2016 D2 | The Moon's size, orbit and the distance to the Sun

Geocentric ephemerides of the Moon for September 2015 are given in the table. Each reading was taken at 00:00 UT. Assume the observer is at the centre of the Earth, and "angular size" means the angular diameter of the object or shadow. A total lunar eclipse occurred this month.

(D2.1) In September 2015, apogee of the lunar orbit is closest to New Moon / First Quarter / Full Moon / Third Quarter. (3)

(D2.2) In September 2015, the ascending node of the lunar orbit on the ecliptic is closest to New Moon / First Quarter / Full Moon / Third Quarter. (4)

(D2.3) Estimate the eccentricity $e$ of the lunar orbit from the data. (4)

(D2.4) In the paper, a composite image of the eclipse shows the Moon at several times. A geometric construction (find the centre of the umbra from chords through the shadow edges) gives the umbra's diameter. Take the result: $\theta_{\rm umbra} \approx 2.76\,\theta_{\rm Moon}$. (8)

(D2.5) The Sun subtends $\theta_{\rm Sun} = 1915.0''$ at Earth on the day of the eclipse. Rays from diametrically opposite edges of the solar disc graze the Earth. Calculate the angular size of the penumbra $\theta_{\rm penumbra}$ in terms of $\theta_{\rm Moon}$. (9)

(D2.6) Let $\theta_{\rm Earth}$ be the angular size of the Earth seen from the centre of the Moon. Express $\theta_{\rm Moon}$ (seen from the Earth's centre on the eclipse day) in terms of $\theta_{\rm Earth}$. (5)

(D2.7) Estimate the radius of the Moon $R_{\rm Moon}$ in km. (3)

(D2.8) Estimate the perigee and apogee distances of the Moon. (4)

(D2.9) Use the data for 10 September to estimate the distance to the Sun. (10)

Use $R_\oplus = 6371$ km.