# Estimating Flight Characteristics of Anomalous Unidentified Aerial Vehicles

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## Abstract

**:**

## 1. Introduction

## 2. Case Studies

#### 2.1. Bethune Encounter (1951)

#### 2.2. Probability Densities

#### 2.3. Japan Air Lines Flight 1628 (1986)

#### 2.4. Nimitz Encounters (2004)

#### 2.4.1. Senior Chief Operations Specialist Kevin Day (RADAR)

#### 2.4.2. Commander David Fravor (PILOT)

#### 2.4.3. ATFLIR Video

## 3. Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Histograms of the samples used to estimate the minimum acceleration of the UAP in the Bethune encounter. In these and subsequent plots, the y-axis illustrates the number of samples, which is proportional to the probability. (

**A**). The duration of the maneuver is a truncated Gaussian distribution for $t=1\mathrm{s}\pm 1\mathrm{s}$. (

**B**). The altitude of the UAV is a truncated Gaussian with $h=9800\phantom{\rule{0.166667em}{0ex}}\mathrm{ft}\pm 200\phantom{\rule{0.166667em}{0ex}}\mathrm{ft}$. (

**C**). The horizontal distance traveled was modeled using a Gaussian distribution of angles as described in the text. (

**D**). The extreme acceleration calls for a logarithmic scale in the histogram above. The most probable acceleration is approximately ${10}^{3.23}\approx 1700\phantom{\rule{0.166667em}{0ex}}\mathrm{g}$.

**Figure 2.**(

**A**). An illustration of the behavior of the UAV in the vicinity of JAL 1628. The UAV and the airplane are approximately to scale, while the distance between them is not. (

**B**). Modeling the UAV as traveling across the diameter of the circle, the acceleration was estimated to be $68\pm 7\phantom{\rule{0.166667em}{0ex}}\mathrm{g}$. (

**C**). Modeling the UAV as moving in a circular motion and focusing only on the centripetal acceleration, resulted in $84\pm 8\phantom{\rule{0.166667em}{0ex}}\mathrm{g}$.

**Figure 3.**An analysis of Senior Chief Day’s radar observations. (

**A**). The posterior probability indicates the maximum likelihood estimate of the acceleration to be ${5600}_{-1190}^{+2270}$ $\mathrm{g}$. (

**B**). The accelerations obtained by sampling resulted in the most probable acceleration of ${5370}_{-820}^{+1430}$ $\mathrm{g}$ while the mean acceleration is $5950\phantom{\rule{0.166667em}{0ex}}\mathrm{g}$ (black dotted line). (

**C**). The power output of the UAP, assumed to have a mass of $1000\phantom{\rule{0.166667em}{0ex}}\mathrm{kg}$, as a function of time indicates a peak power of about $1100\phantom{\rule{0.166667em}{0ex}}\mathrm{GW}$.

**Figure 4.**An analysis of CDR Fravor’s encounter. (

**A**). Truncated Gaussian distribution of Fravor’s visual acuity based on ${1/30}^{\circ}\pm {1/60}^{\circ}$. (

**B**). Gaussian distribution of distances based on the visual acuity distribution in A. (

**C**). The distribution of times based on $1\pm 1\phantom{\rule{0.166667em}{0ex}}\mathrm{s}$. (

**D**). The distribution of accelerations has a maximum at ${150}_{-80}^{+140}$ $\mathrm{g}$ (red lines) and a mean of $550\phantom{\rule{0.166667em}{0ex}}\mathrm{g}$ (black dotted line).

**Figure 5.**(

**A**). Frame 19 of the last 32 frames of the Nimitz ATFLIR video. The narrow horizontal and vertical lines intersecting at the right edge of the UAP image indicate the position of the UAP. (

**B**). The pixel intensities along a row of the frame are plotted along with the best Gaussian curve fit. The rightmost edge of the craft is defined as the center position of the Gaussian plus one standard deviation (indicated by the vertical red line).

**Figure 6.**The figures (

**A**–

**D**) illustrate the position of the right edge of the UAV (+) in pixels, the model fits (solid curves) to the UAV positions in the Nimitz ATFLIR video, and the residuals (model minus data) for each of the four models described in (20), (21), (22), and (23), respectively. The model parameter values for each of the models are listed in Table 1 along with the log evidence, logZ, and log likelihood, logL. The log evidence, logZ (Table 1), strongly favors Model 4 (D), which describes the UAV as accelerating at a magnitude of $75.9\pm 0.2\phantom{\rule{0.166667em}{0ex}}\mathrm{g}$ for about $0.53\phantom{\rule{0.166667em}{0ex}}\mathrm{s}$ to the left and away from the observer. Even though the data are well described by Model 4, it appears from the residuals that the UAV may have accelerated and decelerated erratically multiple times.

**Figure 7.**(

**A**). This figure shows the time required to reach relativistic speeds for a craft undergoing constant acceleration at $1000\phantom{\rule{0.166667em}{0ex}}\mathrm{g}$. In less than $24\phantom{\rule{0.166667em}{0ex}}\mathrm{hrs}$, such a craft would exceed $90\%$ the speed of light. (

**B**). This figure shows the travel time to various distances assuming that the craft accelerates at a constant rate for half of the trip and decelerates at the same rate for the second half. The four star systems indicated are each believed to host one or more planets within the habitable zone. At an acceleration of $100\phantom{\rule{0.166667em}{0ex}}\mathrm{g}$ a craft could travel to Proxima Centuri, $4.37\phantom{\rule{0.166667em}{0ex}}\mathrm{LY}$ distant, in about one and a half months for the travelers. For those of us on Earth, or anywhere else in the galactic frame, the trip would take over four years.

**Table 1.**Kinematic Models for Nimitz Video (Model 4 (

**bold**) was found to be most probable by a factor of $exp\left(1200\right)$ based on the log evidence (logZ) with an overall acceleration of $75.9\pm 0.2\phantom{\rule{0.166667em}{0ex}}\mathrm{g}$).

Model | logZ | LogL | ${\mathit{a}}_{\mathit{x}}$ (g) | ${\mathit{a}}_{\mathit{z}}$ (g) | ${\mathit{x}}_{\mathit{o}}$ (m) | ${\mathit{z}}_{\mathit{o}}$ (m) |
---|---|---|---|---|---|---|

Model 1 | −253,640 | −253,614 | $-71.1\pm 0.7$ | – | $-15.40\pm 0.04$ | 119,700 ± 1200 |

Model 2 | −236,950 | −236,287 | $7.564\pm 0.002$ | $99.994\pm 0.005$ | $-13.36\pm 0.04$ | 12,193 ± 1 |

Model 3 | −53,282 | −53,261 | $-40.2\pm 3.8$ | – | $-4.02\pm 0.05$ | 49,700 ± 4800 |

Model 4 | −52,084 | −52,031 | −35.64 ± 0.08 | 67.04 ± 0.18 | −3.89 ± 0.05 | 43,870 ± 110 |

**Table 2.**Summary of Considered Cases (Detection Modalities include: Visual Contact from Multiple Pilots (Vps), Passenger/s Visual Contact (Vpa/s), Radar (R), Infrared Video (IR). Estimated accelerations range from about $68\phantom{\rule{0.166667em}{0ex}}\mathrm{g}$ to well over $5000\phantom{\rule{0.166667em}{0ex}}\mathrm{g}$).

Case | Year | Detection Modalities | Refs. | Kinematic Model | Figure | Min. Acceleration |
---|---|---|---|---|---|---|

Bethune | 1951 | Vps,Vpas,R | [14,15] | (3) | Figure 1D | $1700\phantom{\rule{0.166667em}{0ex}}\mathrm{g}$ |

JAL1628 | 1986 | Vps,R | [21] | (3) | Figure 2 | $68\pm 7\phantom{\rule{0.166667em}{0ex}}\mathrm{g}$ |

(9) | Figure 2 | $84\pm 8\phantom{\rule{0.166667em}{0ex}}\mathrm{g}$ | ||||

Nimitz | 2004 | |||||

Day | Vps,R | [22] | (3) | Figure 3B | ${5370}_{-820}^{+1430}$$\phantom{\rule{0.166667em}{0ex}}\mathrm{g}$ | |

Fravor | Vps,R | [22] | (17) | Figure 4C | ${150}_{-80}^{+140}$$\phantom{\rule{0.166667em}{0ex}}\mathrm{g}$ | |

ATFLIR | Vps,R,IR | [22] | (23) | Figure 6D | $75.9\pm 0.2\phantom{\rule{0.166667em}{0ex}}\mathrm{g}$ |

**Table 3.**Distances and Travel Times to Various Star Systems. (For each system, the left column lists the travel time $\tau $ (24) experienced by the travelers in units of days (d) and the right column lists the travel time t (25) experienced by those in the galactic (rest) frame in units of years (y).)

Acceleration | Proxima Centauri | Tau Ceti | Gliese 667C | TRAPPIST-1 | ||||
---|---|---|---|---|---|---|---|---|

4.37 LY | 11.9 LY | 25.05 LY | 39.17 LY | |||||

$\tau $ | t | $\tau $ | t | $\tau $ | t | $\tau $ | t | |

$100\phantom{\rule{0.166667em}{0ex}}\mathrm{g}$ | $43.3\phantom{\rule{0.166667em}{0ex}}\mathrm{d}$ | $4.389\phantom{\rule{0.166667em}{0ex}}\mathrm{y}$ | $50.4\phantom{\rule{0.166667em}{0ex}}\mathrm{d}$ | $11.919\phantom{\rule{0.166667em}{0ex}}\mathrm{y}$ | $55.3\phantom{\rule{0.166667em}{0ex}}\mathrm{d}$ | $23.619\phantom{\rule{0.166667em}{0ex}}\mathrm{y}$ | $58.8\phantom{\rule{0.166667em}{0ex}}\mathrm{d}$ | $39.019\phantom{\rule{0.166667em}{0ex}}\mathrm{y}$ |

$300\phantom{\rule{0.166667em}{0ex}}\mathrm{g}$ | $17.0\phantom{\rule{0.166667em}{0ex}}\mathrm{d}$ | $4.377\phantom{\rule{0.166667em}{0ex}}\mathrm{y}$ | $19.4\phantom{\rule{0.166667em}{0ex}}\mathrm{d}$ | $11.907\phantom{\rule{0.166667em}{0ex}}\mathrm{y}$ | $21.0\phantom{\rule{0.166667em}{0ex}}\mathrm{d}$ | $23.607\phantom{\rule{0.166667em}{0ex}}\mathrm{y}$ | $22.2\phantom{\rule{0.166667em}{0ex}}\mathrm{d}$ | $39.007\phantom{\rule{0.166667em}{0ex}}\mathrm{y}$ |

$500\phantom{\rule{0.166667em}{0ex}}\mathrm{g}$ | $10.9\phantom{\rule{0.166667em}{0ex}}\mathrm{d}$ | $4.374\phantom{\rule{0.166667em}{0ex}}\mathrm{y}$ | $12.4\phantom{\rule{0.166667em}{0ex}}\mathrm{d}$ | $11.904\phantom{\rule{0.166667em}{0ex}}\mathrm{y}$ | $13.3\phantom{\rule{0.166667em}{0ex}}\mathrm{d}$ | $23.604\phantom{\rule{0.166667em}{0ex}}\mathrm{y}$ | $14.0\phantom{\rule{0.166667em}{0ex}}\mathrm{d}$ | $39.004\phantom{\rule{0.166667em}{0ex}}\mathrm{y}$ |

$1000\phantom{\rule{0.166667em}{0ex}}\mathrm{g}$ | $6.0\phantom{\rule{0.166667em}{0ex}}\mathrm{d}$ | $4.372\phantom{\rule{0.166667em}{0ex}}\mathrm{y}$ | $6.7\phantom{\rule{0.166667em}{0ex}}\mathrm{d}$ | $11.902\phantom{\rule{0.166667em}{0ex}}\mathrm{y}$ | $7.2\phantom{\rule{0.166667em}{0ex}}\mathrm{d}$ | $23.602\phantom{\rule{0.166667em}{0ex}}\mathrm{y}$ | $7.5\phantom{\rule{0.166667em}{0ex}}\mathrm{d}$ | $39.002\phantom{\rule{0.166667em}{0ex}}\mathrm{y}$ |

$5000\phantom{\rule{0.166667em}{0ex}}\mathrm{g}$ | $1.4\phantom{\rule{0.166667em}{0ex}}\mathrm{d}$ | $4.370\phantom{\rule{0.166667em}{0ex}}\mathrm{y}$ | $1.56\phantom{\rule{0.166667em}{0ex}}\mathrm{d}$ | $11.900\phantom{\rule{0.166667em}{0ex}}\mathrm{y}$ | $1.66\phantom{\rule{0.166667em}{0ex}}\mathrm{d}$ | $23.600\phantom{\rule{0.166667em}{0ex}}\mathrm{y}$ | $1.73\phantom{\rule{0.166667em}{0ex}}\mathrm{d}$ | $39.000\phantom{\rule{0.166667em}{0ex}}\mathrm{y}$ |

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Knuth, K.H.; Powell, R.M.; Reali, P.A.
Estimating Flight Characteristics of Anomalous Unidentified Aerial Vehicles. *Entropy* **2019**, *21*, 939.
https://doi.org/10.3390/e21100939

**AMA Style**

Knuth KH, Powell RM, Reali PA.
Estimating Flight Characteristics of Anomalous Unidentified Aerial Vehicles. *Entropy*. 2019; 21(10):939.
https://doi.org/10.3390/e21100939

**Chicago/Turabian Style**

Knuth, Kevin H., Robert M. Powell, and Peter A. Reali.
2019. "Estimating Flight Characteristics of Anomalous Unidentified Aerial Vehicles" *Entropy* 21, no. 10: 939.
https://doi.org/10.3390/e21100939