Chapter 2, Problem 14P

How many precession periods are in one cycle of Earth’s axis inclination variation? In one cycle of Earth’s orbit eccentricity variation? In the time span shown in Figure 2-11b, how many periods or fractions of periods did the Earth’s axis precess, nod, and Earth’s orbit change shape? Of the three periods, which is likely to have the most effect on the changes shown in Figure 2–11?

To determine

The number of precession periods in inclination change of one cycle of Earth’s axis, eccentricity variation of Earth’s orbit, and number of periods or period fractions of precision of Earth’s axis, nod, and change in shape of Earth’s orbit in time span given in figure 2.11b and also identify which one has more influence on changes plotted in figure 2.11.

Answer to Problem 14P

The number of precession periods in inclination change of one cycle of Earth’s axis is $1.6$.

The number of periods of eccentricity variation of Earth’s orbit is $3.8$.

Number of periods for a cycle of precision of Earth’s axis over time span given in 2.11b is $15.4$.

Number of periods for a cycle of nod over time span given in 2.11b is $9.8$.

Number of periods for a cycle of change in shape of Earth’s orbit in time span given in figure 2.11b is $4$ and the Earth’s axis inclination cycle has greater influence on pattern in figure 2.11.

Explanation of Solution

The precession period of Earth is around $26,000\text{years}$. Time taken to complete one complete cycle of inclination variation of axis of Earth is around $41,000\text{years}$. During this cycle, tilt of axis of Earth changes between $22°$ and $25°$ approximately.

Write the equation to find the number of precision period in a complete cycle of tilt change of Earth’s axis.

    $n_1=\frac{T_{\text{axis}}}{T_{\text{precession}}}$                                                                                                                (I)

Here, $n_1$ is the number of precision periods, $T_{\text{axis}}$ is the time taken to complete one complete cycle of inclination variation of axis of Earth, and $T_{\text{precession}}$ is the precession period of Earth.

Time taken to complete one complete cycle of eccentricity variation of Earth’s orbit is almost $100,000\text{years}$. During this period, shape of earth’s orbit changes between $0\%-3\%$.

Write the equation to find the number of precision period in one complete cycle of eccentricity variation of Earth’s orbit.

    $n_2=\frac{T_{\text{ecc}\text{.var}}}{T_{\text{precession}}}$                                                                                                            (I)

Here, $n_2$ is the number of precision period in one complete cycle of eccentricity variation of Earth’s orbit and $T_{\text{ecc}\text{.var}}$ is the time taken to complete one complete cycle of eccentricity variation of Earth’s orbit.

The time span shown in figure 2.11b is $400,000\text{years}$.

Write the equation to find number of precession periods or period fractions during $400,000\text{years}$.

    $n_3=\frac{400,000}{T_{\text{precession}}}$                                                                                                             (III)

Here, $n_3$ is the number of precession periods or period fractions during $400,000\text{years}$.

Write the equation to find number of nod periods or period fractions during $400,000\text{years}$.

    $n_4=\frac{400,000}{T_{\text{axis}}}$                                                                                                       (IV)

Here, $n_4$ is the number of nod periods or period fractions during $400,000\text{years}$.

Write the equation to find number of periods for change in shape of Earth’s orbit or period fractions during $400,000\text{years}$.

    $n_5=\frac{400,000}{T_{\text{axis}}}$                                                                                                        (V)

Here, $n_5$ is the number of periods for change in shape of Earth’s orbit period fractions during $400,000\text{years}$.

Conclusion:

Substitute $41,000\text{years}$ for $T_{\text{axis}}$ and $26,000\text{years}$ for $T_{\text{precession}}$ in equation (I) to find $n_1$.

    $\begin{array}{c}{n}_1=\frac{41,000\text{years}}{26,000\text{years}}\\ \approx 1.6\end{array}$

Substitute $100,000\text{years}$ for $T_{\text{ecc}\text{.var}}$ and $26,000\text{years}$ for $T_{\text{precession}}$ in equation (II) to find $n_2$.

    $\begin{array}{c}{n}_2=\frac{100,000\text{years}}{26,000\text{years}}\\ =3.8\end{array}$

Substitute $26,000\text{years}$ for $T_{\text{precession}}$ in equation (III) to find $n_3$.

    $\begin{array}{c}{n}_3=\frac{400,000\text{years}}{26,000\text{years}}\\ =15.4\end{array}$

Substitute $41,000\text{years}$ for $T_{\text{axis}}$ in equation (IV) to find $n_4$.

    $\begin{array}{c}{n}_4=\frac{400,000\text{years}}{41,000\text{years}}\\ =9.8\end{array}$

Substitute $100,000\text{years}$ for $T_{\text{ecc}\text{.var}}$ in equation (V) to find $n_5$.

    $\begin{array}{c}{n}_5=\frac{400,000\text{years}}{100,000\text{years}}\\ =4\end{array}$

Figure 2.11 plots the temperature variation in Earth over many years. The prime important reason for seasons is the tilting of axis of Earth. So the changes in figure 2.11 is more influenced by $T_{\text{axis}}$.

The number of precession periods in inclination change of one cycle of Earth’s axis is $1.6$,the number of periods of eccentricity variation of Earth’s orbit is $3.8$ ,number of periods for a cycle of precision of Earth’s axis over time span given in 2.11b is $15.4$, number of periods for a cycle of nod over time span given in 2.11b is $9.8$, number of periods for a cycle of change in shape of Earth’s orbit in time span given in figure 2.11b is $4$, and the Earth’s axis inclination cycle has greater influence on pattern in figure 2.11.