of the satellite fleet.

13 This includes cases where $\underset{\_}{\lambda }=0$ and/or $\overline{\lambda }=\mathrm{\infty }$.

14 This assumption will be abandoned in Section 8.

15 Either incumbent operators will expand their fleet or entrants will build up their own fleet.

16 The second-order condition $2p/(\delta +{\lambda }^{\ast }+{\mu }^{\ast }{)}^3-c^{″}({\lambda }^{\ast })<0$ is sufficient for design choice $\lambda ={\lambda }^{\ast }$ to be a local maximum.

17 Alternatively, one may propose to maximize the yearly net revenue $\left(p/\right(\lambda +\mu )-c(\lambda \left)\right)q-\mathrm{d}r$. Our approach embeds it as a special case, for the two objectives coincide when $\delta =0$.

18 Below, notations ${\mu }^o$, ${\mu }_{\lambda }^o$, ${\mu }_q^o$ and ${\mu }_r^o$ should be read as $\mu ({\lambda }^o,q^o,r^o)$, ${\mu }_{\lambda }({\lambda }^o,q^o,r^o)$, ${\mu }_q({\lambda }^o,q^o,r^o)$ and ${\mu }_r({\lambda }^o,q^o,r^o)$, respectively.

19 A proof is given in Appendix A3.

20 We are aware that implementing an optimal regulation would require an international treaty, which is out of reach in the short term. Yet, we believe that its description remains helpful to guide the debate. Besides, to supplement it, we simulate alternative policies in Section 7. We thank two anonymous referees for suggesting this extension.

21 As debris is the elementary vector of the externality, one might be tempted to regulate the space sector using a Pigouvian tax, levied on each marginal debris emitted, reflecting the associated intertemporal damage. In practice, such a system is not realistic, because the fragments generated by a satellite destruction are not detectable by radar below a size of 10 cm and can potentially cause cascading collisions.

22 Clearly, we find Equations (5(5) $\frac{p}{\delta +\lambda +{\mu }^{\ast }}-c\left(\lambda \right).$(5) ), (6(6) $-\frac{p}{{(\delta +{\lambda }^{\ast }+{\mu }^{\ast })}^2}-c^{\prime }\left({\lambda }^{\ast }\right)=0.$(6) ) and (7(7) $\frac{p}{\delta +{\lambda }^{\ast }+{\mu }^{\ast }}-c\left({\lambda }^{\ast }\right)=0.$(7) ) as a special case when $l=\tau =0$.

23 If ${\mu }_q^{\ast }+ϵ{\mu }_r^{\ast }<0$, part (i) of the proposition is unchanged, while part (ii) is reversed.

24 Remember that they vary proportionally, as $r^{\ast }=ϵq^{\ast }$.

25 See Appendix A4 for the derivation and A5 for the calibration.

26 This implicitly implies that $\overline{\lambda }=\mathrm{\infty }$.

27 In accordance with our physical model, we normalized the units by considering satellites of mass equal to 500 kg.

28 The data collected and calibration steps are explained in a supplementary document.

29 Recall that this is the cost of removing 63 % of current inactive objects in one year. At the end of 2021, with a population of approximately 4000 objects, the unit cost of removing one of them would be $2500/(0.63\times 4000)=1$ $m/sat.

30 There are only four exceptions (Adilov, Alexander and Cunningham, 2015, Bernhard, Deschamps and Zaccour, 2022; Klima 2016, 2018), but they are not relevant to our problem. Adilov, Alexander and Cunningham (2015) considers a limited horizon (two periods) and focus on Geosynchronous Earth Orbit. Bernhard, Deschamps and Zaccour (2022) supplies a dynamic duopoly game, but must assume that an international ‘active debris removal agency’ keeps the stock of debris constant at an exogenous level to calculate the (Markovian Nash) equilibrium of the game. Klima (2016, 2018) is irrelevant because the demand side of the market is ignored in these papers.

31 The details are explained in a supplementary document.

32 Note that this setting is equivalent to that of Section 7 in the special case where A = p and B = 0.

33 Drawing on the standard bioeconomic model, we implicitly consider here a removal production function $\mathrm{AE}\left(t\right)y\left(t\right)$, where $A$ is a catchability coefficient, $E\left(t\right)$ is the fleet of cleaning spacecrafts and $y\left(t\right)$ is the population of inactive objects. In our paper, we define the rate of removal $r\left(t\right)=\mathrm{AE}\left(t\right)$ to save notation.

34 The open access equilibrium corresponds to the particular case where $l=\tau =ϵ=0$.

35 All results are reversed if ${\mu }_q^{\ast }+ϵ{\mu }_r^{\ast }<0$.