🌱 Tony's Memex Tony's Intertwingled Memex
Digital Garden | OER Curriculum
Home / Digital Garden / Aerospace Radar Tracking Coverage Sector Area (Aerospace Engineering) Solution

Aerospace Radar Tracking Coverage Sector Area (Aerospace Engineering) Solution

🖨️ Printable Reference / PDF

An aerospace radar systems engineer at the UNLV Howard R. Hughes College of Engineering is configuring an automated airspace surveillance phased-array radar unit at Harry Reid International Airport in Las Vegas.

The primary radar is calibrated to filter out near-field ground clutter, establishing an inner blind-zone radius of rinner=10 kmr_{\text{inner}} = 10\text{ km}, while its transmitter power budget limits solid target detection to a maximum instrumental range of router=80 kmr_{\text{outer}} = 80\text{ km}. To maintain high tracking data-rate updates along a critical high-density arrival corridor, the antenna electronically steers its beam across an azimuth swath from θ1=π6 rad\theta_1 = \frac{\pi}{6}\text{ rad} (30∘30^\circ) to θ2=π3 rad\theta_2 = \frac{\pi}{3}\text{ rad} (60∘60^\circ).

Step-by-Step Solution:

  1. Express Geographic Corridor as Polar Bounded Domain:
    State the Polar Rectangle Definition (defined in Polar Rectangles and Annular Sectors):

    R={(r,θ)∈R2∣rinner≤r≤router, θ1≤θ≤θ2}R = \left\{ (r, \theta) \in \mathbb{R}^2 \mid r_{\text{inner}} \le r \le r_{\text{outer}}, \, \theta_1 \le \theta \le \theta_2 \right\}
    Substitute the radar system operating parameters:
    10≤r≤80 km,π6≤θ≤π3 rad10 \le r \le 80\text{ km}, \quad \frac{\pi}{6} \le \theta \le \frac{\pi}{3}\text{ rad}

  2. Formulate the Double Integral for Coverage Area:
    State the Polar Area Double Integral:

    Area=∬R1 dA=∫θ1θ2∫rinnerrouterr dr dθ\text{Area} = \iint_R 1 \, dA = \int_{\theta_1}^{\theta_2} \int_{r_{\text{inner}}}^{r_{\text{outer}}} r \, dr \, d\theta
    Substitute the specific numerical integration boundaries:
    Area=∫π6π3∫1080r dr dθ\text{Area} = \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \int_{10}^{80} r \, dr \, d\theta

  3. Evaluate the Double Integral:
    Apply Fubini's Separability Theorem in Polar Coordinates (defined in Separability in Polar Integrals):

    Area=(∫π6π3dθ)(∫1080r dr)\text{Area} = \left( \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} d\theta \right) \left( \int_{10}^{80} r \, dr \right)

    Evaluate the radial integral:

    ∫1080r dr=[r22]1080=(80)22−(10)22=6400−1002=63002=3150 km2\int_{10}^{80} r \, dr = \left[ \frac{r^2}{2} \right]_{10}^{80} = \frac{(80)^2}{2} - \frac{(10)^2}{2} = \frac{6400 - 100}{2} = \frac{6300}{2} = 3150\text{ km}^2

    Evaluate the angular integral:

    ∫π6π3dθ=[θ]π6π3=π3−π6=2π−π6=π6 rad\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} d\theta = [\theta]_{\frac{\pi}{6}}^{\frac{\pi}{3}} = \frac{\pi}{3} - \frac{\pi}{6} = \frac{2\pi - \pi}{6} = \frac{\pi}{6}\text{ rad}

    Multiply the evaluated results:

    Area=(π6)×3150=525π km2≈1649.336 km2\text{Area} = \left(\frac{\pi}{6}\right) \times 3150 = 525\pi\text{ km}^2 \approx 1649.336\text{ km}^2

Meaning of the Answer & Real-Life Application:

  • Quantitative Interpretation: The active geographic surveillance coverage area swept by the phased-array beam is exactly 525π km2≈1649.34 km2525\pi\text{ km}^2 \approx 1649.34\text{ km}^2.
  • Engineering Application: In air traffic management and radar receiver architecture, computing the exact ground footprint area allows systems engineers to estimate the expected target population based on peak regional aircraft densities (e.g., aircraft per square kilometer). This directly sizes the real-time Kalman filtering track-processor memory, prevents tracker buffer saturation, and guarantees seamless handoff coordination with adjacent Federal Aviation Administration (FAA) en-route radar sectors across Nevada airspace.