## Step 1: Prove Incremental SFD is a Straight Line in Log-Log Space The incremental SFD is given as: \[ \text{Ni}(x) = C_i x^{-q} \] Take the logarithm of both sides: \[ \log_{10}(\text{Ni}(x)) = \log_{10}(C_i) + \log_{10}(x^{-q}) \] \[ \log_{10}(\text{Ni}(x)) = \log_{10}(C_i) - q \log_{10}(x) \] **Explanation:** In log-log space, plotting \(\log_{10}(\text{Ni}(x))\) vs. \(\log_{10}(x)\) gives a straight line with: - **Slope:** \(-q\) - **Y-intercept:** \(\log_{10}(C_i)\) --- ## Step 2: Derive the Cumulative SFD from the Incremental SFD The cumulative SFD considers all craters larger than \(x\): \[ N(>x) = \int_x^\infty \text{Ni}(x') \, dx' \] \[ N(>x) = \int_x^\infty C_i {x'}^{-q} dx' \] Integrating: \[ N(>x) = C_i \int_x^\infty {x'}^{-q} dx' \] \[ N(>x) = C_i \left[ \frac{{x'}^{-q+1}}{-q+1} \right]_x^\infty \] For \(q > 1\), the upper limit (\(x' \rightarrow \infty\)) goes to : \[ N(>x) = -\frac{C_i}{-q+1} x^{-q+1} \] \[ N(>x) = \frac{C_i}{q-1} x^{-(q-1)} \] Let \(C_c = \frac{C_i}{q-1}\) and \(b = q-1\): \[ N(>x) = C_c x^{-b} \] **Explanation:** We derived the cumulative SFD form and expressed \(C_c\) and \(b\) in terms of the incremental constants. --- ## Final Answers 1. **Incremental SFD plotted in log-log space is a straight line** with slope \(-q\) and y-intercept \(\log_{10}(C_i)\). 2. **Cumulative SFD** is \(N(>x) = \frac{C_i}{q-1} x^{-(q-1)}\), or \(N(>x) = C_c x^{-b}\), where \(C_c = \frac{C_i}{q-1}\) and \(b = q-1\). --- **(c) Sketch Guidance:** Draw two straight lines on a log-log plot of cumulative SFDs. The line with a higher vertical position (larger \(C_c\)) represents an older surface (more craters), and the lower one is a younger surface (fewer craters). Label each line accordingly. *(Not shown here as you must sketch by hand.)*

# Step 1: Prove Incremental SFD is a Straight Line in Log-Log Space The incremental SFD is defined as: \[ \text{Ni}(x) = C_i x^{-q} \] Taking the logarithm of both sides: \[ \log_{10}(\text{Ni}(x)) = \log_{10}(C_i) + \log_{10}(x^{-q}) \] This simplifies to: \[ \log_{10}(\text{Ni}(x)) = \log_{10}(C_i) - q \log_{10}(x) \] ## Explanation In log-log space, the equation \(\log_{10}(\text{Ni}(x))\) plotted against \(\log_{10}(x)\) yields a straight line with: - **Slope:** \(-q\) - **Y-intercept:** \(\log_{10}(C_i)\) --- # Step 2: Derive the Cumulative SFD from the Incremental SFD The cumulative SFD, which counts all craters larger than size \(x\), is expressed as: \[ N(>x) = \int_x^\infty \text{Ni}(x') \, dx' \] Substituting for \(\text{Ni}(x')\): \[ N(>x) = \int_x^\infty C_i {x'}^{-q} \, dx' \] ## Integration Steps Perform the integration: \[ N(>x) = C_i \int_x^\infty {x'}^{-q} \, dx' \] \[ N(>x) = C_i \left[ \frac{{x'}^{-q+1}}{-q+1} \right]_x^\infty \] Assuming \(q > 1\), the upper limit approaches zero, leading to: \[ N(>x) = -\frac{C_i}{-q+1} x^{-q+1} \] Thus, we have: \[ N(>x) = \frac{C_i}{q-1} x^{-(q-1)} \] Letting \(C_c = \frac{C_i}{q-1}\) and \(b = q-1\), we rewrite: \[ N(>x) = C_c x^{-b} \] ## Explanation This derivation provides the cumulative SFD in a new form, with \(C_c\) and \(b\) expressed in terms of constants from the incremental SFD. --- # Final Answers 1. **Incremental SFD:** In log-log space, the plot is a straight line with: - **Slope:** \(-q\) - **Y-intercept:** \(\log_{10}(C_i)\) 2. **Cumulative SFD:** \[ N(>x) = \frac{C_i}{q-1} x^{-(q-1)} \quad \text{or} \quad N(>x) = C_c x^{-b} \] Where: - \(C_c = \frac{C_i}{q-1}\) - \(b = q-1\) --- # Sketch Guidance for Part (c) - Create a log-log plot of cumulative SFDs. - Draw two lines: - **Older Surface:** Higher \(C_c\) (more craters), labeled accordingly. - **Younger Surface:** Lower \(C_c\) (fewer craters), labeled accordingly. *(Sketching should be done by hand as instructed.)*

# Step 1: Prove Incremental SFD is a Straight Line in Log-Log Space The incremental SFD is defined as: \[ \text{Ni}(x) = C_i x^{-q} \] Taking the logarithm of both sides: \[ \log_{10}(\text{Ni}(x)) = \log_{10}(C_i) + \log_{10}(x^{-q}) \] This simplifies to: \[ \log_{10}(\text{Ni}(x)) = \log_{10}(C_i) - q \log_{10}(x) \] ## Explanation In log-log space, the equation \(\log_{10}(\text{Ni}(x))\) plotted against \(\log_{10}(x)\) yields a straight line with: - **Slope:** \(-q\) - **Y-intercept:** \(\log_{10}(C_i)\) --- # Step 2: Derive the Cumulative SFD from the Incremental SFD The cumulative SFD, which counts all craters larger than size \(x\), is expressed as: \[ N(>x) = \int_x^\infty \text{Ni}(x') \, dx' \] Substituting for \(\text{Ni}(x')\): \[ N(>x) = \int_x^\infty C_i {x'}^{-q} \, dx' \] ## Integration Steps Perform the integration: \[ N(>x) = C_i \int_x^\infty {x'}^{-q} \, dx' \] \[ N(>x) = C_i \left[ \frac{{x'}^{-q+1}}{-q+1} \right]_x^\infty \] Assuming \(q > 1\), the upper limit approaches zero, leading to: \[ N(>x) = -\frac{C_i}{-q+1} x^{-q+1} \] Thus, we have: \[ N(>x) = \frac{C_i}{q-1} x^{-(q-1)} \] Letting \(C_c = \frac{C_i}{q-1}\) and \(b = q-1\), we rewrite: \[ N(>x) = C_c x^{-b} \] ## Explanation This derivation provides the cumulative SFD in a new form, with \(C_c\) and \(b\) expressed in terms of constants from the incremental SFD. --- # Final Answers 1. **Incremental SFD:** In log-log space, the plot is a straight line with: - **Slope:** \(-q\) - **Y-intercept:** \(\log_{10}(C_i)\) 2. **Cumulative SFD:** \[ N(>x) = \frac{C_i}{q-1} x^{-(q-1)} \quad \text{or} \quad N(>x) = C_c x^{-b} \] Where: - \(C_c = \frac{C_i}{q-1}\) - \(b = q-1\) --- # Sketch Guidance for Part (c) - Create a log-log plot of cumulative SFDs. - Draw two lines: - **Older Surface:** Higher \(C_c\) (more craters), labeled accordingly. - **Younger Surface:** Lower \(C_c\) (fewer craters), labeled accordingly. *(Sketching should be done by hand as instructed.)*