How was New Horizons able to direct data so precisely back to Earth?












57












$begingroup$


The New Horizons space probe is sending back images of Ultima Thule, 6.4 billion kilometres away.



Barring having a very large power to send information back on a very large angle, it seems to me that the level of precision to send this information back so that it can be captured by an antenna on Earth is prohibitively hard to get. How is NASA accomplishing it?










share|improve this question











$endgroup$








  • 13




    $begingroup$
    Related: it's standard procedure to supercool the receivers back on Earth, in freezers transparent to the wavelength to be received, to drastically reduce thermal noise. They can discern a very faint arriving signal.
    $endgroup$
    – Emilio M Bumachar
    Jan 7 at 10:35






  • 4




    $begingroup$
    @EmilioMBumachar one of the biggest things we've done for detection accuracy is figure out how to link many radar dishes together into large arrays in order to increase the functional aperture of the detector. Basic optics dictates how sharply you can focus is dependent on the aperture of your camera. Only then can the sensors clearly "see" the spacecraft transmitting data. Also there is a high degree of redundancy in the data in order to perform error correction.
    $endgroup$
    – Aaron
    Jan 7 at 18:51








  • 5




    $begingroup$
    Forget New Horizons, we're still getting data from Voyager 2, something like 119AU (~17.8 billion km) away, sent wth 1970s technology... :-)
    $endgroup$
    – T.J. Crowder
    Jan 9 at 8:22


















57












$begingroup$


The New Horizons space probe is sending back images of Ultima Thule, 6.4 billion kilometres away.



Barring having a very large power to send information back on a very large angle, it seems to me that the level of precision to send this information back so that it can be captured by an antenna on Earth is prohibitively hard to get. How is NASA accomplishing it?










share|improve this question











$endgroup$








  • 13




    $begingroup$
    Related: it's standard procedure to supercool the receivers back on Earth, in freezers transparent to the wavelength to be received, to drastically reduce thermal noise. They can discern a very faint arriving signal.
    $endgroup$
    – Emilio M Bumachar
    Jan 7 at 10:35






  • 4




    $begingroup$
    @EmilioMBumachar one of the biggest things we've done for detection accuracy is figure out how to link many radar dishes together into large arrays in order to increase the functional aperture of the detector. Basic optics dictates how sharply you can focus is dependent on the aperture of your camera. Only then can the sensors clearly "see" the spacecraft transmitting data. Also there is a high degree of redundancy in the data in order to perform error correction.
    $endgroup$
    – Aaron
    Jan 7 at 18:51








  • 5




    $begingroup$
    Forget New Horizons, we're still getting data from Voyager 2, something like 119AU (~17.8 billion km) away, sent wth 1970s technology... :-)
    $endgroup$
    – T.J. Crowder
    Jan 9 at 8:22
















57












57








57


4



$begingroup$


The New Horizons space probe is sending back images of Ultima Thule, 6.4 billion kilometres away.



Barring having a very large power to send information back on a very large angle, it seems to me that the level of precision to send this information back so that it can be captured by an antenna on Earth is prohibitively hard to get. How is NASA accomplishing it?










share|improve this question











$endgroup$




The New Horizons space probe is sending back images of Ultima Thule, 6.4 billion kilometres away.



Barring having a very large power to send information back on a very large angle, it seems to me that the level of precision to send this information back so that it can be captured by an antenna on Earth is prohibitively hard to get. How is NASA accomplishing it?







nasa spacecraft new-horizons






share|improve this question















share|improve this question













share|improve this question




share|improve this question








edited Jan 8 at 9:47









Basil Bourque

1725




1725










asked Jan 6 at 18:30









useruser

39125




39125








  • 13




    $begingroup$
    Related: it's standard procedure to supercool the receivers back on Earth, in freezers transparent to the wavelength to be received, to drastically reduce thermal noise. They can discern a very faint arriving signal.
    $endgroup$
    – Emilio M Bumachar
    Jan 7 at 10:35






  • 4




    $begingroup$
    @EmilioMBumachar one of the biggest things we've done for detection accuracy is figure out how to link many radar dishes together into large arrays in order to increase the functional aperture of the detector. Basic optics dictates how sharply you can focus is dependent on the aperture of your camera. Only then can the sensors clearly "see" the spacecraft transmitting data. Also there is a high degree of redundancy in the data in order to perform error correction.
    $endgroup$
    – Aaron
    Jan 7 at 18:51








  • 5




    $begingroup$
    Forget New Horizons, we're still getting data from Voyager 2, something like 119AU (~17.8 billion km) away, sent wth 1970s technology... :-)
    $endgroup$
    – T.J. Crowder
    Jan 9 at 8:22
















  • 13




    $begingroup$
    Related: it's standard procedure to supercool the receivers back on Earth, in freezers transparent to the wavelength to be received, to drastically reduce thermal noise. They can discern a very faint arriving signal.
    $endgroup$
    – Emilio M Bumachar
    Jan 7 at 10:35






  • 4




    $begingroup$
    @EmilioMBumachar one of the biggest things we've done for detection accuracy is figure out how to link many radar dishes together into large arrays in order to increase the functional aperture of the detector. Basic optics dictates how sharply you can focus is dependent on the aperture of your camera. Only then can the sensors clearly "see" the spacecraft transmitting data. Also there is a high degree of redundancy in the data in order to perform error correction.
    $endgroup$
    – Aaron
    Jan 7 at 18:51








  • 5




    $begingroup$
    Forget New Horizons, we're still getting data from Voyager 2, something like 119AU (~17.8 billion km) away, sent wth 1970s technology... :-)
    $endgroup$
    – T.J. Crowder
    Jan 9 at 8:22










13




13




$begingroup$
Related: it's standard procedure to supercool the receivers back on Earth, in freezers transparent to the wavelength to be received, to drastically reduce thermal noise. They can discern a very faint arriving signal.
$endgroup$
– Emilio M Bumachar
Jan 7 at 10:35




$begingroup$
Related: it's standard procedure to supercool the receivers back on Earth, in freezers transparent to the wavelength to be received, to drastically reduce thermal noise. They can discern a very faint arriving signal.
$endgroup$
– Emilio M Bumachar
Jan 7 at 10:35




4




4




$begingroup$
@EmilioMBumachar one of the biggest things we've done for detection accuracy is figure out how to link many radar dishes together into large arrays in order to increase the functional aperture of the detector. Basic optics dictates how sharply you can focus is dependent on the aperture of your camera. Only then can the sensors clearly "see" the spacecraft transmitting data. Also there is a high degree of redundancy in the data in order to perform error correction.
$endgroup$
– Aaron
Jan 7 at 18:51






$begingroup$
@EmilioMBumachar one of the biggest things we've done for detection accuracy is figure out how to link many radar dishes together into large arrays in order to increase the functional aperture of the detector. Basic optics dictates how sharply you can focus is dependent on the aperture of your camera. Only then can the sensors clearly "see" the spacecraft transmitting data. Also there is a high degree of redundancy in the data in order to perform error correction.
$endgroup$
– Aaron
Jan 7 at 18:51






5




5




$begingroup$
Forget New Horizons, we're still getting data from Voyager 2, something like 119AU (~17.8 billion km) away, sent wth 1970s technology... :-)
$endgroup$
– T.J. Crowder
Jan 9 at 8:22






$begingroup$
Forget New Horizons, we're still getting data from Voyager 2, something like 119AU (~17.8 billion km) away, sent wth 1970s technology... :-)
$endgroup$
– T.J. Crowder
Jan 9 at 8:22












1 Answer
1






active

oldest

votes


















85












$begingroup$

The high gain antenna of New Horizon as an opening angle of its beam of about 0.6°. That means, it has to be pointed at Earth with an error margin of 0.3°.
As a practical example, this is more like pointing a torch (flashlight) with a (well focused) beam at a far target than aiming with a tiny Laser spot. For comparison, 0.6° is slightly larger than the apparent size of the Moon in our skies which is 0.5° across.



The antenna is fixed to the spacecraft, so that the whole space probe has to rotate (that's the reason there was no direct data download during the encounter with Pluto). Rotating is accomplished by its on-board thrusters that can be used to adjust rotation very precisely.



Now the only remaining point is to figure out were Earth is located. Luckily somebody installed a bright beacon light relatively close to Earth (better known as the Sun) that can be used to find it. From Pluto Earth is at most 1.3° off to either side of the Sun.



New Horizons is equipped with a star tracker - essentially a camera that takes images of the sky and some software that reads the position and brightness of stars and compares them with a map. If, for any reason, pointing the antenna at Earth fails the space probe can switch to its smaller medium gain antenna which can work even when missing Earth by up to 10°. Essentially this allows to operate in a pure Sun-tracking mode without knowing the precise orientation of the probe.



And, as a last backup, there is also an omnidirectional antenna that can receive commands under almost any conditions to help getting the spacecraft operational again. Due to its low gain, this antenna could be used during the initial phase of the flight only - after traveling more than a few AU the signal received is too weak to be useful.






share|improve this answer











$endgroup$









  • 5




    $begingroup$
    In addition to rate gyros and star trackers, New Horizons is also equipped with bright beacon detectors, aka Sun sensors.
    $endgroup$
    – David Hammen
    Jan 6 at 19:51






  • 1




    $begingroup$
    I highly doubt the omni is in range anymore.
    $endgroup$
    – Joshua
    Jan 6 at 22:55






  • 3




    $begingroup$
    @Joshua it appears not The low gain system was only intended to be used within 1AU. While there probably was some room to push it using a more powerful transmitter at its current distance or ~44AU an increase of ~2000x would be needed vs at 1AU. While Arecibo is a more powerful transmitter than any of NASAs normal radio telescopes, it's not that much more powerful. spaceflight101.com/newhorizons/spacecraft-overview
    $endgroup$
    – Dan Neely
    Jan 6 at 23:51






  • 6




    $begingroup$
    Perhaps, for the benefit of lay readers who are not intimately familiar with angle measures in practice, it should be noted that the 0.6° beam angle is slightly larger in HN's view of the sky than the full moon is in our sky. So it's not like it needs to be aimed with fantastic precision that exceeds what human eyes could do.
    $endgroup$
    – Henning Makholm
    Jan 8 at 13:05






  • 4




    $begingroup$
    @Old_Fossil Only during a few days per year - most of the time Earth would be outside the beam. With high gain antennas there is almost no signal outside the intended opening angle, the falloff is very steep.
    $endgroup$
    – asdfex
    Jan 8 at 17:44











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1 Answer
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active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









85












$begingroup$

The high gain antenna of New Horizon as an opening angle of its beam of about 0.6°. That means, it has to be pointed at Earth with an error margin of 0.3°.
As a practical example, this is more like pointing a torch (flashlight) with a (well focused) beam at a far target than aiming with a tiny Laser spot. For comparison, 0.6° is slightly larger than the apparent size of the Moon in our skies which is 0.5° across.



The antenna is fixed to the spacecraft, so that the whole space probe has to rotate (that's the reason there was no direct data download during the encounter with Pluto). Rotating is accomplished by its on-board thrusters that can be used to adjust rotation very precisely.



Now the only remaining point is to figure out were Earth is located. Luckily somebody installed a bright beacon light relatively close to Earth (better known as the Sun) that can be used to find it. From Pluto Earth is at most 1.3° off to either side of the Sun.



New Horizons is equipped with a star tracker - essentially a camera that takes images of the sky and some software that reads the position and brightness of stars and compares them with a map. If, for any reason, pointing the antenna at Earth fails the space probe can switch to its smaller medium gain antenna which can work even when missing Earth by up to 10°. Essentially this allows to operate in a pure Sun-tracking mode without knowing the precise orientation of the probe.



And, as a last backup, there is also an omnidirectional antenna that can receive commands under almost any conditions to help getting the spacecraft operational again. Due to its low gain, this antenna could be used during the initial phase of the flight only - after traveling more than a few AU the signal received is too weak to be useful.






share|improve this answer











$endgroup$









  • 5




    $begingroup$
    In addition to rate gyros and star trackers, New Horizons is also equipped with bright beacon detectors, aka Sun sensors.
    $endgroup$
    – David Hammen
    Jan 6 at 19:51






  • 1




    $begingroup$
    I highly doubt the omni is in range anymore.
    $endgroup$
    – Joshua
    Jan 6 at 22:55






  • 3




    $begingroup$
    @Joshua it appears not The low gain system was only intended to be used within 1AU. While there probably was some room to push it using a more powerful transmitter at its current distance or ~44AU an increase of ~2000x would be needed vs at 1AU. While Arecibo is a more powerful transmitter than any of NASAs normal radio telescopes, it's not that much more powerful. spaceflight101.com/newhorizons/spacecraft-overview
    $endgroup$
    – Dan Neely
    Jan 6 at 23:51






  • 6




    $begingroup$
    Perhaps, for the benefit of lay readers who are not intimately familiar with angle measures in practice, it should be noted that the 0.6° beam angle is slightly larger in HN's view of the sky than the full moon is in our sky. So it's not like it needs to be aimed with fantastic precision that exceeds what human eyes could do.
    $endgroup$
    – Henning Makholm
    Jan 8 at 13:05






  • 4




    $begingroup$
    @Old_Fossil Only during a few days per year - most of the time Earth would be outside the beam. With high gain antennas there is almost no signal outside the intended opening angle, the falloff is very steep.
    $endgroup$
    – asdfex
    Jan 8 at 17:44
















85












$begingroup$

The high gain antenna of New Horizon as an opening angle of its beam of about 0.6°. That means, it has to be pointed at Earth with an error margin of 0.3°.
As a practical example, this is more like pointing a torch (flashlight) with a (well focused) beam at a far target than aiming with a tiny Laser spot. For comparison, 0.6° is slightly larger than the apparent size of the Moon in our skies which is 0.5° across.



The antenna is fixed to the spacecraft, so that the whole space probe has to rotate (that's the reason there was no direct data download during the encounter with Pluto). Rotating is accomplished by its on-board thrusters that can be used to adjust rotation very precisely.



Now the only remaining point is to figure out were Earth is located. Luckily somebody installed a bright beacon light relatively close to Earth (better known as the Sun) that can be used to find it. From Pluto Earth is at most 1.3° off to either side of the Sun.



New Horizons is equipped with a star tracker - essentially a camera that takes images of the sky and some software that reads the position and brightness of stars and compares them with a map. If, for any reason, pointing the antenna at Earth fails the space probe can switch to its smaller medium gain antenna which can work even when missing Earth by up to 10°. Essentially this allows to operate in a pure Sun-tracking mode without knowing the precise orientation of the probe.



And, as a last backup, there is also an omnidirectional antenna that can receive commands under almost any conditions to help getting the spacecraft operational again. Due to its low gain, this antenna could be used during the initial phase of the flight only - after traveling more than a few AU the signal received is too weak to be useful.






share|improve this answer











$endgroup$









  • 5




    $begingroup$
    In addition to rate gyros and star trackers, New Horizons is also equipped with bright beacon detectors, aka Sun sensors.
    $endgroup$
    – David Hammen
    Jan 6 at 19:51






  • 1




    $begingroup$
    I highly doubt the omni is in range anymore.
    $endgroup$
    – Joshua
    Jan 6 at 22:55






  • 3




    $begingroup$
    @Joshua it appears not The low gain system was only intended to be used within 1AU. While there probably was some room to push it using a more powerful transmitter at its current distance or ~44AU an increase of ~2000x would be needed vs at 1AU. While Arecibo is a more powerful transmitter than any of NASAs normal radio telescopes, it's not that much more powerful. spaceflight101.com/newhorizons/spacecraft-overview
    $endgroup$
    – Dan Neely
    Jan 6 at 23:51






  • 6




    $begingroup$
    Perhaps, for the benefit of lay readers who are not intimately familiar with angle measures in practice, it should be noted that the 0.6° beam angle is slightly larger in HN's view of the sky than the full moon is in our sky. So it's not like it needs to be aimed with fantastic precision that exceeds what human eyes could do.
    $endgroup$
    – Henning Makholm
    Jan 8 at 13:05






  • 4




    $begingroup$
    @Old_Fossil Only during a few days per year - most of the time Earth would be outside the beam. With high gain antennas there is almost no signal outside the intended opening angle, the falloff is very steep.
    $endgroup$
    – asdfex
    Jan 8 at 17:44














85












85








85





$begingroup$

The high gain antenna of New Horizon as an opening angle of its beam of about 0.6°. That means, it has to be pointed at Earth with an error margin of 0.3°.
As a practical example, this is more like pointing a torch (flashlight) with a (well focused) beam at a far target than aiming with a tiny Laser spot. For comparison, 0.6° is slightly larger than the apparent size of the Moon in our skies which is 0.5° across.



The antenna is fixed to the spacecraft, so that the whole space probe has to rotate (that's the reason there was no direct data download during the encounter with Pluto). Rotating is accomplished by its on-board thrusters that can be used to adjust rotation very precisely.



Now the only remaining point is to figure out were Earth is located. Luckily somebody installed a bright beacon light relatively close to Earth (better known as the Sun) that can be used to find it. From Pluto Earth is at most 1.3° off to either side of the Sun.



New Horizons is equipped with a star tracker - essentially a camera that takes images of the sky and some software that reads the position and brightness of stars and compares them with a map. If, for any reason, pointing the antenna at Earth fails the space probe can switch to its smaller medium gain antenna which can work even when missing Earth by up to 10°. Essentially this allows to operate in a pure Sun-tracking mode without knowing the precise orientation of the probe.



And, as a last backup, there is also an omnidirectional antenna that can receive commands under almost any conditions to help getting the spacecraft operational again. Due to its low gain, this antenna could be used during the initial phase of the flight only - after traveling more than a few AU the signal received is too weak to be useful.






share|improve this answer











$endgroup$



The high gain antenna of New Horizon as an opening angle of its beam of about 0.6°. That means, it has to be pointed at Earth with an error margin of 0.3°.
As a practical example, this is more like pointing a torch (flashlight) with a (well focused) beam at a far target than aiming with a tiny Laser spot. For comparison, 0.6° is slightly larger than the apparent size of the Moon in our skies which is 0.5° across.



The antenna is fixed to the spacecraft, so that the whole space probe has to rotate (that's the reason there was no direct data download during the encounter with Pluto). Rotating is accomplished by its on-board thrusters that can be used to adjust rotation very precisely.



Now the only remaining point is to figure out were Earth is located. Luckily somebody installed a bright beacon light relatively close to Earth (better known as the Sun) that can be used to find it. From Pluto Earth is at most 1.3° off to either side of the Sun.



New Horizons is equipped with a star tracker - essentially a camera that takes images of the sky and some software that reads the position and brightness of stars and compares them with a map. If, for any reason, pointing the antenna at Earth fails the space probe can switch to its smaller medium gain antenna which can work even when missing Earth by up to 10°. Essentially this allows to operate in a pure Sun-tracking mode without knowing the precise orientation of the probe.



And, as a last backup, there is also an omnidirectional antenna that can receive commands under almost any conditions to help getting the spacecraft operational again. Due to its low gain, this antenna could be used during the initial phase of the flight only - after traveling more than a few AU the signal received is too weak to be useful.







share|improve this answer














share|improve this answer



share|improve this answer








edited Jan 8 at 14:39

























answered Jan 6 at 18:56









asdfexasdfex

5,7441725




5,7441725








  • 5




    $begingroup$
    In addition to rate gyros and star trackers, New Horizons is also equipped with bright beacon detectors, aka Sun sensors.
    $endgroup$
    – David Hammen
    Jan 6 at 19:51






  • 1




    $begingroup$
    I highly doubt the omni is in range anymore.
    $endgroup$
    – Joshua
    Jan 6 at 22:55






  • 3




    $begingroup$
    @Joshua it appears not The low gain system was only intended to be used within 1AU. While there probably was some room to push it using a more powerful transmitter at its current distance or ~44AU an increase of ~2000x would be needed vs at 1AU. While Arecibo is a more powerful transmitter than any of NASAs normal radio telescopes, it's not that much more powerful. spaceflight101.com/newhorizons/spacecraft-overview
    $endgroup$
    – Dan Neely
    Jan 6 at 23:51






  • 6




    $begingroup$
    Perhaps, for the benefit of lay readers who are not intimately familiar with angle measures in practice, it should be noted that the 0.6° beam angle is slightly larger in HN's view of the sky than the full moon is in our sky. So it's not like it needs to be aimed with fantastic precision that exceeds what human eyes could do.
    $endgroup$
    – Henning Makholm
    Jan 8 at 13:05






  • 4




    $begingroup$
    @Old_Fossil Only during a few days per year - most of the time Earth would be outside the beam. With high gain antennas there is almost no signal outside the intended opening angle, the falloff is very steep.
    $endgroup$
    – asdfex
    Jan 8 at 17:44














  • 5




    $begingroup$
    In addition to rate gyros and star trackers, New Horizons is also equipped with bright beacon detectors, aka Sun sensors.
    $endgroup$
    – David Hammen
    Jan 6 at 19:51






  • 1




    $begingroup$
    I highly doubt the omni is in range anymore.
    $endgroup$
    – Joshua
    Jan 6 at 22:55






  • 3




    $begingroup$
    @Joshua it appears not The low gain system was only intended to be used within 1AU. While there probably was some room to push it using a more powerful transmitter at its current distance or ~44AU an increase of ~2000x would be needed vs at 1AU. While Arecibo is a more powerful transmitter than any of NASAs normal radio telescopes, it's not that much more powerful. spaceflight101.com/newhorizons/spacecraft-overview
    $endgroup$
    – Dan Neely
    Jan 6 at 23:51






  • 6




    $begingroup$
    Perhaps, for the benefit of lay readers who are not intimately familiar with angle measures in practice, it should be noted that the 0.6° beam angle is slightly larger in HN's view of the sky than the full moon is in our sky. So it's not like it needs to be aimed with fantastic precision that exceeds what human eyes could do.
    $endgroup$
    – Henning Makholm
    Jan 8 at 13:05






  • 4




    $begingroup$
    @Old_Fossil Only during a few days per year - most of the time Earth would be outside the beam. With high gain antennas there is almost no signal outside the intended opening angle, the falloff is very steep.
    $endgroup$
    – asdfex
    Jan 8 at 17:44








5




5




$begingroup$
In addition to rate gyros and star trackers, New Horizons is also equipped with bright beacon detectors, aka Sun sensors.
$endgroup$
– David Hammen
Jan 6 at 19:51




$begingroup$
In addition to rate gyros and star trackers, New Horizons is also equipped with bright beacon detectors, aka Sun sensors.
$endgroup$
– David Hammen
Jan 6 at 19:51




1




1




$begingroup$
I highly doubt the omni is in range anymore.
$endgroup$
– Joshua
Jan 6 at 22:55




$begingroup$
I highly doubt the omni is in range anymore.
$endgroup$
– Joshua
Jan 6 at 22:55




3




3




$begingroup$
@Joshua it appears not The low gain system was only intended to be used within 1AU. While there probably was some room to push it using a more powerful transmitter at its current distance or ~44AU an increase of ~2000x would be needed vs at 1AU. While Arecibo is a more powerful transmitter than any of NASAs normal radio telescopes, it's not that much more powerful. spaceflight101.com/newhorizons/spacecraft-overview
$endgroup$
– Dan Neely
Jan 6 at 23:51




$begingroup$
@Joshua it appears not The low gain system was only intended to be used within 1AU. While there probably was some room to push it using a more powerful transmitter at its current distance or ~44AU an increase of ~2000x would be needed vs at 1AU. While Arecibo is a more powerful transmitter than any of NASAs normal radio telescopes, it's not that much more powerful. spaceflight101.com/newhorizons/spacecraft-overview
$endgroup$
– Dan Neely
Jan 6 at 23:51




6




6




$begingroup$
Perhaps, for the benefit of lay readers who are not intimately familiar with angle measures in practice, it should be noted that the 0.6° beam angle is slightly larger in HN's view of the sky than the full moon is in our sky. So it's not like it needs to be aimed with fantastic precision that exceeds what human eyes could do.
$endgroup$
– Henning Makholm
Jan 8 at 13:05




$begingroup$
Perhaps, for the benefit of lay readers who are not intimately familiar with angle measures in practice, it should be noted that the 0.6° beam angle is slightly larger in HN's view of the sky than the full moon is in our sky. So it's not like it needs to be aimed with fantastic precision that exceeds what human eyes could do.
$endgroup$
– Henning Makholm
Jan 8 at 13:05




4




4




$begingroup$
@Old_Fossil Only during a few days per year - most of the time Earth would be outside the beam. With high gain antennas there is almost no signal outside the intended opening angle, the falloff is very steep.
$endgroup$
– asdfex
Jan 8 at 17:44




$begingroup$
@Old_Fossil Only during a few days per year - most of the time Earth would be outside the beam. With high gain antennas there is almost no signal outside the intended opening angle, the falloff is very steep.
$endgroup$
– asdfex
Jan 8 at 17:44


















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