# Conditions for high-resolution bistatic radar observations of Apophis in
2029

Agustín Vallejo, Jorge I. Zuluaga, Germán Chaparro
SEAP/FACom, Instituto de Física - FCEN, Universidad de Antioquia Calle 70 No.
52-21, Medellín, Colombia.
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

On April 13, 2029, asteroid Apophis will pass within six Earth radii
($\sim$31000 km above the surface), in the closest approach of this asteroid
in recorded history. This event provides unique scientific opportunities to
study the asteroid, its orbit, and surface characteristics at an exceptionally
close distance. In this paper, we perform a novel synthetic geometrical,
geographical and temporal analysis of the conditions under which the asteroid
can be observed from Earth, with a particular emphasis on the conditions and
scientific opportunities for bistatic radar observations, the most feasible
radar technique applicable during such a close approach. For this purpose, we
compile a list of present and future radio observatories or radio facilities
around the globe which could participate in bistatic radar observation
campaigns during the close approach of Apophis. We estimate signal-to-noise
ratios, apparent sky rotation, surface coverage and other observing
conditions. We find that a global collaboration of observatories across
Australia, Africa, Europe and America will produce high-resolution delay-
Doppler radar images with signal-to-noise ratios above $10^{8}$, while
covering $\sim$85% of the asteroid surface. Moreover, if properly coordinated,
the extreme approach of the asteroid might allow for radio amateur detection
of the signals sent by large radio observatories and citizen science projects
could then be organized. We also find that for visual observations, the Canary
Islands will offer the best observing conditions during the closest approach,
both for professionals as well as for amateurs. The apparent size of Apophis
will be 2-3 times larger than typical seeing, allowing for resolved images of
the surface.

###### keywords:

minor planets, asteroids: individual: 99942 Apophis – techniques: radar
astronomy – ephemerides

††pubyear: 2021††pagerange: Conditions for high-resolution bistatic radar
observations of Apophis in 2029–Conditions for high-resolution bistatic radar
observations of Apophis in 2029

##  1 Introduction

Previous observations of the trajectory of asteroid 99942 Apophis have shown
that on April 13, 2029, it will pass within six Earth radii from our planet.
Since its discovery in 2004, the probability of impact has been reduced from
2.7% to practically nil (Chesley, 2005; Wlodarczyk, 2013). In particular,
recent radar observations of Apophis have mostly ruled out this risk (Brozovic
et al., 2022). However, Apophis still provides unique opportunities in terms
of asteroid studies, which are usually limited to very distant and faint
objects, and relatively small ones passing close-by (in the tens of meters
range), theoretical modeling, or morphology estimations (Herique et al.,
2020). This time, the asteroid will come to us.

Pravec et al. (2014) and others have revealed Apophis to be an elongated
object of $340\pm 40$ m on its semimajor axis, axial anisotropy $a/b$ of 1.44,
and rotational period of $\sim$30.56 h. Observations of Apophis using
Goldstone and Arecibo by Brozović et al. (2018) have given further clues to
its particular shape, albedo, and other physical features such as its radar
cross-section. While they report no Yarkovsky acceleration, Tholen &
Farnocchia (2020) did detect a semimajor axis drift of about $\sim$170 m per
year. These observations were all prior to the close encounter of March 2021
at 0.12 au (Chesley, 2005), which will probably settle the issue of the role
of the Yarkovsky effect.

It has been predicted that for the 2029 fly-by, Apophis’ spin will
significantly change from its current state (Scheeres et al., 2005; Souchay,
J. et al., 2018), while its shape will remain mostly unaffected by tidal
forces (Yu et al., 2014). Regarding solar radiation pressure, it has been
shown that its effect is minimal on the orbit of Apophis (Giorgini et al.,
2008; Žižka & Vokrouhlický, 2011). Therefore, taking into account known non-
gravitational effects, the ephemeris can be calculated with an uncertainty
smaller than a few kilometers111NASA JPL Horizon data retrieved at  which
gives us a unique opportunity to prepare collaborative strategies for
observing the asteroid well in advance of its fly-by.

Although optically resolved observations of Apophis will be possible during
its closest approach (see section 2), it will be visible mostly from the day-
side of the planet, thus preventing high-quality observations. Fortunately,
Apophis will fly over some of the most important radar facilities and radio
telescopes on Earth, such as Goldstone, Very Large Array (VLA), next-
generation VLA (ngVLA), Haystack Ultrawideband Satellite Imaging Radar
(HUSIR), Green Bank (which is currently being fitted with a new, 100 kW
transmitter, see Lazio et al. 2021), and also China’s Five-hundred-meter
Aperture Spherical radio Telescope (FAST).

A combination of bistatic radar observations of this asteroid from multiple
radio facilities will allow for a detailed map of the asteroid down to the
highest possible range resolution, which is currently 1.875 m/px using the
Goldstone Deep Space Station-13 antenna (DSS-13, Naidu et al. 2016). In fact,
the achievable range resolution could potentially be close to two orders of
magnitude better if using satellite imaging radar facilities which have not
yet seen an asteroid pass within their range. This would allow for a detailed
study of asteroid properties for which radar astronomy is well suited for:
high-cadence measurements of its spin state, surface dust properties, and
possible tidal effects caused by its close approach such as asteroid-quakes.
Additionally, due to the short round-trip time at the closest approach,
bistatic radar has the best potential for achieving a high delay-Doppler
frequency resolution with an integration time that is not restricted by the
TX-RX switch time of monostatic observations.

The case for planetary radar observations has been strongly supported by
Brozovic et al. (2018) and Virkki et al. (2020). In particular, Herique et al.
(2020) has advocated for the use of this technique for monitoring observations
of Apophis. In the case of the Apophis 2029 fly-by and other objects
experiencing close encounters with Earth, radar signals have round-trip travel
times that are too short for monostatic radar observations. This means that
radar observation campaigns for the 2029 Apophis fly-by have to be mostly
bistatic, i.e. where the transmitting and receiving antennas are not the same.

For this reason we propose a sequential bistatic radar observation plan during
the day of closest approach of Apophis, calculating the expected radar
parameters such as the apparent asteroid sky rotation, signal-to-noise ratio
(SNR) per run, etc., based on its expected trajectory, spin state, and
observation conditions to an unprecedented level of detail. We identify the
best pairings of radar facilities and suitable radiotelescopes in terms of the
best achievable SNR and range resolution.

This paper is organized as follows: In section 2 we describe the details of
the 2029 fly-by: the approaching conditions of Apophis around Earth, its
magnitude, apparent size, distances, etc. A detailed analysis of the
conditions for making paired radar observations using radio antennas around
the world is described in section 3. The technique of bistatic radar
observations is described in section 4. In section 5 we discuss on the impact
the rotation of Apophis would have on radar observations. Along the paper we
emphasize on how this flight represents a potential citizen science
opportunity, and how even amateur radio astronomers can take part in this
important event.

##  2 Approaching window and visual observations

Radar observations of Apophis will start to be feasible from early March up to
June 2029 (a couple of months before and after its closest approach) when the
asteroid will come at a distance of $\sim 2000\;R_{\oplus}$ ($\sim 0.009$ AU
or $\sim 33$ times the Earth-Moon distance) from our planet (Brozović et al.,
2020). However, at its closest approach Apophis will be at a distance of near
$5\;R_{\oplus}$ from Earth. Around 4 hours before its closest approach (April
13, 21:50 UTC222All the information about Apophis trajectory used in this work
was obtained from JPL’s Horizons web service, and processed in Python with
Astropy and Cartopy libraries.), the apparent rotation rate of the asteroid
(see section 5) will increase, while the distance to Earth will decrease at a
fast pace, increasing significantly the signal-to-noise ratio for bistatic
radar observations (see section 4). Coincidentally, around 8 hours before
closest approach, when the asteroid is at a distance of ${\sim}30R_{\oplus}$,
its visual magnitude will reach the human eye limit. For these two reasons, we
have chosen the period of time when the asteroid is inside $30R_{\oplus}$,
that covers from April 13, 14:00 UTC to April 14, 06:00 UTC, as the time-
window to focus our analysis (we will call it the approaching window ).

In Figure 1 we show the path of the asteroid in the approaching window, both
in an Earth-centered reference frame (fixed on the stars) and in a rotating
reference frame (fixed on the Earth). As we see there, Apophis closest
approach will occur at $6.0\;R_{\oplus}$ with respect to Earth’s center,
precisely over a point on the Earth’ surface with longitude (lon.)
$43.4^{\circ}W$ and latitude (lat.) $28.9^{\circ}N$ (a point on the Atlantic
ocean, northwest of Canary Islands).

Figure 1: Distance of asteroid Apophis to Earth’s center in earth radii during
the hours around the closest approach. Left panel shows the right ascension of
the asteroid and the geocentric distance (projection of an hyperbola onto the
equatorial plane). Right panel shows the projected geographical longitude of
the Apophis as seen from a reference frame fixed on the rotating Earth. At its
closest approach the asteroid will fly-by over the atlantic (lon.
$43.4^{\circ}W$, lat. $28.9^{\circ}N$) at a distance of $6.0R_{\oplus}$ around
18:50 local time (21:50 UT). The yellow circle represents the Sun’s position
in each frame of reference. The points in the paths are 1h marks.

The asteroid will be visible to the naked eye mostly over Europe, Africa and
western Asia from the beginning of the approaching window and up to 2 hours
after the closest approach (see Figure 2). Then, the asteroid will experience
a rapid decrease in apparent brightness due to phase variations and the fact
that it will be flying over the sun-illuminated side of the Earth. Only radar
observations will still be feasible then.

Figure 2: Apparent magnitude of Apophis on the hours around closest approach
(21:45 UTC). As it gets closer, it will become brighter, reaching a minimum
visual magnitude of 3.7 at 20:35 UTC. Right after closest approach the
asteroid will go into the illuminated side of Earth. Alongside the top x axis,
the projected asteroid’s latitude is described for each time to determine at
which meridian the asteroid will be visible.

In Figure 3 we show the projection of Apophis ’ path over the surface of the
Earth during the approaching window (i.e. when the asteroid is closer than
$30R_{\oplus}$). We also show the sub-asteroid coordinates, the distance and
apparent magnitude of the asteroid on 2 hour intervals in Table 1. As we see
in the map, the projected path during the approaching window will start over
Australia and then it will move rapidly towards southeast Africa and cross the
Atlantic ocean towards America. At the end of the approaching window the
asteroid will be flying over northeast Asia.

Figure 3: Projection of Apophis’ path over Earth’s surface along with major radio telescopes in the world (stars are transmitting facilities whereas blue circles are receiving facilities) across its path. The red circle is the point of closest approach over lon. $43.4^{\circ}W$, lat. $28.9^{\circ}N$ and distance of $6.0R_{\oplus}$. The trajectory was divided in four stages based on geographical sub-asteroid point: Australia, Africa (and Europe), USA, and Asia. (See section 3) Time [UTC] | Lon. [deg] | Lat. [deg] | Dist. [$R_{\oplus}$] | V mag.
---|---|---|---|---
2029-Apr-13 14:00 | 147.4 | -23.5 | 28.5 | 5.8
2029-Apr-13 16:00 | 113.1 | -21.0 | 21.7 | 5.2
2029-Apr-13 18:00 | 75.5 | -15.9 | 15.0 | 4.5
2029-Apr-13 20:00 | 27.9 | -2.3 | 8.8 | 3.8
2029-Apr-13 21:45 | -43.4 | 28.9 | 6.0 | 4.4
2029-Apr-14 00:00 | -151.3 | 36.3 | 10.3 | 10.0
2029-Apr-14 02:00 | 159.8 | 30.1 | 16.7 | 14.0
2029-Apr-14 04:00 | 122.4 | 26.6 | 23.4 | 17.0
2029-Apr-14 06:00 | 88.4 | 24.5 | 30.2 | 18.0
Table 1: Apophis projected position, distance and (geocentric) apparent
magnitude on 2h intervals during the selected time-window.

One of the best suited close-approach observation regions on Earth are the
Canary Islands. In Figure 4 we show the elevation and apparent magnitude of
Apophis as seen from the top of mount Teide in Tenerife333We have created a
website that provides the visual observing conditions for any location on
Earth https://agustinvallejo.github.io/apophis.html (an ideal place for
professional as well as amateur observations). Using the size estimation of
Brozović et al. (2018) of $340\pm 40$ m the maximum apparent size of Apophis
will be of $1.82\pm 0.21^{\prime\prime}$ which is 2-3 times larger than the
local seeing 444For typical observing conditions in the El Teide or Roque de
los Muchachos sites see
https://www.ing.iac.es/astronomy/observing/conditions/enofolleto.pdf, page 6
(last accessed ).. Although the optical observing window is much shorter than
for radar observations, optical observations will help in constraining the
shape in addition to the radar data.. Table 2 also shows a list of significant
optical telescopes that are well-located for observing the close approach.

Name | Location | Longitude (°) | Latitude (°) | Diameter (m)
---|---|---|---|---
Devasthal | Nainital district, Kumaon division, Uttarakhand, India | 79.684 | 29.36 | 3.60
BTA-6 | Caucasus Mountains, Russia | 41.44 | 43.65 | 6.00
SALT | Great Karoo, Sutherland, South Africa | 20.81 | -32.38 | 10.50
Calar Alto Observatory | Almería, Spain | -2.55 | 37.22 | 3.50
William Herschel | Santa Cruz de Tenerife, Canary Islands, Spain | -17.88 | 28.76 | 4.20
Gran Telescopio de Canarias | La Palma, Canary Islands, Spain | -17.89 | 28.76 | 10.40
Galileo National Telescope | La Palma, Canary Islands, Spain | -17.89 | 28.76 | 3.58
Table 2: Some of the optical telescopes that could make observations of the
asteroid in visible light, sorted by longitude. Figure 4: Elevation and
apparent magnitude of Apophis during its closest approach to Earth in April
13, 2029, as seen in Teide (Canary Islands).

##  3 Coordinated radar observations

One hour after the closest approach, Apophis will be over North America, and
although it will be daytime, the asteroid will be flying overhead the area of
the Goldstone complex and the Green Bank radio observatory. This will be one
of the ideal moments for bistatic radar observations (see section 4) as these
radio facilities have large antennas, significant experience with planetary
radar astronomy and a clear opportunity for observing the asteroid (at that
time Apophis will still be at ${\sim}7\;R_{\oplus}$ from Earth’s center, or
${\sim}6\;R_{\oplus}$ from the telescopes).

As a matter of fact (as shown in Figure 3), most of the largest radio
facilities on Earth with transmission capabilities (stars in the map) will be
able to participate in radar observations of the asteroid. However, during the
approaching window, the close proximity of the asteroid and the fast movement
of it through the sky will make monostatic radar observations hard to achieve:
if a single facility has transmission (TX) as well as receiving (RX)
capabilities, it will be very difficult for it to switch modes in time (Virkki
et al., 2020). However, although most radio telescope facilities do not have
TX capabilities, a significant number have significant RX sensitivity. This is
why bistatic radar observations around the globe would offer the best
opportunities to observe the closest approach of Apophis.

The next significant moment of the fly-by will occur 7 hours after the closest
approach, when the asteroid passes over NE Asia. In particular, it will fly
very close to the zenith of the FAST radio telescope. Since FAST is not fully
steerable, the fact that the receding, still close Apophis will fly almost
vertically over FAST is truly fortunate.

In order to organize a global radar observational effort, we propose to split
the approaching window in four stages. At each stage we identify the best
pairings of radar facilities and observatories for performing bistatic radar
observations. In Table 3 we show the location, and the RX and TX properties of
the radio observatories included in this work. For performing the pairing
analysis, we will refer to Figure 5, where we show the elevation of the
asteroid as a function of time for all the observatories listed in Table 3.
Additionally, in Table 4 we pair some of the most important radio facilities
to show which ones would be available at the same time for bistatic
observations.

Furthermore, simultaneous bistatic observations (e.g. when DSS-13 and DSS-14
transmit at the same time) will let us explore the asteroid beyond its shape
from the radar albedo and polarization at different wavelengths. This will
help constrain the surface composition and regolith grain size distribution
(Virkki & Muinonen, 2016).

In the following sections we will describe the observation plan along with the
achievable per run SNR for significant bistatic radar pairs in each region.
Here each _run_ refers to the minimum integration time needed for achieving
full Doppler frequency resolution (see Equation 3).

Stage | Name | Lon. [deg] | Lat. [deg] | D [m] |  $P_{\mathrm{tx}}$ [kW] |  $F_{\mathrm{tx}}$ [GHz] ($\lambda$ [cm]) |  $F_{\mathrm{rx}}$ [GHz]
---|---|---|---|---|---|---|---
1.1 | Canberra DSS-43 | 148.98 | -35.40 | 70 | 80 | 7.2 (4.2) | –
1.2 | Parkes | 148.26 | -33.00 | 64 | – | – | 0.8-22
2.1 | Madrid DSS-63 | -4.25 | 40.43 | 70 | 20 | 7.2 (4.2) | –
2.2 | Lovell | -2.31 | 53.24 | 76 | – | – | 0.4-6
2.3 | Effelsberg | 6.88 | 50.52 | 100 | – | – | 0.3-95
2.4 | AVN-Ghana | 0.31 | 5.74 | 32 | – | – | 4-8
2.5 | TIRA (*) | 7.12 | 50.66 | 34 | 1000 | 22.5 (1.33) | –
2.6 | Yevpatoria RT-70 | 33.18 | 45.18 | 70 | 100 | 5 (6) | –
2.7 | Sardinia | 9.25 | 39.48 | 64 | – | – | 0.3-116
3.1 | Goldstone DSS-13 | -116.89 | 35.43 | 34 | 80 | 7.2 (4.2) | –
3.1 | Goldstone DSS-14 | -116.89 | 35.43 | 70 | 450 | 8.6 (3.5) | –
3.2 | Arecibo (defunct) | -66.75 | 18.34 | 305 | 2000 | 2.4 (13) | –
3.3 | Green Bank (**) | -79.84 | 38.43 | 100 | 100 | 35 (0.85) | 0.3-116
3.4 | Very Large Array | -107.62 | 34.08 | 25 ea. | – | – | 0.06-50
3.5 | Haystack (HUSIR) (*) | -71.42 | 42.68 | 36.6 | 250 | 10 (3) | –
4.1 | Galenki RT-70 | 131.76 | 44.03 | 70 | 80 | 5 (6) | –
4.2 | Usuda Deep Space Center | 138.36 | 36.13 | 64 | 20 | 8 (3.7) | –
4.3 | FAST | 106.86 | 25.65 | 500 | – | – | 0.07-3
Table 3: Relevant radio facilities in the asteroid’s path. Those with listed power are the transmitting facilities considered here. (*) The average power for pulsed radars is an order of magnitude lower than the peak power referenced here. (**) The quoted $F_{\mathrm{tx}}$ value of 35 GHz for the GBT is a preliminary value, and could change in the future. RX / TX | Canberra DSS-43 | Madrid DSS-63 | Goldstone | Green Bank | Galenki RT-70 | Usuda
---|---|---|---|---|---|---
Parkes | X |  |  |  | X | X
Lovell |  | X | X | X |  |
Effelsberg |  | X | X | X |  |
AVN-Ghana |  | X |  | X |  |
VLA |  |  | X | X |  |
Green Bank |  |  | X |  |  |
FAST |  |  |  |  | X | X
Table 4: Available pairing of some of the proposed radio observatories based
on elevation and available window of observation. In the first column are the
receivers and on the first row are the possible transmitters. Figure 5:
Proposed observation plan of Apophis closest approach, which will get almost
complete coverage of the fly-by. a) the expected elevations of the asteroid as
seen from the transmitting facilities. b) the elevations from the receiving
end. We suggest the 4-stage plan of the transit: the first one being Canberra
DSS-43 paired with Parkes, as they have optimal view of the asteroid, then
with the African and European dishes (Madrid DSS-63 with AVN-Ghana, Lovell or
Effelsberg). This stage will see the asteroid at its closest. Third stage will
be Goldstone with Green Bank or VLA, which will also get the closest point.
Fourth stage is FAST with Asian or Russian antennas as transmitters. Note that
the y-axis goes from 15° to 90°, because the asteroid being over the horizon
does not guarantee a successful observation, all antennas have a minimum
elevation, most of them around 15°. FAST can only point above $\sim$45°,
depending on the mode of observation.

###  3.1 Stage 1. Australia

At the beginning of the approaching window, Apophis will fly directly over
Australian territory. This moment will offer a good opportunity to transmit
from Canberra’s DSS-43 antenna. This is one of the largest steerable antennas
in the hemisphere (70 m diameter) with a recently installed 100 kW
transmitter. An S-band 400 kW555See the DSN 101 10-m Subnet Telecommunications
Interfaces report https://deepspace.jpl.nasa.gov/dsndocs/810-005/101/101E.pdf
(last accessed ). and an X-band 80 kW666See the DSN 301 Coverage and Geometry
Report https://deepspace.jpl.nasa.gov/dsndocs/810-005/301/301N.pdf (last
accessed ). transmitter are also available. The Parkes radio telescope will
also be under the asteroid’s path, as well as any of the Australia Telescope
Compact Array (ATCA) elements, which we do not consider in our calculations
but could plausibly be used as RX.

Apophis will be over the horizon with respect to DSS-43 and Parkes from 9:00
UTC to 19:00 UTC, and the distance will decrease from $40R_{\oplus}$ to
$10R_{\oplus}$ during this time window. The flight path will allow for an
elevation of almost 80°.

Last, but not least, this stage could also be ideal for visual observations of
Apophis , as it will be close to midnight in western Australia. Because at
this stage the visual magnitude of the asteroid will be very close to the
human eye limit, most telescopes will be able to observe the approaching
object.

###  3.2 Stage 2. Africa and Europe

The asteroid will continue its course westward and fly over Africa towards the
Canary Islands. In this stage, the best antenna for transmission will be
Madrid’s DSS-63. Similar to the one in Canberra, DSS-63 has a 70m diameter and
20 kW of available TX power. At apogee, DSS-63 will see the asteroid at 60∘ of
elevation (see Figure 5).

At the reception end, there are a few alternatives in Europe as well as in
Africa. In Europe there are the Effelsberg (100 m) and Lovell (76 m) radio
telescopes, both of which will see a maximum elevation of about 45∘. In
Africa, for example, the Ghana Kuntunse radio telescope (32 m) will see the
asteroid nearly in the zenith. In the case that the Square Kilometer Array is
operational in 2029 it could plausibly participate in these observations on
the RX end.

Other radio telescope arrays such as any of the European VLBI Network (EVN)
antennas could also be used for this observation. For this, either the Madrid
antenna, the Tracking and Imaging Radar (TIRA) in Germany or the Yevpatoria
RT-70 antenna (Zakharov et al., 2018) could be used for transmission to
increase the number of antenna pairs for bistatic observations.

This stage spans from 18:00 UTC to 23:00 UTC. During this time the asteroid
will be fully visible for countries in Europe and Africa, even with the naked
eye (see Figure 2). The geocentric distance during this stage, will change
from $15\;R_{\oplus}$ to $6\;R_{\oplus}$ at 21:45 UTC and finally
$7R_{\oplus}$ at the end of this stage.

###  3.3 Stage 3. Americas

Just after the closest approach, the asteroid will continue its path over
North America. Unfortunately (for visual observers) it is going to be daytime
in this hemisphere and no visual observations will be attainable. However, the
asteroid will fly right over important American radio telescopes. The
Goldstone complex is an ideal facility for transmitting radar signals to the
asteroid surface, with its DSS-13 and DSS-14 antennas (TX frequency: 7190 MHz
and 8560 MHz, respectively). Previous radar observations have also used DSS-14
for mapping the shape and surface characteristics of Apophis at 18.75 m/px,
since its 70 m dish and 500 kW transmitting power allowed for an SNR of $\sim
23$/day (Brozović et al., 2018). However, for the 2029 approach the expected
per run SNR for the smaller TX-capable antenna (DSS-13, 34 m dish) will be of
the order of $10^{7}$ (see Figure 7). Thus, DSS-13 with its planned 7190 MHz
transmitter, with an 80 MHz decoder bandwidth Brozovic et al. (2022) will
yield a range resolution of 1.875 m/px (see Table 3) .

The Green Bank Telescope (GBT, $D=$100 m) is also currently being fitted with
TX capabilities (Lazio et al., 2021) with a 100 kW transmitter at 35 GHz
planned to be installed before the Apophis approach. Any of the VLA or VLBA
dishes would be able to act as RX at the planned TX frequencies mentioned
above.

We should note that sub-m range resolution could be achieved when using
Haystack Ultrawideband Satellite Imaging Radar, HUSIR (Murray & Kennedy, 2020)
reaching an instantaneous777For pulsed radars the subsecond (e.g., 1/60 Hz)
duty cycle duration might be more appropriate, but at this point we only quote
this as a representative SNR value. (1 second) SNR of up to $\sim\times
10^{7}$ for monostatic observations, with a 1024 MHz decoder bandwidth
allowing for a range resolution of 25 cm/px at X-band (10 GHz), and with a
$T_{\text{sys}}=100$ K. The average SNR will be an order of magnitude lower
due to HUSIR being a pulsed radar facility instead of continuous-wave. HUSIR
is also equipped with a W-band (96 GHz, 0.3 cm) transmitter with an 8 GHz
effective bandwidth, allowing for a range resolution in the order of 3 cm/px
(Camp & Mayhan, 2000; Kempkes et al., 2006; MacDonald et al., 2014). It should
be also be noted that although this instrument is capable of satellite
imaging, its asteroid imaging capabilities have not yet been fully tested at
this resolution.

To match the best possible resolution, X-band observations require a rotation
of at least 3.44∘ over the integration time (Camp & Mayhan, 2000), which for
the case of Apophis is of the order of $\sim 20$ min. W-band observations
would require 8x more integration time.

Although HUSIR is capable of monostatic radar observations at very close
distances, it is limited to observing anything within 50,000 km (for X-band)
and 5,000 km for W-band. Therefore, we propose using GBT as RX in conjunction
with HUSIR (as TX) for bistatic observations, although keeping in mind that
the GBT dish has a very low aperture efficiency at W-band. Also, the W-band
suffers a lot from atmospheric absorption, so the system temperature should be
about an order of magnitude more than the usual 25 K. However, the achievable
per-run SNR could be up to $\sim 10^{8}$, given $T_{\text{sys}}=25$ K for
bistatic observations, or $\sim 10^{7}$, for $T_{\text{sys}}=250$ K.

Another ideal transmitting antenna at this stage would have been the Arecibo
Telescope, whose location is very close to the flight path of the asteroid. It
will be a great loss if a similar observatory were not to be built at the same
place in the following years, although we are not aware of any such plan.

###  3.4 Stage 4. East Asia

As the asteroid is finally leaving Earth’s proximity on the morning of April
14th, it will fly over the 500 m FAST radio telescope. One candidate for
transmitting is the Japanese Usuda radio station (64 m diameter and 20 kW of
power). Another plausible TX facility is the 70 m Galenki (Ussuriysk), Russia
antenna with 80 kW of transmitting power. It should be noted FAST is limited
in terms of frequency capabilities, as it can currently only observe up to 3
GHz (S-band), whereas the Russian antennas do not transmit at S-band but at 5
GHz. The Apophis flyby can thus be an important scientific case for expansion
and improvement of these facilities. Therefore, this stage is only feasible if
those upgrades are implemented and completed before the flyby.

##  4 Sensitivities and resolution

Asteroid radar observation is a technique that depends on many geometrical and
physical factors (Ostro, 1993; Hudson, 1994; Ostro et al., 2002). Due to the
high level of control with respect to technical aspects, this technique can be
considered to be no more complicated than obtaining visual observations, as
the illumination is always well constrained and the corresponding phase angle
is near $0^{\circ}$ even for bistatic observations. In this section, we will
use the ephemeris of the 2029 Apophis approach to estimate relevant quantities
involved in bistatic radar observations of the asteroid. Our main goal here is
to identify the scientific opportunities of a coordinated global collaboration
for mapping a significant fraction of the surface of Apophis at unprecedented
SNR and range resolution.

The delay-Doppler technique is often used (see e.g. Ostro et al. 2002) for
imaging the surface of the asteroid and/or reconstructing its 3D shape. For
this purpose, signals are sent from a transmitting antenna (transmitting
frequency $F_{\mathrm{tx}}$) to the asteroid and its echo is registered in a
receiving station. Range information is obtained by time-modulating the signal
via binary phase coding or frequency modulation (e.g. chirp). Thus, for an
appropriate SNR the maximum spatial resolution is determined by the decoder
bandwidth of the transmitter. The Doppler shifted, echoed signal is analysed
in time and frequency domain to construct an image data set containing
dynamical and spatial information (for a review of the technique see Ostro
1993; Margot 2021 and references therein).

One of the first and foremost opportunities of the Apophis approach will be to
constrain its orbit. Being a Potentially Hazardous Asteroid (PHA), all the
information that we can obtain to further constrain the non-gravitational
forces acting on the asteroid, is required to predict future approaches and
assess the probability of an impact.

In order to achieve this, radar observations measure the round-trip time (the
delay) of the signal $\tau_{\mathrm{eph}}(t)=2d(t)/c$, where $d(t)$ is the
instantaneous distance between the target and the radar, and the Doppler
frequency shift (DFS), $\nu_{\mathrm{eph}}(t)=2F_{\mathrm{tx}}v(t)/c$ of the
center-of-mass of the asteroid, where $v(t)$ is the instantaneous target-radar
relative radial velocity. In Figure 6 we show the typical value of these two
critical quantities for observations during the approaching window using the
DSS-14 and DSS-13 antennas at the Goldstone complex.

Figure 6: Round-trip time $\tau_{\mathrm{eph}}(t)$ and center-of-mass Doppler
frequency shift $\nu_{\mathrm{eph}}$, for the radar signal in the case of
bistatic radar observations using the antennas DSS 14 and DSS-13 of the
Goldstone Complex.

The expected range resolution for each observation depends on the sampling
capabilities of the transmitting radar system. For instance, when using the
pulse compression technique (Naidu et al., 2013) the range resolution $\delta
r$ is calculated from the decoder effective bandwidth $B_{\mathrm{eff}}$ as
follows,

$\delta r=\frac{c}{2}\frac{1}{B_{\mathrm{eff}}}\ .$ (1)

The echoed signal will produce a delay dispersion of $\sim D_{||}/(2\delta
r)$, where $D_{||}$ is the spatial “length” of the asteroid in a direction
parallel to the viewing line (which for the case of Apophis would be in the
300-370 m range depending on its relative orientation), and a Doppler shift
dispersion (or bandwidth),

$\Delta\nu=\frac{2D\omega\cos\delta}{\lambda}\ .$ (2)

Here, D is the spatial “width” of the asteroid, measured orthogonal to the
line of sight (since Apophis is not spherical, in general we can expect
$D_{||}$ to be different to $D$). $\delta$ is the target subradar latitude and
$\omega$ is the apparent angular frequency of the asteroid which will change
during the passage due to the combined effects of its intrinsic rotation and
the apparent rotation due to the change in viewing perspective as it
approaches Earth (see below). The subradar latitude $\delta$ will also change
during the passage due to the asteroid orbit not being coplanar with the
Earth’s equator. This means that the Doppler bandwidth will not remain
constant during the passage. Here we consistently update this value for our
per-run SNR calculation (see Equation 3). To sample this bandwidth with an
adequate frequency resolution, the integration time per run needs to be at
least,

$\Delta t_{\text{run}}=\Delta\nu^{-1}\ .$ (3)

For monostatic observations, the TX-RX switch time limits the highest possible
frequency resolution to $(\tau_{\mathrm{eph}}-t_{\text{switch}})^{-1}$. This
restriction does not apply to bistatic observations, and the integration time
is only limited by the time the asteroid can be tracked by the telescope pair,
i.e. one observation track.

The achievable SNR will depend on many factors, involving the TX/RX
instrumentation as well as the target properties. For instance, one of the
main quantities is the total received echo power, which depends on the antenna
gains $G$, both for the transmitting and receiving antennas (Naidu et al.,
2016),

$G=4\pi\eta A/\lambda^{2}\ .$ (4)

Here $\eta$ is the aperture efficiency, $A$ is the physical area of the
antenna, and $\lambda$ is the radar operating wavelength (which depends on the
configuration of the transmitting antenna). The aperture efficiency for most
paraboloidal antennas considered here (see Table 3) is in the 0.6-0.7 range
(Naidu et al., 2016), although FAST is on the lower end of this range (Dong &
Han, 2013). Once the transmitted power $P_{\mathrm{tx}}$ is set, the received
power $P_{\mathrm{rx}}$ can then be calculated as (Naidu et al., 2013),

$P_{\mathrm{rx}}=\frac{G_{\mathrm{rx}}G_{\mathrm{tx}}\lambda^{2}\sigma}{(4\pi)^{3}R_{\mathrm{rx}}^{2}R_{\mathrm{tx}}^{2}}\,P_{\mathrm{tx}}\
.$ (5)

Here $\sigma$ is the radar cross-section of the target object, which has been
measured as $\sigma\approx{0.023}$ km2 in previous observations (Brozović et
al., 2018), and $R_{\mathrm{tx}}$, $R_{\mathrm{tx}}$ are the distances from
the TX and RX antennas to the target, which are usually assumed to be
approximately the same, e.g. $R_{\mathrm{rx}}\approx R_{\mathrm{tx}}$ (Naidu
et al., 2016). In the case of the 2029 Apophis passage, however, the short
distance range involved does not allow for this approximation, and a proper
calculation of both quantities is required. It should be noted that the
transmitted peak power for pulsed radar systems such as HUSIR and TIRA is not
the same as the effective average power, as discussed above.

The SNR is the echo-power standard $z$-score of the signal, i.e. the number of
standard deviations above the mean noise power,

$\mathrm{SNR}=\frac{P_{\mathrm{rx}}}{\Delta P_{\mathrm{rms}}}\ .$ (6)

Here $\Delta P_{\mathrm{rms}}$ is the noise standard deviation which depends
on the system temperature of the receiving telescope $T_{\mathrm{sys}}$, the
Doppler bandwidth $\Delta\nu$, and the integration time $\Delta t$,

$\Delta P_{\mathrm{rms}}=kT_{\mathrm{sys}}\sqrt{\frac{\Delta\nu}{\Delta t}}\
.$ (7)

Here $k$ is the Boltzmann constant.

Replacing Eqs. (2), (5) and (7) into Eq. (6) the final expression for the
signal-to-noise ratio of bistatic radar observations becomes,

$\mathrm{SNR}=P_{\mathrm{tx}}\frac{G_{\mathrm{rx}}G_{\mathrm{tx}}\lambda^{5/2}}{(4\pi)^{3}kT_{\mathrm{sys}}}\sqrt{\frac{\Delta
t}{2D\omega\cos\delta}}\frac{\sigma}{R_{\mathrm{rx}}^{2}R_{\mathrm{tx}}^{2}}\
.$ (8)

In Figure 7 we show our simulated values of the expected frequency integrated
SNR per run (see Equation 3) for bistatic radar observations of the 2029
Apophis approach for different observatory pairs and calculated with Equation
8 and the antenna properties listed in Table 3. We assume a nominal system
temperature of 25 K. This temperature may be in some cases higher by a factor
of 2, or more when specified, which means that our calculated SNR may be off
by the same factor. We do not consider this to be a significant problem since
the lowest estimated SNR is in the order of $10^{7}$, for the case of the
DSS-13 + VLA pairing. Even in the lowest SNR case (observations performed
between DSS-13 and a VLA element), values of $10^{4}$ can be achievable, which
is well suited for radar observations. With Arecibo out of operation,
observations with FAST + Usuda will provide the highest SNR per run during the
whole set of observations.

Figure 7: Expected values of the frequency integrated SNR per run for
different pairs of radio observatories during the 2029 Apophis approaching
window. The grey dotted lines correspond to times in which Apophis is under
the minimum elevation for either of the paired antennas. The run time is
calculated here according to Equation 3.

##  5 Rotation, axis orientation and coverage

Using previous observations of Apophis and theoretical models, several authors
have been able to constrain the rotational state of the asteroid (Brozović et
al., 2018). It is now known that the spin axis of Apophis (which is also the
longest one) probably points towards RA $118.8^{\circ}$ and DEC
$-79.4^{\circ}$ in J2000 coordinates (Pravec et al., 2014; Souchay, J. et al.,
2018). Moreover, Apophis seems to be a non-principal axis rotator with a main
sidereal rotation period of 30.56 h (Brozović et al., 2018) which corresponds
to an intrinsic angular speed $\omega_{\mathrm{s}}=11.7$ deg/h. In Figure 8 we
schematically show the expected orientation of Apophis spin during its 2029
approach.

Figure 8: Schematic illustration of the trajectory of Apophis during the 2029
approach, showing the most probable orientation of its spin axis. For
illustration purposes, Apophis size has been increased by a factor of 30 000,
and its geometrical main axis pointing in the same direction as the
rotational, although there is no strong evidence of the long principal axis
being aligned with the spin axis. The asteroid main axis of rotation is
pointing at roughly the southern celestial pole (J2000 RA,DEC =
$118.8^{\circ},-79.4^{\circ}$).

In order to take into account the expected spin state on the observations
(delay-Doppler observations and surface coverage), we need to predict how the
surface will be rotating with respect to the radar plane-of-sky (POS)
reference frame. Although its current rotation speed is well constrained,
according to Souchay, J. et al. (2018) significant variations in obliquity and
precession in longitude may occur due to gravitational torques during the
close approach. However, the single most significant effect altering the
observed rotational speed will be the change in the Earth-facing side of the
asteroid due to the change in perspective as the asteroid swings by. Thus, the
swiftness of the asteroid approach will temporarily increase its apparent
rotational speed as seen from Earth, which will increase the surface coverage
due to the change in visual perspective. The apparent angular speed of the
asteroid in the POS can be obtained from the vector sum (Ostro et al., 2002),

$\vec{\omega}_{\mathrm{app}}=\vec{\omega}_{\mathrm{s}}+\vec{\omega}_{\mathrm{sky}}\
.$

Here $\vec{\omega}_{\mathrm{s}}$ is the sidereal (intrinsic) spin-vector and
$\vec{\omega}_{\mathrm{sky}}$ is the sky angular speed which is given by,

$\vec{\omega}_{\mathrm{sky}}=\dot{\hat{e}}\times{\hat{e}}\ .$

Here, $\hat{e}=\vec{r}/r$ is the target-to-radar unit vector, and
$\dot{\hat{e}}$ is its vectorial speed that can be calculated from ephemeris
using,

$\dot{\hat{e}}=\frac{\vec{v}}{r}-\frac{\vec{r}}{r^{2}}\dot{r}\ .$

In Figure 9 we show the value of $\vec{\omega}_{\mathrm{sky}}$ and
$\vec{\omega}_{\mathrm{app}}$, as a function of time for the hypothetical case
of a radar in the center of the Earth.

Figure 9: Apparent rotational speed of the asteroid in the hours around the
closest approach (solid line). The rotational speed of the asteroid is a
combination of its intrinsic rotation ($\omega_{\mathrm{s}}$) and the sky
angular speed of the asteroid during its closest approach
($\omega_{\mathrm{sky}}$).

As expected, given the extremely close and rapid approach of Apophis ,
$\omega_{\mathrm{sky}}$ will reach values ${\sim}4$ times larger than the
intrinsic rotational velocity of the asteroid. However, since the spin axis of
the asteroid is almost orthogonal to its orbital plane, the apparent rotation
will possibly compensate its intrinsic rotation. As a result, during 10 of the
14 hours of the approaching window the asteroid will likely rotate slower than
usual. At certain times, its surface will even seem static with respect to
Earth (minimum values in the solid blue line in Figure 9).

However, during the closest approach (between UTC 20:00 and 23:59 of April 13)
the sky rotation is higher than its intrinsic rotation and the asteroid will
seem to be rotating ${\sim}2-3$ times faster. This will increase the Doppler
shift dispersion (see Equation 2). More interestingly, this boost in the
apparent rotation will allow observations to cover a large fraction of the
asteroid surface at potentially better range resolutions than in observing
campaigns previous to this approach.

If we manage to perform the coordinated radar observations program proposed in
this paper, an interesting question to raise is what fraction of the asteroid
surface will be covered by observations. Since the approaching window
(${\sim}14$ hours) is almost half of a full rotation of the asteroid
(${\sim}30$ hours) and the apparent rotation happens in the opposite direction
(see above), it would be expected that not all the surface will be imaged at
the plausible resolution shown here.

In order to predict the radar coverage of the asteroid’s surface, we simulated
the Earth’s apparent sky motion as seen from many points on Apophis’ surface.
For this purpose, we used a spherical model of the asteroid, create a grid of
local coordinates, and calculate for each point and at every time during the
approaching window, if the geocenter was above the local horizon. In Figure 10
we show the resulting coverage map.

Figure 10: Surface coverage of Apophis assuming a continuous observation
during the 14 hours around the closest approach. The reference meridian
(lonA=0) is defined by the sub-Earth direction at the time of closest
approach. A value of 50% over a point in the map means that this point can be
observed during half of the approaching window.

The Apophis geographical coordinates were defined with respect to its axis of
rotation (defining the latitude) and the direction of Earth in the moment of
closest approach (defining longitude). The prime meridian in this system pass
through the sub-Earth point on the surface of the asteroid at the time of
closest approach, namely 2029-04-13 at 21:45 UTC.

Due to the inclination of the asteroid rotation axis, the coordinates of the
sub Earth point at that time will be lon${}_{A}=0^{\circ}$,
lat${}_{A}=26.4^{\circ}$ (white circle in Figure 10). All the radio
observatories included in this work will be projected within an area of
$10^{\circ}$ around this point and hence they will cover approximately the
same area of the asteroid, if visible from their location.

It should be noted that Apophis will not do a full revolution in the time
window described in this paper, and in the best case scenario 85% of the
surface will be partially observable during the 14 hour window of interest.
The coverage of this missing zone will have to rely on observations taken in
the months before and after the closest approach, during which a high SNR will
be achievable for many of the radar pairs suggested here, although possibly
not at the highest possible range resolution.

Finally, we also note that Apophis may experience a spin rate change during
its close approach due to tidal effects. Thus, the actual intrinsic spin axis
of Apophis may be different than we assume here. However, given that the main
source of apparent rotation is the change of perspective due to the sky speed
of the asteroid during its rapid orbital approach, the apparent spin axis will
not be too different from our theoretical estimations shown here.

##  6 Summary and Discussion

The 300 m asteroid Apophis will fly within six earth radii from geocenter on
April 13th 2029. Although there is no collision risk, the possibility is not
yet ruled out for future encounters. Further studies of the asteroid surface,
shape and orbital changes have yet to be made, and it is during this close
encounter when the scientific opportunities to study this object are the
greatest. We present in this paper a sequence of possible radar observations
of Apophis within a 14 hr window near the closest approach and evaluate which
radio facilities will be optimal for complementary observations of the
asteroid. Given that the next windows of opportunity for radar ranging during
a close passage of Apophis will be in 2044 and 2051 and at much larger
distances (Sokolov et al., 2021), we should take full advantage of this
opportunity.

We use trajectory and ephemeris data provided by the JPL Horizons Web Service
to visualize the path of Apophis around the world (section 2). We show that it
will fly over some of the world’s most important radio telescopes, such as
Parkes, GBT, FAST, among others, as well as over antennas with powerful
transmitting capabilities: Canberra DSS-43, Madrid DSS-63 (Imbriale et al.,
2001), HUSIR and Goldstone DSS-13, DSS-14, among others. Based on this
information we propose a 4-stage observation plan using TX/RX capable antenna
pairs under the asteroid path in order to obtain continuous and overlapping
information of this object.

The 4-stage observation plan will help scheduling radar observations of the
asteroid in the near future, as well as coordinating scientific efforts in
countries under the flight path. For example: In stages 1 and 2 (Australia and
Africa/Europe) Apophis will be visible even with the naked eye in the night
sky. During stages 2 and 3 (Africa/Europe and USA) the apparent rotation of
Apophis will increase 2.5x, and the per frequency bin SNR will correspondingly
increase with respect to similar pairings in other stages. As for stage 4
(east Asia) the asteroid is going to fly over the Chinese 500 m dish (FAST),
which can be paired with the TX capable deep space antennas in Usuda, Japan,
or in Galenki (Ussuriysk), Russia if pairing frequency issues are resolved.

Following this plan, the Goldstone DSS-14 + Green Bank pairing will yield the
highest possible per run, frequency integrated SNR ($\sim 10^{10}$), but the
DSS-13 + VLA and HUSIR + GBT pairings will yield the best range resolutions,
reaching near-cm/px at per run, integrated SNRs above $\sim 10^{5-6}$ (Figure
7). We have found suitable pairings around the world that ensure that bistatic
radar pairs are always mapping the asteroid during its close approach, thus
achieving an unprecedented multi-wavelength coverage of the asteroid. This
will drastically improve our knowledge of its astrometry, shape, composition,
and regolith size distribution.

We have shown that powerful antennas are necessary for optimal bistatic
observations of Apophis. With this in mind, we strongly advise in favour of
the reconstruction of a next generation-Arecibo Radio Telescope such as the 1
MW-transmitter instrument proposed by Roshi et al. (2021), as its geographical
location and transmitting power would make it the optimal transmission
facility for this task, and the data gathered by the US bistatic radar pairs
would be the best we could get out of this asteroid’s close flight.

We also encourage the use of the coordinate system proposed in section 5. This
would facilitate joint efforts in describing and modeling the shape of
Apophis.

Finally, we consider the fly-by of Apophis as a perfect opportunity to engage
the general public with science, and we advise the scientific community and
scientific journalists to start planning for this date. In countries around
the stages 1 and 2 (see section 3) everyone can experience the naked-eye sight
of a 350 m asteroid, which will offer unprecedented astrophotography
opportunities of an asteroid.

In countries where daylight obstructs the view, citizenship science projects
can be proposed based on the way bistatic radar works: If a powerful enough
antenna is bouncing waves off Apophis, even small antennas might be able to
detect signals coming off it. This is because the expected SNRs will be so
high (our lowest SNR for DSS-13 + VLA is in the order $10^{7}$) that even with
a 1-m size antenna an SNR (per 2 minutes integration time) above $10^{2}$
could be expected, assuming a $T_{\text{sys}}\sim 300$ K. This also applies
for other bistatic radar pairs in regions of the world. And although it is
unlikely that amateur antennas will have powerful enough decoders to delay-
Doppler resolve Apophis, the SNR may be just high enough so amateur radio
astronomers who prepare their instrumentation for pair-specific frequencies
may be able to pick up bounce signals.

##  7 Data Availability

All the computations provided in this paper can be reproduced using the
scripts and tools available in the GitHub repo https://github.com/seap-
udea/Apophis2029. They can also be executed as a Google Colaboratory using the
link https://bit.ly/Apophis2029-Colab2.

##  8 Acknowledgements

Most of the computations that made possible this work were performed with
Astropy (Astropy Collaboration et al., 2018), Cartopy (Met Office, 2015), and
their related tools and libraries, iPython (Pérez & Granger, 2007), Jupyter
(Pérez & Granger, 2007), Matplotlib (Hunter et al., 2007), scipy and numpy
(Van Der Walt et al., 2011). The authors are grateful with the anonymous
referee, whose comments and suggestions significantly improved this work. We
especially appreciate the useful insights and and numerous comments of Dr.
Joseph Lazio.

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Asymptotic Symmetries of Yang-Mills with Theta Term and Monopoles

Carlos Cardona111E-mail address<EMAIL_ADDRESS>
Physics Department, Universidad del Valle222Temporary affiliation

Calle 13 Number 100-00, Santiago de Cali, Colombia.

Abstract

In this short note we suggest that the singular behavior of large gauge
transformations preserving the vacuum at null infinity in Yang-Mills theory
implies monopoles into the bulk, as well as that the inclusion of a theta term
induces a decoupling between holomorphic and anti-holomorphic currents
associated to those large gauge transformations

###### Contents

  1. 1 Introduction
  2. 2 Large Singular Gauge Transformations and Monopoles in Maxwell Theory
  3. 3 Non-Abelian Gauge Theory with Theta Term

##  1 Introduction

Recently there has been a renewed interest in the study of the asymptotic
symmetries that preserve the Minkowskian structure of asymptotically flat
space-times. It has been known for a long time [1, 2] that this is a infinite
symmetry, composed by Lorentz symmetry plus the so-called supertranslations.
In retarded or advanced coordinates the asymptotic region has the topology of
$\mathbb{R}\times S^{2}$, where supertraslations act on the null direction
$\mathbb{R}$ and are infinite, and the Lorentz group corresponds to the
$SL(2,\mathbb{C})$ preserving the structure of $S^{2}$. This set of symmetries
is known as Bondi-van der Burg-Metzner-Sachs (BMS). However, very recently it
has been shown that the BMS group can be enhanced by allowing singular
transformations on the sphere at spatial infinity [3, 4]. Using this fact
Strominger and Cachazo have found a new Weinberg’s soft theorem that contains
an additional subleading soft factor that depends on the angular momentum of
the particle emitting the soft graviton [5]. Subleading soft factors for
double soft emission in a variety of theories have also been found very
recently in [6].

By considering the analogous situation in Yang-Mills theory, Strominger and
collaborators have managed to get the soft theorem for gluons and photons [7,
8, 9] as the Ward identity corresponding to the asymptotic singular gauge
symmetry, which are the analogous of the asymptotic BMS diffeomorphisms in
gravity. The subleading factors correcting the Weinberg’s soft theorem in
Yang-Mills were obtained in [10].

As asymptotic gauge transformations act on the two dimensional $S^{2}$, it is
natural to expect that the asymptotic symmetry $SL(2,\mathbb{C})$ gets
enhanced to a Kac-Moody symmetry as long as singular transformations are
allowed, which was actually shown in [8] for Yang-Mills. However, it was found
that the usual holomorphic and anti-holomorphic currents do not decouple, and
hence is only possible to define a single holomorphic (or anti-holomorphic)
current algebra.

Asymptotic symmetries at the boundary are particularly important in the
context of the AdS/CFT correspondence, whose roots can be traced back to the
work of Brown and Henneaux where it was observed that for a theory of gravity
in three dimensional $AdS$, the algebra of asymptotic symmetries at the
boundary is Virasoro [12]. This has also played a central role in the
understanding of the $AdS_{3}/CFT_{2}$ correspondence.

In this note, we would like to suggest that by allowing singular gauge
transformations at infinity in Yang-Mills theory, one should induce monopoles.
Additionally, by adding a theta term to the non-abelian Yang-Mills Lagrangian,
we will show that the surface term coming from it allows us to define two
independent holomorphic and anti-holomorphic current algebras at different
values of the theta parameter, just as in two-dimensional WZW models, which
was actually the expected situation.

This paper is organized as follows, in section two we discuss how the singular
gauge transformations at null infinity imply the existence of monopoles at the
bulk for a $U(1)$ theory, then in section three we show the same phenomena for
the non-abelian case, but we also discuss how the inclusion of a theta term
induces a decoupling between the holomorphic and anti-holomorphic currents
associated to the large gauge transformations.

##  2 Large Singular Gauge Transformations and Monopoles in Maxwell Theory

Let us star considering a $U(1)$ gauge theory coupled to some massless charged
scalars,

$\mathcal{S}=-\frac{1}{4e^{2}}\int
d^{4}x\sqrt{g}\left(F_{\mu\nu}F^{\mu\nu}+\mathcal{D}_{\mu}\Phi\mathcal{D}^{\mu}\Phi\right)$
(1)

We rewrite the space-time metric $ds^{2}=\eta_{\mu\nu}dx^{\mu}dx^{\nu}$ in
terms of retarded coordinates

$v=t-r,\quad x^{1}+ix^{2}=\sqrt{2\gamma_{z\bar{z}}}rz,\quad
x^{3}=\sqrt{\frac{\gamma_{z\bar{z}}}{2}}(1-z\bar{z})\,.$ (2)

In the particular case of the stereographic projection, we can take on the
north chart $z={\rm e}^{i\phi}\tan(\theta/2)$, leading to the round metric on
the $S^{2}$ sphere $\gamma_{z\bar{z}}=\frac{2}{(1+z\bar{z})^{2}}$. The space-
time metric looks like,

$ds^{2}=-dv^{2}-2dvdr+2r^{2}\gamma_{z\bar{z}}dzd\bar{z}\,.$ (3)

The spatial infinity $r=\infty$ is commonly denoted $\mathcal{I}^{+}$ and has
the topology $S^{2}\times\mathbb{R}$ with coordinates $(v,z,\bar{z})$ and has
its null boundaries at $v=\pm\infty$ which are denoted by
$\mathcal{I}^{+}_{\pm}$. Similarly we have the advanced coordinates

$u=t+r,\quad x^{1}+ix^{2}=-\sqrt{2\gamma_{z\bar{z}}}rz,\quad
x^{3}=-\sqrt{\frac{\gamma_{z\bar{z}}}{2}}(1-z\bar{z})\,.$ (4)

which give rise to

$ds^{2}=-du^{2}-2dudr+2r^{2}\gamma_{z\bar{z}}dzd\bar{z}\,.$ (5)

Its topology is again $S^{2}\times\mathbb{R}$ and the null boundary
$v=\pm\infty$ is denoted by $\mathcal{I}^{-}_{\pm}$.

We would like to consider configurations approaching the vacuum at the
boundaries ${\cal I}^{\pm}_{\pm}$,

$F_{\mu\nu}=0,\quad\mathcal{D}_{\mu}\Phi=0\,.$ (6)

We work in the radial and asymptotic retarded gauge

$A_{r}=0,\quad A_{v}|_{{\cal I}^{+}}=0\,,$ (7)

and expand the fields in the asymptotic region as in [7],

$\displaystyle A_{z}(r,v,z,\bar{z})$ $\displaystyle=$
$\displaystyle\sum_{0}^{\infty}\frac{A_{z}^{(n)}(v,z,\bar{z})}{r^{n}}\,,$ (8)
$\displaystyle j^{E}_{v}(r,v,z,\bar{z})$ $\displaystyle=$
$\displaystyle\sum_{0}^{\infty}\frac{j_{v}^{(n)}(v,z,\bar{z})}{r^{n+2}}\,.$
(9)

Using the Maxwell equations in the gauge (7)333As in [9, 7] we take
$j^{E}_{\nu}=(j^{E}_{u},0,0,0)$ for the sake of simplicity.
$\nabla^{\mu}F_{\mu\nu}=j^{E}_{\nu}$, we get [7, 9]

$\displaystyle\partial_{v}(\partial_{z}A_{\bar{z}}+\partial_{\bar{z}}A_{z})$
$\displaystyle=$ $\displaystyle er^{2}\gamma_{z\bar{z}}j^{E}_{v}\,,$ (10)
$\displaystyle\partial_{r}(\partial_{z}A_{\bar{z}}+\partial_{\bar{z}}A_{z})$
$\displaystyle=$ $\displaystyle 0\,,$ $\displaystyle
r^{2}\partial_{r}(\partial_{r}A_{z}-2\partial_{v}A_{z})])$ $\displaystyle=$
$\displaystyle 0$

Also, the gauge conditions (7) left a residual gauge freedom at ${\cal I}^{+}$
generated by an arbitrary function $\epsilon(z,\bar{z})$ and hence the vacuum
conditions (6) implies a flat field strength $F_{z\bar{z}}=0$ at spatial
infinity,

$A_{z}|_{{\cal
I}^{+}_{\pm}}=A_{z}(\infty,z,\bar{z})=\partial\epsilon(z,\bar{z})\,.$ (11)

Putting this condition into the equations of motion (10) and using the
expansion (8) one can write the leading field $A_{z}^{(0)}$ and
$A_{\bar{z}}^{(0)}$ in terms of the leading current as,

$\displaystyle\partial_{\bar{z}}A^{(0)}_{z}$ $\displaystyle=$
$\displaystyle\frac{e}{2}\gamma_{z\bar{z}}\int dvj^{(0)}_{v}(v,z,\bar{z})\,.$
(12) $\displaystyle\partial_{z}A^{(0)}_{\bar{z}}$ $\displaystyle=$
$\displaystyle 0\,.$ (13)

For a set of massless point electric particles crossing ${\cal I}^{+}$ we use
the current,

$j^{(0)}_{v}=\sum_{k}q_{k}\delta(v-v_{k})\frac{\delta(z-z_{k})}{\gamma_{z\bar{z}}}\,,$
(14)

such that (12) is solved by444At the sphere this solution is unique, unlike to
surfaces of higher genus. I thank to Humberto Gomez for point it out to me.,

$A^{(0)}_{z}(\infty,z,\bar{z})=\frac{e}{2}\sum_{k}\frac{q_{k}}{z-z_{k}}\,.$
(15)

Since it is a singular gauge potential, the flux over the asymptotic sphere of
the magnetic field is non-zero, and hence there should be Dirac monopoles. The
usual Dirac monopole discussed in the literature has the following form in
stereographic coordinates555 This is the form over the north chart,

$A_{+}=\frac{1}{4\pi r}\frac{{\rm e}^{i\phi}}{z}\,.$ (16)

the potential is in this case singular at the south pole of $S^{2}$ which in
turn induce a magnetic flux over the sphere. The potential (15) is then the
usual Dirac monopole at conformal infinity and with $\phi=0$. Let us compute
the magnetic flux produced by the potential (15) over the $S^{2}$ at null
infinity,

$\int_{S^{2}}B=\int_{\partial S^{2}}A_{z}=\frac{e}{2}\sum_{k}q_{k}\oint_{{\cal
C}_{k}}\frac{dz}{z-z_{k}}=\frac{e}{2}\sum_{k}q_{k}\,.$ (17)

Since the magnetic flux should be an integer we obtain the Dirac quantization
condition666For a total magnetic charge $g=1$.

$Q=4\pi n\,,$ (18)

where we have defined the total charge as $Q=\sum_{k}eq_{k}$. The usual
interpretation is that some Dirac strings pinch the sphere at the points
$z_{k}$.

##  3 Non-Abelian Gauge Theory with Theta Term

In this section we re-do the analysis of the previous section for a Yang-Mills
theory with gauge group ${\cal G}$ and scalar matter, however our main concern
is to investigate the effect of a theta term on the asymptotic symmetries. Let
us start reviewing the case without theta term in order to elucidate the
effect of the topological contribution. Although we believe the results are
more general, let us take the following action for the sake of clarity,

$\mathcal{L}=-\frac{1}{4}{\rm
Tr}(F_{\mu\nu}F^{\mu\nu})-\frac{1}{2}\mathcal{D}_{\mu}\phi\mathcal{D}^{\mu}\phi-V(\phi)\,.$
(19)

Our conventions for this theory are given by ${\rm
Tr}[T^{a}T^{b}]=+\delta^{ab}$,
$\mathcal{D}^{\mu}=\partial^{\mu}-ig_{YM}[A^{\mu},.]$ and
$[D_{\mu},D_{\nu}]=-ig_{YM}F_{\mu\nu}$ with

$F_{\mu\nu}=\mathcal{D}_{[\mu}A_{\nu]}-ig_{YM}[A_{\mu},A_{\nu}]=F^{a}_{\mu\nu}T^{a}\,.$
(20)

We would like to consider a configuration such that the fields approach the
vacuum at $\mathcal{I}^{\pm}_{\pm}$, where the fields will take the following
values,

${F_{\mu\nu}^{a}=0},\quad\mathcal{D}\phi^{a}=0,\quad V(\phi)=0\,.$ (21)

In the particular Georgi-Glashow model where ${\cal G}=SU(2)$ and
$V=\frac{g_{YM}^{2}}{4}(\phi^{2}-\phi_{0}^{2})^{2}$, the vacuum breaks the
gauge symmetry down to $U(1)$, and the function $\phi(z,\bar{z})$ maps the
$S^{2}$ at null infinity $(r=\infty,u=\infty)$ to the $S^{2}$ which leaves the
vacuum $\phi^{2}=\phi_{0}^{2}$ invariant. Mappings from $S^{2}$ to $S^{2}$ are
characterized by the integers $\Pi_{2}(S^{2})=\mathbb{Z}$, moreover, the
symmetry breaking at null infinity implies the existence of ’t Hooft-Polyakov
monopoles [13, 14] the same way the singular asymptotic behavior in the
abelian case implies Dirac monopoles, whose magnetic charges are quantized
corresponding to the integers $\Pi_{2}(S^{2})$.

As in the abelian case we would like to work in the asymptotic retarded gauge

$A_{v}|_{\mathcal{I}^{+}}=A_{r}=0\,.$ (22)

Such that the leading terms of the field strength at $\mathcal{I}^{+}$ are

$F_{vr}=0,\quad F_{vz}=\partial_{u}A_{z},\quad
F_{z\bar{z}}=\partial_{z}A_{\bar{z}}-\partial_{\bar{z}}A_{z}-ig_{YM}[A_{z},A_{\bar{z}}]\,.$
(23)

This implies that at $\mathcal{I}^{+}$ there is still a residual gauge freedom
given by transformations which are independent of $(v,r)$, i.e,

$A_{\mu}|_{\mathcal{I}^{+}}\to
A_{\mu}|_{\mathcal{I}^{+}}+i\mathcal{U}(z,\bar{z})\partial_{\mu}\mathcal{U}(z,\bar{z})^{-1}\,.$
(24)

with $\mathcal{U}(z,\bar{z})\in{\cal G}$. Those transformations have been
called large gauge symmetries in [8].

The equations of motion of the action (19) are

$\displaystyle\mathcal{D}^{\mu}F_{\mu\nu}$ $\displaystyle=$ $\displaystyle
g_{YM}\left([\mathcal{D}_{\nu}\phi,\phi]+j^{E}_{\nu}\right)\equiv
g_{YM}J^{E\phi}_{\nu}$ $\displaystyle\mathcal{D}_{\mu}\mathcal{D}^{\mu}\phi$
$\displaystyle=$ $\displaystyle-V(\phi)\,.$ (25)

Under conditions (21) in the retarded gauge (22) the above equations takes the
form,

$\displaystyle\partial_{v}(\partial_{z}A_{\bar{z}}+\partial_{\bar{z}}A_{z}+ig_{YM}[A_{z},A_{\bar{z}}])$
$\displaystyle=$ $\displaystyle g_{YM}r^{2}\gamma_{z\bar{z}}j^{E}_{v}\,,$ (26)
$\displaystyle\partial_{r}(\partial_{z}A_{\bar{z}}+\partial_{\bar{z}}A_{z}+ig_{YM}[A_{z},A_{\bar{z}}])$
$\displaystyle=$ $\displaystyle g_{YM}r^{2}\gamma_{z\bar{z}}j^{E}_{r}\,,$ (27)
$\displaystyle
r^{2}\partial_{r}(\partial_{r}A_{z}-2\partial_{v}A_{z})+\partial_{z}(\gamma^{z\bar{z}}ig_{YM}[A_{z},A_{\bar{z}}])$
$\displaystyle=$ $\displaystyle g_{YM}r^{2}\gamma_{z\bar{z}}j^{E}_{z}$ (28)

Once the vacuum $F_{\mu\nu}=0$ is reached at spatial infinity $S^{2}$, the
condition
$F_{z\bar{z}}=\partial_{z}A_{\bar{z}}-\partial_{\bar{z}}A_{z}-ig_{YM}[A_{z},A_{\bar{z}}]=0$
implies that $A_{z}$ and $A_{\bar{z}}$ become pure gauge

$A_{z}|_{\mathcal{I}^{+}_{+}}=i\mathcal{U}\partial_{z}\mathcal{U}^{-1}\,.$
(29)

Using this fact, one can solve (26) for the leading modes $A^{(0)}_{z}$ in
terms of the leading color current as in the abelian case, i.e, for a set of
massless point particles crossing ${\cal I}^{+}$ one can write,

$\partial_{\bar{z}}A_{z}^{(0)}(\infty,z,\bar{z})=\frac{g_{YM}}{2}\sum_{k}\epsilon_{k}(z,\bar{z})\delta^{2}(z-z_{k})\,,$
(30)

where $\epsilon(z,\bar{z})$ corresponds to the infinitesimal generator for
large gauge transformations in the representation of the charged particle
insertion at $z_{k}$. The potential is singular at the point insertions which
induce a “magnetic” color flux through the $S^{2}$ at null infinity. The
necessary condition to have ’t Hooft-Polyakov monopoles is that the field
configuration reach the vacuum at spatial infinity, so the case we have
considered here is a particular case where the fields reach the vacuum at null
infinity.

Away from the charge insertions or without charge insertions at all, the
potential $A_{z}(\infty,z,\bar{z})$ is holomorphic, as in the abelian case but
with an important difference. Naively one could expect $A_{\bar{z}}$ to be
anti-holomorphic as in the $U(1)$ case, but since $F_{z\bar{z}}=0$ then
$\partial_{z}A_{\bar{z}}\neq 0$. It was noticed quite elegantly in [8] that in
the limit where two gluons become soft, the result depends on the order of the
limits, which in turns implies that the Kac-Moody algebra based on the group
${\cal G}$ associated to the $A_{z}$ is coupled to the one associated to the
current $A_{\bar{z}}$. This situation is similar to the one found in two-
dimensional WZW models, where in order to guarantee a pair of independent
holomorphic and an anti-holomorphic Kac-Moody algebras it is needed to
introduce a total derivative to the lagrangian, which supplements the
equations of motions on the boundary at infinity. In four dimensions the
natural candidate to be added to the Yang-Mills action (19) is a theta term,

$S_{\theta}=\frac{\theta}{32\pi^{2}}\int d^{4}x{\rm
Tr}F_{\mu\nu}{}^{*}F^{\mu\nu}$ (31)

where ${}^{*}F^{\mu\nu}=\frac{1}{2}\epsilon^{\mu\nu\rho\sigma}F_{\rho\sigma}$
and this action in four dimensions is a total derivative that might be
evaluated at $r=\infty$,

$\int d^{4}x{\rm Tr}F_{\mu\nu}{}^{*}F^{\mu\nu}=2\int_{\mathbb{R}\times
S^{2}}dudzd\bar{z}\,{\rm
Tr}\,\epsilon^{rmnl}(A_{m}\partial_{n}A_{l}+A_{m}A_{n}A_{l})\,.$ (32) 777In
this expression, $r$ stands for the coordinate.

This induces a Chern-Simons term on ${\cal I}^{+}$ while $\theta$ should be an
integer multiple of $2\pi$ in order to preserve gauge invariance. At ${\cal
I}^{+}$ the potential is a pure gauge
$A_{z,(\bar{z})}=i\mathcal{U}\partial_{z,(\bar{z})}\mathcal{U}^{-1}$ such that
the theta action reduces to

$S_{\theta}=\frac{\theta}{16\pi^{2}}\int_{\mathbb{R}\times
S^{2}}dudzd\bar{z}\,{\rm
Tr}\,\epsilon^{rmnl}(\mathcal{U}\partial_{m}\mathcal{U}^{-1}\,\mathcal{U}\partial_{n}\mathcal{U}^{-1}\,\mathcal{U}\partial_{l}\mathcal{U}^{-1})$
(33)

The variation of the $\theta$ term is again a total derivative which lead us
to

$\delta S_{\theta}=-\frac{\theta}{4\pi}\int
d^{2}z\epsilon^{rvlm}(\mathcal{U}^{-1}\delta
U\partial_{l}A_{m})=-\frac{\theta}{4\pi}\int
dvd^{2}z\epsilon^{rvlm}\partial_{v}(\mathcal{U}^{-1}\delta
U\partial_{l}A_{m})\,.$ (34)

Therefore we should sum up to the equations (26) the following term,

$-\frac{\theta}{4\pi}\partial_{v}(\epsilon^{rvlm}\partial_{l}A_{m})$ (35)

So the asymptotic equation (26) translate to

$\partial_{\bar{z}}A_{z}\left(1+\frac{\theta}{4\pi}\right)+\partial_{z}A_{\bar{z}}\left(1-\frac{\theta}{4\pi}\right)+ig_{YM}[A_{z},A_{\bar{z}}]=g_{YM}\gamma_{z\bar{z}}\int
dvj^{E}_{v}\,,$ (36)

or using the flat condition $F_{z\bar{z}}=0$ and the current (14), we can
rewrite,

$\partial_{\bar{z}}A_{z}\left(2+\frac{\theta}{4\pi}\right)-\partial_{z}A_{\bar{z}}\left(\frac{\theta}{4\pi}\right)=\frac{g_{YM}}{2}\sum_{k}\epsilon_{k}(z,\bar{z})\delta^{2}(z-z_{k})\,,$
(37)

At $\theta=0$, $A_{z}$ is holomorphic and at $\theta=-8\pi$ $A_{\bar{z}}$ is
anti-holomorphic away from the punctures on the sphere. Therefore, including a
theta term to the Yang-Mills action, has the effect of decoupling the
holomorphic and anti-holomorphic Kac-Moody algebras at the boundary for
particular values of the $\theta$ parameter, as we have mentioned at the
introduction.

Due to the appearance of the holomorphic-antiholomorphic Kac-Moody algebras,
the S-matrix elements would resemble correlations functions of some two-
dimensional euclidean WZW model based on the gauge group ${\cal G}$ and
therefore it would be interesting to look for a string theory realization of
it in the way already suggested by Strominger[7].

The addition of a theta term breaks CP symmetry so it also would be
interesting to see the consequences of this breaking on the S-matrix from the
asymptotic symmetries perspective. Particularly, in the analysis of [8] it was
important to consider CPT-invariant gauge transformations which allow to
relate the asymptotic symmetries at past infinity with the ones at future
infinity, so maybe it will be important to see how this matching conditions
should be modified in the present case.

Acknowledgements I would like to thank to Freddy Cachazo and Humberto Gomez
for carefully reading, correcting and making comments on this manuscript. I
also thank to the Physics Department of the Universidad del Valle for the
hospitality during the completion of this work.

## References

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  * [8] T. He, P. Mitra, and A. Strominger, “2D Kac-Moody Symmetry of 4D Yang-Mills Theory,” arXiv:1503.02663 [hep-th].
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###### Proof of Theorem 10.16.

By taking a suitable open cover of $X$, we may and will assume that $H$ is
defined by $f\in\mathcal{O}_{X}(X)$. By Proposition 10.12, the assertion in
the theorem is equivalent to the fact that for every $q\in{\mathbf{Z}}_{\geq
0}$ and $\alpha\in{\mathbf{Q}}\cap[0,1)$, we have $\partial_{t}^{q}\delta\in
V^{>\alpha}$ if and only if $q+\alpha<\widetilde{\alpha}(X,H)$. Note also that
by Corollary 5.15, we have $\partial_{t}^{q}\delta\in V^{>\alpha}$ if and only
if all roots of $b_{\partial_{t}^{q}\delta}$ are $<-\alpha$. Since $\alpha<1$,
it follows from Theorem 10.17 that this is the case if and only if all roots
of $\widetilde{b}_{f}$ are $<-q-\alpha$, or equivalently,
$\widetilde{\alpha}(X,H)>q+\alpha$. ∎

We now give a lower bound for $\widetilde{\alpha}(X,H)$ in terms of a log
resolution. This was proved in [MP] using techniques involving mixed Hodge
modules; the more elementary proof that we give here, based on the Kashiwara-
Lichtin estimate for roots of $b$-functions, is from [DM]. Note that if $H$ is
not reduced, then $\operatorname{lct}(X,H)<1$, hence
$\widetilde{\alpha}(X,H)=\operatorname{lct}(X,H)$ can be computed in terms of
a log resolution via (82).

######  Theorem 10.18.

Let $X$ be a smooth, irreducible algebraic variety and $H$ a nonempty reduced
hypersurface in $X$. Let $\pi\colon Y\to X$ be a log resolution of $(X,H)$
such that the strict transforms of the irreducible components of $H$ on $Y$
are disjoint. If we write $\pi^{*}(H)=\sum_{i=1}^{N}a_{i}E_{i}$ and
$K_{Y/X}=\sum_{i=1}^{N}k_{i}E_{i}$, then

(87) $\widetilde{\alpha}(X,H)\geq\min\big{\\{}\tfrac{k_{i}+1}{a_{i}}\mid
E_{i}\,\,\text{exceptional}\big{\\}}.$

###### Proof.

After taking a suitable open cover of $X$, we may and will assume that $H$ is
defined by $f\in\mathcal{O}_{X}(X)$. Let $\lambda$ be the right-hand side of
(87) and let us write $\lambda=q+\beta$, with $q\in{\mathbf{Z}}_{\geq 0}$ and
$\beta\in(0,1]$. We have $\widetilde{\alpha}(X,H)\geq\lambda$ if and only if
every root of $\widetilde{b}_{f}(s-q)$ is $\leq-\beta$; by Theorem 10.17, this
is equivalent to $b_{\partial_{t}^{q}\delta}(s)$ having all roots
$\leq-\beta$. However, by Theorem 5.23 that such a root is either $-m$, for
some positive integer $m$ (in which case it is clearly $\leq-\beta$) or of the
form $q-\tfrac{k_{i}+\ell}{a_{i}}$, for some positive integer $\ell$ and some
$i$ with $E_{i}$ exceptional (in which case, it is $\leq q-\lambda=-\beta$).
This completes the proof. ∎

######  Remark 10.19 (Computation of the minimal exponent as the minimum on a
log resolution).

Unlike in the case of formula (82) that computes $\operatorname{lct}(X,H)$ in
terms of any log resolution, we can’t hope that the inequality in (87) is an
equality for an arbitrary resolution. Indeed, if the minimum on the right-hand
side of (87) is $>1$ (so $\widetilde{\alpha}(X,H)>1$), then one can construct
a sequence of blow-ups with smooth centers of codimension $2$, such that the
corresponding minima converge to $1$. Indeed, let $E_{j}$ be the strict
transform of an irreducible component of $H$ (so that $a_{j}=1$ and $k_{j}=0$)
and let $E_{i}$ be an exceptional divisor that intersects $E_{j}$. We consider
the blow-up $Y_{1}\to Y$ of $Y$ along a connected component of $E_{i}\cap
E_{j}$, and let $G_{1}$ be the exceptional divisor. We consider next the blow-
up $Y_{2}\to Y_{1}$ of $Y_{1}$ along the intersection of $G_{1}$ with the
strict transform of $E_{j}$ and let $G_{2}$ be the exceptional divisor, etc.
An easy computation shows that the coefficient of $G_{\ell}$ in the inverse
image of $H$ on $Y_{\ell}$ is $a_{i}+\ell$ and its coefficient in
$K_{Y_{\ell}/X}$ is $k_{i}+\ell$. Note that
$\lim_{\ell\to\infty}\tfrac{k_{i}+1+\ell}{a_{i}+\ell}=1$.

However, it is an interesting question whether given a reduced hypersurface
$H$ on $X$, there is a log resolution $\pi\colon Y\to X$ of $(X,H)$ such that
we have equality in (87).

One can show that certain properties of the log canonical threshold extend to
the minimal exponent $\widetilde{\alpha}(X,H)$, see [MP]. Two important ones
concern the behavior with respect to restriction to a smooth hypersurface and
semicontinuity in families. These properties can be proved by following the
corresponding proofs for log canonical thresholds via multiplier ideals in
[Lazarsfeld, Chapter 9] once we have a family of ideals that detect the
minimal exponent. This was done in [MP] using the _Hodge ideals_ $I_{p}(\alpha
H)$ constructed in [MP2] and [MP3] (here $p\in{\mathbf{Z}}_{\geq 0}$ and
$\alpha$ is a positive rational number). One can attach such ideals to any
reduced666Note that if $H$ is not reduced, then ${\rm lct}(X,H)<1$ and thus
$\widetilde{\alpha}(X,H)={\rm lct}(X,H)$. Therefore, in order to extend
properties of the log canonical threshold to the minimal exponent, it is
enough to consider the case when $H$ is reduced. hypersurface $H$ in $X$ by
using the Hodge filtration on suitable Hodge modules associated to a local
equation of $H$. By making use of results on mixed Hodge modules, it was shown
in [MP4] and [MP3] that these ideals satisfy properties that parallel those of
multiplier ideals. On the other hand, it was shown in [MP, Theorem A’] that if
$\lambda=p+\alpha$, where $p\in{\mathbf{Z}}_{\geq 0}$ and $\alpha\in(0,1]$,
then $I_{p}(\alpha
H)\cdot\mathcal{O}_{H}=\widetilde{\mathcal{J}}\big{(}X,(\lambda-\epsilon)H\big{)}\cdot\mathcal{O}_{H}$
for $0<\epsilon\ll 1$. It thus follows from Proposition 10.12 that
$\widetilde{\alpha}(X,H)\geq p+\alpha$ if and only if $I_{p}(\alpha
H)=\mathcal{O}_{X}$. As we have already pointed out, Schnell and Yang have
recently developed in [SchnellYang] a theory of twisted Hodge modules and used
it to show that the ideals $\widetilde{\mathcal{J}}(X,\lambda H)$ themselves
satisfy properties analogous to those of multiplier ideals, so one can also
use these invariants in order to prove the properties of the minimal exponent.

Finally, we mention that there is a notion of minimal exponent associated to
local complete intersection subvarieties $Z$ inside a smooth variety $X$, see
[CDMO]. This is defined using the $V$-filtration associated to a local system
of equations of $Z$ and many of the properties extend from the case of
hypersurfaces to the case of arbitrary local complete intersections.

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BriançonJ.GrangerM.MaisonobePh.Sur le polynome de bernstein des singularités
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