The Range of North Korean ICBMs: Update

A.  Introduction

This is an update of an earlier post (Post #13) published in August 2017.  On November 28, 2017, North Korea conducted a test flight of a heavier, longer-range missile (Hwasong-15) that appears to be capable of delivering a nuclear weapon anywhere in North America and Europe.  Moreover, additional information about all of North Korea’s missile tests is now publicly available, allowing a simpler and better estimate of missile range to be calculated.  In this post, we outline this new approach, and include an analysis of the Hwasong-15 test flight.

B.  The “Effective Launch Speed”

The flight of an ICBM has three successive parts: the boost phase, while the missile’s rocket engines are firing; the unpowered midcourse phase, during which the missile is coasting under the influence of gravity; the terminal, or re-entry phase, when the warhead re-enters Earth’s atmosphere and converges on its target.  The midcourse phase occupies most of the flight time and is the easiest to observe.  Since August, the apogee (highest altitude) $\boldsymbol{h_{max}}$ reached in midcourse has been made public for each test flight.  With this new information, we can calculate an effective launch velocity and from it estimate the maximum range of the missile.  The numerical calculation of launch velocity based on total flight time, which we used in the previous post, is no longer necessary.

Recall that, for political reasons, North Korea uses highly “lofted” – nearly vertical  trajectories to test its ICBMs.  In the following, we will ignore the small horizontal component of the missile’s velocity when analyzing its test flight.  As before, we will also ignore the Earth’s rotation and neglect atmospheric drag.  (Air drag has little effect on the trajectory during the boost and midcourse phases because air density decreases rapidly with altitude, and throughout its boost phase, the ICBM is moving relatively slowly as it passes through the atmosphere.)

A missile’s midcourse trajectory is pre-determined by its burnout velocity $\boldsymbol{\vec{v}_{bo}}$ and altitude $\boldsymbol{h_{bo}}$ reached at the end of its boost phase.  Ignoring the small horizontal velocity of the missile, energy conservation requires

$\large \boldsymbol{\frac{E}{m}=\frac{1}{2}v_{bo}^{2}-\frac{Gm_{E}}{r_{E}+h_{bo}}=-\frac{Gm_{E}}{r_{E}+h_{max}}}$

(1)

where E is the total mechanical energy of the missile after burnout, m is the missile’s mass at burnout, G is the gravitational constant, and $\boldsymbol{m_{E}}$ and $\boldsymbol{r_{E}}$ are the mass and radius of Earth.  In the aftermath of a missile test, the only information publicly available are the total flight time and apogee $\boldsymbol{h_{max}}$: we do not know $\boldsymbol{v_{bo}}$ or $\boldsymbol{h_{bo}}$.  So imagine instead that the missile was launched at sea level by an instantaneous rocket burst (like an artillery shell), and it reached the same apogee as in the real flight test.  The “effective launch speed”  $\boldsymbol{v_{0}}$ due to this instantaneous burst obeys a relation akin to Eqn. 1,

$\large \boldsymbol{\frac{E}{m}=\frac{1}{2}v_{0}^{2}-\frac{Gm_{E}}{r_{E}}=-\frac{Gm_{E}}{r_{E}+h_{max}}}$.

(2)

For example, the Hwasong-14 missile launched on July 4 rose to a height of 2802 km (Ref. 1). Using Eqn. 2, its effective launch speed was $\boldsymbol{v_{0}=6.17\: \mathrm{km/s = 0.553 \boldsymbol{\, v_{esc}}}}$, where $\boldsymbol{v_{esc}=\sqrt{2Gm_{E}/r_{E}}}$ is the escape velocity (11.17 km/s) from the surface of the planet.

C.  An Initial Estimate of Maximum Range

The previous section dealt only with near-vertical test flights.  Using those results, we can now estimate the range of the same missile when it is launched on a minimum energy trajectory (MET), i.e., the path that maximizes its range.  The physics is the same as we presented in Section C of the August post.  The missile trajectory is a segment of an ellipse with one focus ($\boldsymbol{F_{1}}$) located at the Earth’s center (Figure 2), and a semi-major axis a derived from conservation of energy.  In terms of the effective launch speed $\boldsymbol{v_{0}}$,

$\large \boldsymbol{\frac{E}{m}=\frac{1}{2}v_{0}^{2}-\frac{Gm_{E}}{r_{E}}=-\frac{Gm_{E}}{2a}}$.

(3)

Solving for a and using $\boldsymbol{u_{0}=v_{0}/v_{esc}}$, we obtain

$\large \boldsymbol{\frac{2a}{r_{E}}=\frac{1}{1-u{_{0}^{2}}}}$.

(4)

In Figure 2, the launch point is P and the target point is Q, $\boldsymbol{F_{2}}$ is the second (empty) focus, and θ is the angle between $\boldsymbol{\overline{F_{1}P}}$ (or $\boldsymbol{\overline{F_{1}Q}}$) and the major axis of the ellipse.  The missile’s range is maximized when θ is a maximum, which occurs when $\boldsymbol{\overline{F_{2}P}}$ is perpendicular to the major axis.  Since $\boldsymbol{\overline{F_{1}P}+\overline{F_{2}P}=2a}$,

$\large \boldsymbol{sin(\theta _{max})=\frac{2a-r_{E}}{r_{E}}=\frac{u_{0}^{2}}{1-u_{0}^{2}}}$

(5)

and the maximum range is

$\large \boldsymbol{R_{max}=2r_{E}\theta _{max}}$.

(6)

To illustrate, for the Hwasong-14 launched on July 4, we found that $\boldsymbol{u_{0}=0.553}$, so $\boldsymbol{\theta _{max}=0.455\: \mathrm{(rad)}=26.1^{\circ}}$ and  $\boldsymbol{R_{max}=5810\, \mathrm{km}}$.  However, this estimate is significantly shorter than the published range of 6700 km (Ref. 2), and we will correct it in Section D.

Using the reflection property of the ellipse, the launch angle (measured relative to the local horizontal), is given by

$\large \boldsymbol{\varphi _{0}=\frac{\pi }{4}-\frac{\theta _{max}}{2}}$.

(7)

To illustrate, for the July 4 missile test, $\boldsymbol{\theta _{max}=0.455\: \mathrm{(rad)}}$, so $\boldsymbol{\varphi _{0}=.558\: \mathrm{(rad)}=32.0^{\circ}}$.  See the August post for more details.

D.  An Improved Estimate of Missile Range

The range calculated using the above procedure  will be consistently shorter than the figure quoted by arms control experts, because the launch velocity  calculated for the near-vertical test flight is not equal to $\boldsymbol{v_{0}}$ for the same missile on a MET.  Let’s look at this in more detail.  In a typical ICBM flight, the missile first rises vertically from its launch pad (or silo), then is tipped by a few degrees toward its target by firing auxiliary steering thrusters or, in more sophisticated designs, by briefly diverting (gimbaling) the main engine exhaust off-axis.  The rocket thrust is then re-aligned with the missile axis, which then rotates to the desired launch angle $\boldsymbol{\varphi _{0}}$ using gravity to alter its direction.  This so-called “gravity turn” minimizes stress on the missile body, and is a good approximation to what actually occurs in practice (Ref. 3).  Figure 3 shows boost phase trajectories of a missile undergoing a gravity turn for several values of $\boldsymbol{\varphi_{0}}$, plus a free body diagram for the missile .  The equations of motion for the missile during this maneuver are:

$\large \boldsymbol{a_{\parallel }=\frac{dv}{dt}=\frac{F_{th}}{m}-gsin\varphi }$

$\large \boldsymbol{a_{\perp }=v\frac{d\varphi }{dt}=-gcos\varphi }$

(8)

where $\boldsymbol{F_{th}}$ is the rocket thrust, m is the instantaneous missile mass, and φ is the instantaneous angle  ($\boldsymbol{0\leq \varphi \leq \varphi _{0}}$) between the velocity vector $\boldsymbol{\vec{v}}$ and the horizontal direction.  $\boldsymbol{a_{\parallel }}$ and $\boldsymbol{a_{\perp }}$ are the components of the acceleration parallel and perpendicular to $\boldsymbol{\vec{v}}$.  The “gravity loss” term $\boldsymbol{-gsin\varphi }$ reduces the burnout velocity, and its integrated effect increases with the launch angle $\boldsymbol{\varphi _{0} }$ and the duration of the boost phase. Since $\boldsymbol{F_{th}}$, mφ, and $\boldsymbol{g=g(h)}$ are functions of time, the two (coupled) Eqns. 8 above must be solved numerically to find the burnout speed $\boldsymbol{v_{bo}}$ and altitude $\boldsymbol{h_{bo}}$  (Fig. 3).  Clearly, the more we know about a missile’s design, the more accurately we can model its boost phase behavior.

We want to find the $\boldsymbol{\varphi _{0}}$-dependence of $\boldsymbol{v_{bo}}$ and $\boldsymbol{h_{bo}}$, and thereby obtain a $\boldsymbol{\varphi _{0}}$-dependent correction to the effective launch speed $\boldsymbol{v_{0}}$.  While the design details of North Korea’s Hwasong-14 and Hwasong-15 ICBMs are secret, they are known to be two-stage liquid-fueled rockets.  So let us instead study the boost phase behavior of two similar missiles of known design.  One well-characterized missile is the Titan-II, a silo-based American ICBM of 1960s vintage (Ref. 4).  A second missile design comes from a study of the feasibility of ballistic missile defense, conducted by the American Physical Society in 2004 (Ref. 5).  This missile, called Model L in the study, has a significantly higher thrust and shorter burnout time than the Titan-II, resulting in a higher burnout velocity.  Remarkably, in spite of their different designs, the $\boldsymbol{\varphi _{0}}$-dependent correction to $\boldsymbol{v _{0}}$ is almost exactly the same for both missiles.

Using Eqns. 8, we simulated the boost phase of both missiles for seven values of $\boldsymbol{\varphi _{0}}$  .  In each simulation, the missile left its launch pad and traveled straight upwards for 10 s before being given a small angular “kick” to begin the gravity turn.  The kick angle (< 5°) was selected by trial and error to produce the desired launch angle  $\boldsymbol{\varphi _{0}}$ at burnout ($\boldsymbol{11.25^{\circ}\leq \varphi _{0}\leq 90^{\circ}}$).  For each missile and launch angle, $\boldsymbol{v_{bo}}$ and $\boldsymbol{h_{bo}}$ were recorded, and an equivalent launch speed $\boldsymbol{v_{0}}$ was calculated by combining Eqns. 1 and 2:

$\large \boldsymbol{v_{0}^{2}=v_{bo}^{2}+2Gm_{E}\left ( \frac{1}{r_{E}}-\frac{1}{r_{E}+h_{bo}} \right )}$.

(9)

For example, in Section C we found $\boldsymbol{v_{0}(90^{\circ})=6.17\: \mathrm{km/s}}$ and $\boldsymbol{\varphi _{0}=32^{\circ}}$ in our analysis of of the July 4 missile test.  According to Fig. 4, we should add $\boldsymbol{\Delta v_{0}=0.31\pm 0.01\: \mathrm{km/s}}$ to $\boldsymbol{v_{0}(90^{\circ})}$ to improve our estimate of range.  Using Eqn. 5, the corrected launch speed $\boldsymbol{{v}'_{0}=6.48\: \mathrm{km/s}}$ yields a new value $\boldsymbol{{\theta }'_{max}=30.9^{\circ}}$ and a new range estimate $\boldsymbol{{R}'_{max}=6790\pm 40\: \mathrm{km}}$.  This agrees well with the estimate (≈ 6700 km) adopted by arms control analysts.

The new value $\boldsymbol{{\theta }'_{max}}$ changes the value of $\boldsymbol{\varphi _{0}}$, which in turn influences $\boldsymbol{\Delta v_{0}}$.  This suggests that we iterate our calculations until they converge.  We think this is unwarranted, however, because we have introduced an unavoidable error in $\boldsymbol{\Delta v_{0}}$ by replacing the Hwasong missiles with the Model L or Titan-II.

Table 1 summarizes our results for the North Korean ICBM tests of 2017, and compares them against the range estimates appearing in the news media.  Also included is the May 14 test of a Hwasong-12 intermediate range ballistic missile (IRBM).  Note that the uncertainty increases with launch speed: as $\boldsymbol{v_{0}\rightarrow 7.90\: \mathrm{km/s}}$, which is the speed needed for low Earth orbit (i.e., unlimited range), our range estimate becomes more sensitive to errors in $\boldsymbol{v_{0}}$.  Perhaps, for a similar reason, the maximum range of the Hwasong-15 has only been published as a lower limit.

References

1. Ankit Panda, “The Hwasong-15: Anatomy of North Korea’s New ICBM,” The Diplomat, December 6, 2017, https://thediplomat.com/2017/12/the-hwasong-15-the-anatomy-of-north-koreas-new-icbm/

2. Published estimates of missile range are originally due to D. Wright, All Things Nuclearhttp://allthingsnuclear.org/dwright/ , May 13, July 3, July 28, September 14 and November 28, 2017

3.  Glen J. Culler and Burton D. Fried, “Universal Gravity Turn Trajectories,” J. Appl. Phys. 28, 672-676 (1956)

4.  D. K. Stumpf, “Titan-II: A History of a Cold War Missile Program,” U. Arkansas Press (2000), Chapter 8

5.  D. K. Barton et al, “Report of the APS Study Group on Boost-Phase Intercept Systems for National Missile Defense,” Rev. Mod. Phys76, S1 (2004)