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NASA reports this photo was taken 19 Sep 1977 by Voyager 1:

Voyager 1 Takes the First Image of the Earth-Moon System in a Single Frame

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?

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    $\begingroup$ Focus is irrelevant at astronomical distances. You focus your camera at infinity. $\endgroup$
    – TonyK
    Jul 9 at 19:48
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    $\begingroup$ I'm asking out of curiosity about this particular photo. From my perspective, answers can use whatever method they like, whether it be a once-off argument for this photo, or a general solution to "which is closer: the earth or the moon?". (The reality is, I didn't know what kind of answers to expect when asking.) $\endgroup$ Jul 9 at 23:12
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    $\begingroup$ Why would Voyager being launched from Earth make it more probable that Earth is closer in this photo? $\endgroup$ Jul 10 at 0:32
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    $\begingroup$ When someone asks a question at Stack Exchange, they're often requested to "show effort" or "explain their thoughts". Thus I did so. I'm aware my statements are vague and unscientific (as I point out). My intuition tells me: if we take a sphere centered at the origin [earth] and a non-origin point x [moon], then inside the sphere, there is a greater volume of points closer to the origin than x (or a similar claim holds, if we just consider the surface of the sphere). $\endgroup$ Jul 10 at 1:12
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    $\begingroup$ I borrowed your image and used it here astronomy.stackexchange.com/a/49826/7982 thanks! $\endgroup$
    – uhoh
    Jul 10 at 7:32

6 Answers 6

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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.

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    $\begingroup$ Marginally closer, by about 3.3%. That wouldn't be apparent to the unaided eye. $\endgroup$
    – CJ Dennis
    Jul 10 at 3:23
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    $\begingroup$ Are the position vectors relative to the bodies' surfaces or center of mass? $\endgroup$
    – BMF
    Jul 10 at 4:33
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    $\begingroup$ Center of mass. $\endgroup$ Jul 10 at 16:36
  • $\begingroup$ @GregMiller What units, if discernable, applies to 1.22E+07? $\endgroup$ Jul 12 at 13:38
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    $\begingroup$ @ThorbjørnRavnAndersen Solar System Barycenter. $\endgroup$ Jul 12 at 15:41
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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.

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    $\begingroup$ Not an expert, but won't you need to take geometric distortion into account for this calculation to work? Voyager had a wide angle camera. $\endgroup$ Jul 9 at 19:04
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    $\begingroup$ Given the great distance from Earth/Moon, this picture was taken, I find the 27.3/22.9 gap to big to be explained only on account of the relatively slightly closer Earth. $\endgroup$
    – d_e
    Jul 9 at 19:06
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    $\begingroup$ @Jean-MariePrival: You are most likely right; I should have specified that this whole method is only approximate. Then again, we’re not trying to determine the exact distance of each body… $\endgroup$ Jul 9 at 19:12
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    $\begingroup$ @d_e: I haven’t done 3D calculations. Other effects (like the camera lens, as pointed out by Jean-Marie) can be at play indeed. $\endgroup$ Jul 9 at 19:13
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    $\begingroup$ +1 for centering on using the picture and not outside tools $\endgroup$
    – Mike M
    Jul 10 at 16:26
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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!


Earth is ~360,000 km closer

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:

Horizons setup for this data

and here part of the output from Horizons for each body:

Voyager 1 from the geocenter (Earth):

*******************************************************************************
 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
******************************************************************************************************************************************************************************

and Voyager 1 from the lunacenter (the Moon):

*******************************************************************************
 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
******************************************************************************************************************************************************************************
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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.

Earth & Moon distance to Voyager 1, 1977-18

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.

Earth, Moon to Voyager 1, Sep 1977

So Voyager 1 was equidistant from the Earth & Moon some time around the 12th and the 26th.

Update

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.

Moon & Voyager 1 trajectories

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.

  • Orbit - right mouse, or left mouse + ctrl/meta/shiftKey
    • touch: two-finger rotate
  • Zoom - middle mouse, or mousewheel
    • touch: two-finger spread or squish
  • Pan - left mouse, or arrow keys
    • touch: one-finger move

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.

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  • $\begingroup$ Hmm, so 10 hours after launch, probe was closer to the moon than earth? Interesting. $\endgroup$ Jul 12 at 13:36
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    $\begingroup$ @chux Indeed! It takes less than half a day to get past the lunar orbit when you're going faster than 10 km/s (relative to the Earth). Here's the Earth distance data for the 1st 2 days. ssd.jpl.nasa.gov/api/… $\endgroup$
    – PM 2Ring
    Jul 12 at 21:48
  • $\begingroup$ And here's the corresponding Moon data: ssd.jpl.nasa.gov/api/… Both queries use a Step Size of 1 hour, but you can easily edit that. $\endgroup$
    – PM 2Ring
    Jul 12 at 21:50
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Here is my answer, which I think involes the easiest possible method of getting a rough answer.

I measured the distants between the tips of the crescents of he Moon and the Earth on the Screen of my computer. The tips of the Moon's cresent were about 3 milimeters apart, and the tips of the Earth's cresent were about 15 milimeters 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 camers than the Earth.

So the Earth is closer than the Moon in the photo.

The quesiton doesn't ask for how much closer, and I won't botther 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 makng a more accurate calculation possible.

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  • $\begingroup$ +1 making this estimation so simple one could memorize a rule of thumb (by knowing both radii approximately) $\endgroup$
    – ojdo
    Jul 12 at 9:25
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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).

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    $\begingroup$ I might had better days, but I don't quite follow the logic that leads to "Thus, the Moon must be on the farther side of its orbit"; moreover, taken at the face value, the 5 deg inclination is pretty obscure reference here: for it is about total plane of the orbit, and here we are looking at a point a orbit, so unless 19 Sep 1977 was probed we cannot really say what is the ecliptic latitude of the Moon. In theory it might be that all three on the same ecliptic plane. $\endgroup$
    – d_e
    Jul 9 at 12:54
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    $\begingroup$ Yes, 3d visualization is always tough, and explaining it in words is even harder! The key with the 5 degrees is that it’s small, i.e. not 90 degrees. It doesn’t matter if it’s 0, +5, or -5 at this moment; either way, it’s small. From the illumination, we can see that Voyager is closer to being in the Earth’s orbital plane than above it. (From above, you always get half lit, half dark.). And we know that the Moon is close to that plane, too - that’s all I was going for. I welcome clearer / more intuitive answers! $\endgroup$ Jul 9 at 13:14
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    $\begingroup$ Thanks for the clarification. I believe I understand what you have said that Voyager should be very close to the Earth-Sun-Moon plane (in other words the ecliptic latitude of Voyager viewed from Earth should be small). Unfortunately for me, it makes me feel even worse for I cannot see how this means that Moon is farther. If the Moon were closer what should have we seen? $\endgroup$
    – d_e
    Jul 9 at 14:45
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    $\begingroup$ I'm not seeing it either. The moon being closer wouldn't change the observed illumination by much. $\endgroup$ Jul 9 at 18:09
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    $\begingroup$ I checked Horizons and indeed Earth is closer, but I also don't follow your explanation. Right now the Moon is at roughly 1st quarter as seen from Earth it seems, but if instead it were at 3rd quarter (and the inclination had precessed) the Moon could look very similar to this except be closer to Voyager rather than further. $\endgroup$
    – uhoh
    Jul 9 at 20:48

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