NASA reports this photo was taken 19 Sep 1977 by Voyager 1:
It doesn't strike me as obvious how to determine which celestial body is closer to Voyager 1 as the photo was taken. My (not very scientific) guess is that the Earth is closer, since the moon seems more out of focus. It also seems more probable, since it was launched from earth.
Question: In this image taken by Voyager 1, which is closer: the earth or the moon?
I used JPL Horizons to get the position vectors of each relative to the SSB on Sep 19, 1977.
Voyager X = 1.547492527774134E+08 Y = 2.045859856469853E+06 Z = 8.442122223290936E+05
Earth X = 1.503771470116906E+08 Y =-9.323322057091754E+06 Z =-1.007092461168021E+04
Moon X = 1.502869015003825E+08 Y =-9.680376320152178E+06 Z = 1.932598843803350E+04
Subtracting Voyager's position from each, and computing $ \sqrt{x^2+y^2+z^2} $ yields:
Earth 1.22E+07km
Moon 1.26E+07km
So, the Earth is closer.
Another way to look at it is by linear measurements. Using Photoshop, and averaging three measures each of the Moon and the Earth (with the “Ruler” tool), I get 12.56 pixels for the Moon and 54.67 pixels from the Earth (both measured from “horn” to “horn”). This means the Moon’s angular size is 22.9% of the Earth’s angular size on that image.
Now, the Moon has a radius of 1,737.4 km, and the Earth’s radius is 6,371 km. This means that in actual sizes, the Moon is 27.3% the radius of the Earth.
Since on the image, the Moon’s size is less that 27.3% that of the Earth (as 22.9 < 27.3), I conclude that the Moon is further than the Earth on the image. This is in agreement with Eric Jensen’s answer above. Q.E.D.
note: this confirms @GregMiller's answer I started typing it last night and fell asleep; woke up and finished it and clicked "post" then to discovered a bunch of new answers.
For those who haven't used Horizons before perhaps this will help get you started; it's fun!
In the photo the Moon's apparent distance from the center of the Earth, projected into the plane of the photo is only about 4 Earth diameters or about 51,000 km. This is only a small fraction of their actual distance of about 360,000 to 390,000 km, so one of them is a lot closer to Voyager than the other to be seen so closely together.
Checking Horizons for 19 and 20 September 1977 it's clear that the Moon is between 365,450 and 353,603 km further from Voyager 1 than the Earth.
In this image taken by Voyager 1, which is closer: the earth or the moon?
A: Earth is closer.
Here is how I set up Horizons to calculate:
and here part of the output from Horizons for each body:
*******************************************************************************
Revised: Jan 21, 2015 Voyager 1 Spacecraft (interplanetary) / (Sun) -31
http://www.jpl.nasa.gov/missions/voyager-1/
BACKGROUND
See the web-page above for extensive information on the Voyager Mission.
SPACECRAFT TRAJECTORY:
This trajectory is composed of two merged sections:
#1) 1977-Sep-5 to 1986-Jan-1:
A patched conic mission-design type trajectory in which the conics
were constructed to approximately match specific events (such
satellite encounters), providing a rough accuracy.
#2) 1986-Jan-1 to 2031-Jan-1 (pfile_a54206u)
Time-extended trajectory provided by Voyager Navigation
Ephemeris / WWW_USER Sat Jul 9 13:34:31 2022 Pasadena, USA / Horizons
Target body name: Voyager 1 (spacecraft) (-31) {source: Voyager_1}
Center body name: Earth (399) {source: Voyager_1}
Center-site name: GEOCENTRIC
Start time : A.D. 1977-Sep-19 00:00:00.0000 UT
Stop time : A.D. 1977-Sep-20 00:00:00.0000 UT
Step-size : 1440 minutes
Target pole/equ : No model available
Target radii : (unavailable)
Center geodetic : 0.00000000,0.00000000,0.0000000 {E-lon(deg),Lat(deg),Alt(km)}
Center cylindric: 0.00000000,0.00000000,0.0000000 {E-lon(deg),Dxy(km),Dz(km)}
Center pole/equ : ITRF93 {East-longitude positive}
Center radii : 6378.1 x 6378.1 x 6356.8 km {Equator, meridian, pole}
Target primary : Earth
Vis. interferer : MOON (R_eq= 1737.400) km {source: Voyager_1}
Rel. light bend : Sun, EARTH {source: Voyager_1}
Rel. lght bnd GM: 1.3271E+11, 3.9860E+05 km^3/s^2
Atmos refraction: NO (AIRLESS)
RA format : HMS
Time format : CAL
EOP file : eop.220707.p220930
EOP coverage : DATA-BASED 1962-JAN-20 TO 2022-JUL-07. PREDICTS-> 2022-SEP-29
Units conversion: 1 au= 149597870.700 km, c= 299792.458 km/s, 1 day= 86400.0 s
Table cut-offs 1: Elevation (-90.0deg=NO ),Airmass (>38.000=NO), Daylight (NO )
Table cut-offs 2: Solar elongation ( 0.0,180.0=NO ),Local Hour Angle( 0.0=NO )
Table cut-offs 3: RA/DEC angular rate ( 0.0=NO )
Date__(UT)__HR:MN R.A._____(ICRF)_____DEC APmag S-brt delta deldot S-O-T /r S-T-O Sky_motion Sky_mot_PA RelVel-ANG Lun_Sky_Brt sky_SNR
$$SOE
1977-Sep-19 00:00 04 26 24.32 +25 45 43.6 n.a. n.a. 1.1824945971E+07 10.0591463 107.2217 /L 68.5794 0.1500504 259.33422 89.183432 n.a. n.a.
1977-Sep-20 00:00 04 26 07.77 +25 45 01.6 n.a. n.a. 1.2692956351E+07 10.0362045 108.2591 /L 67.2734 0.1658961 259.18231 89.028742 n.a. n.a.
$$EOE
*******************************************************************************
Revised: Jan 21, 2015 Voyager 1 Spacecraft (interplanetary) / (Sun) -31
http://www.jpl.nasa.gov/missions/voyager-1/
BACKGROUND
See the web-page above for extensive information on the Voyager Mission.
SPACECRAFT TRAJECTORY:
This trajectory is composed of two merged sections:
#1) 1977-Sep-5 to 1986-Jan-1:
A patched conic mission-design type trajectory in which the conics
were constructed to approximately match specific events (such
satellite encounters), providing a rough accuracy.
#2) 1986-Jan-1 to 2031-Jan-1 (pfile_a54206u)
Time-extended trajectory provided by Voyager Navigation
Ephemeris / WWW_USER Sat Jul 9 09:07:30 2022 Pasadena, USA / Horizons
Target body name: Voyager 1 (spacecraft) (-31) {source: Voyager_1}
Center body name: Moon (301) {source: Voyager_1}
Center-site name: BODYCENTRIC
Start time : A.D. 1977-Sep-19 00:00:00.0000 UT
Stop time : A.D. 1977-Sep-20 00:00:00.0000 UT
Step-size : 1440 minutes
Target pole/equ : No model available
Target radii : (unavailable)
Center geodetic : 0.00000000,0.00000000,0.0000000 {E-lon(deg),Lat(deg),Alt(km)}
Center cylindric: 0.00000000,0.00000000,0.0000000 {E-lon(deg),Dxy(km),Dz(km)}
Center pole/equ : MEAN_ME (high precision) {East-longitude positive}
Center radii : 1737.4 x 1737.4 x 1737.4 km {Equator, meridian, pole}
Target primary : Earth
Vis. interferer : EARTH (R_eq= 6378.137) km {source: Voyager_1}
Rel. light bend : Sun, EARTH {source: Voyager_1}
Rel. lght bnd GM: 1.3271E+11, 3.9860E+05 km^3/s^2
Atmos refraction: NO (AIRLESS)
RA format : HMS
Time format : CAL
EOP file : eop.220707.p220930
EOP coverage : DATA-BASED 1962-JAN-20 TO 2022-JUL-07. PREDICTS-> 2022-SEP-29
Units conversion: 1 au= 149597870.700 km, c= 299792.458 km/s, 1 day= 86400.0 s
Table cut-offs 1: Elevation (-90.0deg=NO ),Airmass n.a. , Daylight (NO )
Table cut-offs 2: Solar elongation ( 0.0,180.0=NO ),Local Hour Angle( 0.0=NO )
Table cut-offs 3: RA/DEC angular rate ( 0.0=NO )
Date__(UT)__HR:MN R.A._____(ICRF)_____DEC APmag S-brt delta deldot S-O-T /r S-T-O Sky_motion Sky_mot_PA RelVel-ANG Lun_Sky_Brt sky_SNR
$$SOE
1977-Sep-19 00:00 04 26 39.10 +25 31 06.2 n.a. n.a. 1.2190396108E+07 10.0456822 107.0775 /L 68.5903 0.9274265 91.964274 84.803988 n.a. n.a.
1977-Sep-20 00:00 04 28 07.07 +25 34 16.4 n.a. n.a. 1.3046559309E+07 9.7765479 107.7115 /L 67.6819 0.7438366 89.544459 85.414311 n.a. n.a.
$$EOE
The Earth was closer.
I can't find the time when that picture was taken. That NASA press release is dated Sep 19, 1977, but the consensus seems to be that the photo was taken around the 18th of September.
However, they do say that Voyager 1 was around 11.66 million km from the Earth at the time of the photo, which puts it around 1977-Sep-18 19:30 UTC, according to Horizons.
At that time, the distance from Voyager 1 to the Moon was ~12.03 million km.
Here's my plotting script.
Here are direct links for the raw Horizons data:
Earth
Moon
FWIW, here's a plot for September, starting about 10 hours after launch.
So Voyager 1 was equidistant from the Earth & Moon some time around the 12th and the 26th.
Here's a plot of the trajectories of the Moon (orange) and Voyager 1 (green) relative to the Earth (blue) over that time interval, with a time step of 1 day. Hopefully, this plot makes it clear that Voyager was closer to the Earth than to the Moon during that interval.
The frame is the ICRF, essentially the J2000.0 frame. The XY plane is (parallel to) the J2000 ecliptic, and the grey line emanating from the Earth is parallel to the ICRF X axis, and points towards the J2000 First Point of Aries (the March equinox point). We're looking "down" from north of the ecliptic. The Moon is moving anticlockwise. Voyager is moving at quite a speed, so in one day it traverses a distance equal to the diameter of the Moon's orbit. It's above the ecliptic, moving north.
You can see an interactive 3D view in this orbit plotting script.
That script can be used to create trajectory plots of any number of bodies that Horizons knows about. Any major body can be the center, and some minor bodies can be too. Use the SSD Small-Body Database Lookup tool to get the official name or SPKID for small bodies. Major body SPKIDs can be found here. Please see the Horizons manual for further info, including info on specifying dates and time steps.
palette colours may be specified using standard X11 / Web colour names, or 3 or 6 digit hex codes, eg #F00 or #FF0000 is bright red.
Separate body names / ids and palette colours with commas. You can give more colours than bodies, if you don't supply a colour for a body it will not be plotted.
To disable the plotting of labels, set label_step to 0.
If you select the perspective plotting mode you may need to increase the dot size value considerably.
The program maintains a cache of its last 20 queries of Horizons data, so if you only change "cosmetic" options the plot can be generated without re-fetching the data from Horizons.
Here are the 3D interface controls.
Here's an enhanced version of that script, which allows you to select the orientation of the XY plane. The default is Ecliptic, the other options are Frame for the J2000 Earth equatorial plane, or Body Equator, for the equatorial plane of the central body. In all cases, the actual plane plotted is the plane parallel to the selected plane which passes through the center body.
I set the defaults to show Uranus and a few of its moons, since the Uranus equatorial plane is quite tilted relative to the ecliptic and to Earth's equator.
Here is my answer, which I think involves the easiest possible method of getting a rough answer.
I measured the distances between the tips of the crescents of the Moon and the Earth on the Screen of my computer. The tips of the Moon's crescent were about 3 millimeters apart, and the tips of the Earth's crescent were about 15 millimeters apart.
So the Earth appears to have about 5 times the apparent diameter as the Moon.
The average or mean radius of the Moon is given as 1,737.4 kilometers, or 0.2727 that of the Earth. So the real diameter of the Earth is about 3.667 times the real diameter of the moon.
https://en.wikipedia.org/wiki/Moon
So if the apparent diameter of the Earth appears to exceed the apparent diameter of the moon by significantly more than 3.667 times, the Earth must be significantly closer to the camera than the Moon is, and the moon must be significantly farther from the camera than the Earth.
So the Earth is closer than the Moon in the photo.
The question doesn't ask for how much closer, and I won't bother calculating. No doubt a better image of the photo would enable a more accurate ratio between the apparent diameters of the Earth and the Moon to be measured, thus making a more accurate calculation possible.
The Earth is a bit closer than the Moon in this picture. From the illumination pattern on the Earth and Moon, we can see that the Sun is off to the right. And the Moon’s orbit around the Earth is tilted only slightly (about 5 degrees) compared to the Earth’s orbit around the Sun. Thus, the Moon must be on the farther side of its orbit (farther from Voyager 1 than Earth is) to give the observed perspective.
It’s not a big difference, though. The caption says that the photo was taken at 7.25 million miles from Earth, and the Moon is on average only 239,000 miles from Earth, so the difference is only about 3% (i.e. 0.239/7.25).
Either way, must an Answer rely on the information contained in the picture, or is external data such as the known sizes or positions of Earth and Moon to be considered?
– Robbie Goodwin Jul 09 '22 at 22:57